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Fractions

3 A pepperoni pizza is cut into 6 equal pieces while a sausage pizza of the same size is cut into 8 equal pieces. If Martin eats 2 pieces of the pepperoni pizza and Olivia eats 3 pieces of the sausage pizza, who has eaten more? How can you prove this?

84

Maths Quest 7 for Victoria

Understanding fractions Antonio makes pizzas. Besides selling whole pizzas, he often sells part of a pizza to his customers. For example, this pizza has been cut into 3 equal pieces. If a customer wants only 2 of these slices, then Antonio is selling 2--3- of the pizza to the customer.

Fractions are used to describe parts of a whole. The numerator or top number of the fraction shows how many parts there are, while the denominator or bottom number shows the number of parts the whole has been divided into. So 2--3- means that the whole thing has been divided into 3 equal parts and that we are concerned with 2 parts. The horizontal bar separating the numerator from the denominator is called the vinculum. A proper fraction is a fraction with the numerator smaller than the denominator, like 2--3- . An improper fraction has a numerator that is larger than the denominator, like 5--4- . A mixed number is made up of a number and a fraction, like 1 3--5- .

WORKED Example 1 What fraction of the rectangle shown has been shaded?

THINK 1 2 3

Count how many equal parts the rectangle has been divided into. How many parts are shaded? Write the answer in a sentence.

WRITE Total number of parts = 8 3 parts are shaded. 3 --- of the rectangle has been shaded. 8

Chapter 3 Fractions

85

Fractions can be marked on a number line in the 0 1– 2– 1 11– 12– 2 2 1– 3 3 3 3 3 same way as counting numbers. The number line above can be used to count in fractional steps. The fractions marked in bold are found by adding 1 to the previous fraction or mixed number so that the following sequence of numbers is obtained: 1 --- , 3

1 1--3- , 2 1--3- , 3 1--3- , 4 1--3- , . . .

WORKED Example 2 Copy the number line shown, filling in the missing numbers. Use it to find the next 3 terms in the given sequence: 3--4- , 1, 1 1--4- , _, _, _. THINK 1 2 3 4

0

1– 4

1– 2

3– 4

1

13–4

WRITE

Copy the number line and fill in the missing numbers. Circle the 3 given numbers in the sequence on the number line. On the number line, find the next 3 terms which continue the sequence. Write the sequence with the missing numbers.

0

1– 4

1– 2

3– 4

1 11–4 11–2 13–4

2

0

1– 4

1– 2

3– 4

1 11–4 11–2 13–4

2

3 --- , 4

1, 1 1--4- , 1 1--2- , 1 3--4- , 2

Finding equivalent fractions Antonio cuts a pizza into 2 equal pieces as shown. Each piece is exactly half of the pizza. A lady with a small child comes into the shop and wants to buy some pizza. She decides to cut one of the halves into 2 smaller, equal pieces as shown.

Now the child eats 2 quarters of a pizza and the mother eats half a pizza. If you look carefully you will see that each person has the same amount of pizza. We can say that 1 2 1 2 --- is equivalent to --- or --- = --- . 2 4 2 4 Equivalent fractions can be found by multiplying or dividing both the numerator and the denominator by the same number as long as the number is not zero. Therefore, 1 1×2 2 --- = ------------ = --2 2×2 4 So

and

1 1×3 3 --- = ------------ = --- . 2 2×3 6

1 2 3 --- = --- = --- . 2 4 6

86

Maths Quest 7 for Victoria

WORKED Example 3

For the equivalent fractions 4--5- = numerator and denominator.

8 -----10

find the number which has been used to multiply both

THINK 1

2

What number is 4 multiplied by to equal 8? (4 × _ = 8) What number is 5 multiplied by to equal 10? (5 × _ = 10) Write the answer.

WRITE 4 4×2 8 --- = ------------ = -----5 5 × 2 10

The number used is 2.

WORKED Example 4

From the list, find those fractions which are equivalent to 1--2- : 3--9- , 3--6- , THINK 1

Multiply the numerator and denominator of 1--2- by the numerator of the first fraction (3) to check whether the new fraction is in the list.

2

Multiply the numerator and denominator of 1--2- by the next different numerator (9) and check whether the new fraction is in the list.

3

Continue until all fractions have been considered.

4

Write the equivalent fractions.

9 10 7 17 ------ , ------ , ------ , ------ . 18 15 14 34

WRITE 1×3 3 ------------ = --2×3 6

1×9 9 ------------ = -----2 × 9 18

1 × 10 10 --------------- = -----2 × 10 20 7 1×7 ------------ = -----2 × 7 14 1 × 17 17 --------------- = -----2 × 17 34 Equivalent fractions of

1 --2

are 3--6- ,

WORKED Example 5

Write 3 equivalent fractions for 2--3- . THINK 1

Multiply both the numerator and denominator by 2, 3 and 4.

2

Write the equivalent fractions.

WRITE 2 2×2 2×3 2×4 --- = ------------ = ------------ = -----------3 3×2 3×3 3×4 2 4 6 8 --- = --- = --- = -----3 6 9 12

9 7 17 ------ , ------ , ------ . 18 14 34

Chapter 3 Fractions

87

Note: There are other fractions which are equivalent to 2--3- . These can be found by multiplying the numerator and denominator by other numbers. Answers to exercises (at the back of the book) will always give the 3 smallest equivalent fractions. Equivalent fractions can also be shown using diagrams. Equivalent fractions for 2--3are shown below.

2– 3

2– 3

= 4–6

= 6–9

2– 3

8 =— 12

Notice that the same amount of the rectangle has been shaded in each case.

Comparing fractions If Antonio’s pizza is cut into quarters it would be easy to see that 1 piece is less than 2 pieces, or that 1--4- of the pizza is less than 2--4- . If Antonio has another pizza of the same size but cuts it into thirds, it can still be easily seen that 1--4- is less than 1--3- . This is because a pizza divided into 4 equal pieces will give smaller pieces than one divided into 3 equal pieces. If the numerators are the same, the smaller fraction is the one with the larger 3 3 - is less than --- . If the numerators denominator. For example, 1--6- is greater than 1--7- and ----10 4 are different, another method is required to compare fractions. Fractions which do not have the same denominator can be compared by using equivalent fractions. We need to rewrite each fraction so that it has the same denominator.

WORKED Example 6 Which is the bigger fraction,

2 --3

or 3--5- ?

THINK 1

Find the lowest common multiple of the denominators. That is, find multiples of the larger denominator. Continue until there is one which is divisible by the other denominator.

2

Write each fraction as an equivalent fraction using this number (15) as the denominator. Decide which is bigger by comparing the numerators of the equivalent 9 ------ and ------ ). fractions ( 10 15 15

3

WRITE Multiples of 5 are 5, 10, 15 . . . 15 is divisible by 3 so 15 is the lowest common multiple.

2 2 × 5 10 3 3×3 9 --- = ------------ = ------ and --- = ------------ = -----5 5 × 3 15 3 3 × 5 15 2 --3

is bigger than 3--5- .

88

Maths Quest 7 for Victoria

The answer to worked example 6 can be written as 2--3- > 3--5- . The symbol ‘>’ means bigger than or greater than. The symbol ‘’ means greater than, the symbol ‘ or or < sign between each pair of fractions to make a true statement. a

8 --5

d

15 -----2

1 2--5-

19 -----2

8 1--2-

c

33 -----32

3 1 ----32

5 1--2-

e 3 3--4-

19 -----4

f

7 1--7-

65 -----7

8 --3

-----h 1 16 17

g 2 1--3-

b

35 -----17

1 1 Write down the equivalent fractions from this list: 123 31 10 16 19 8 6 3 1 2 --- , --- , ------ , ------ , ------ , ------ , ------ , ------ , --------- , --------- . 4 6 12 25 32 41 64 76 124 490

2 Write 3 equivalent fractions for 2--5- . 3 Write

12 -----20

in simplest form.

4 On a recent quiz, Simone answered 6 out of the 50 questions incorrectly. What fraction of the test did Simone answer correctly? Express your answer in simplest form. 5 Write

26 -----3

as a mixed number.

6 Change 1 1--4- to an improper fraction. 7 Write form.

12 -----9

as a mixed number in simplest

8 Change 3 6--7- to an improper fraction. 9 Rewrite these fractions in ascending (from smallest to largest) order: 2 14 --- , ------ , 5 5

1 3--5- , 3 3--5- ,

15 ------ , 5

4 2--5- ,

28 ------ , 5

6 1--5- .

10 A catering company provided 12 quiches which were each cut into 8 pieces for a luncheon. At the end of the function there were 12 pieces of quiche left over. How many whole quiches were eaten?

3.1

100

Maths Quest 7 for Victoria

Adding and subtracting fractions Fractions can be added and subtracted if they have the same denominator. Remember to simplify your answer if possible.

WORKED Example 14 Find

5 --9

+ 1--9- .

THINK

WRITE 5 --9

1

Write the question.

2

If the denominators are the same, add or subtract the numerators. Simplify the answer if possible.

3

+

1 --9

=

6 --9

=

2 --3

If fractions do not have the same denominator, use equivalent fractions to make the denominators the same.

WORKED Example 15 Find

5 --6

−

1 ------ , 12

expressing the answer in simplest form.

THINK

WRITE

1

Find multiples of each denominator.

2

Find the lowest common multiple (LCM) of 6 and 12 (12).

3

Rewrite the fractions if necessary so that they both have the LCM as the new denominator.

4

Add or subtract the numerators.

5

Simplify your answer if necessary by cancelling by the highest common factor (9 ÷ 3 = 3, 12 ÷ 3 = 4).

Multiples of 6 are 6, 12 . . . Multiples of 12 are 12 . . . LCM is 12. 5 1 --- – -----6 12 5×2 1 = ------------ – -----6 × 2 12 10 1 = ------ – -----12 12 9 = -----12 93 = ------ 4 12 =

3 --4

With practice many of these steps can be done in your head and the solution written as follows. 5 --6

−

1 -----12

= = =

10 -----12 9 -----12 3 --4

−

1 -----12

Chapter 3 Fractions

101

Adding and subtracting mixed numbers There are a number of ways of adding and subtracting mixed numbers. Use the method which you prefer. All mixed numbers can be added and subtracted by first changing them to improper fractions. Remember to simplify the answer if possible.

WORKED Example 16

Find 2 3--4- + 1 1--2- . THINK

WRITE

1

Write the question.

2 3--4- + 1 1--2-

2

Change to improper fractions.

=

11 -----4

+

3 --2

3

Write both fractions with the same denominator using equivalent fractions.

=

11 -----4

+

6 --4

4

Add the fractions.

=

17 -----4

5

Write the answer as a mixed number if appropriate.

= 4 1--4-

Alternatively, you can add the whole numbers first and then work out the fractional parts. So,

2 3--4- + 1 1--2- = (2 + 1) + ( 3--4- + 1--2- ) = 3 + ( 3--4- + 2--4- ) =3+

5 --4

= 3 + 1 1--4= 4 1--4This alternative method also works for subtracting fractions, unless the second fraction part is bigger than the first fraction part. Then use the method of equal addition. (Also see the section on subtraction in chapter 1.)

WORKED Example 17

Find 3 1--5- − 1 3--4- . THINK

WRITE

1

Write the question.

3 1--5- − 1 3--4-

2

Write both fraction parts with the same denominator (20). Add the same number of parts to both fractions to make 5 - . the second fraction a whole number ----20 Subtract and simplify if necessary.

4 15 - − 1 -----= 3 ----20 20

3 4

9 - − 2 = 3 ----20 9 = 1 ----20

102

Maths Quest 7 for Victoria

Graphics Calculator tip!

Adding and subtracting fractions

As with any calculation involving fractions, if you wish to have an answer expressed as a fraction then each calculation needs to end by pressing MATH , selecting 1:Frac and pressing ENTER . The calculation for worked ÷ 6 − example 15 would be entered as 5 MATH , select 1:Frac and press ENTER . When entering mixed numbers, it is a good idea to use brackets. This allows the correct order of operations to occur, particularly when subtracting. The calculations for worked examples 16 and 17 can be viewed in the screen shown. Note that the answers are given as improper fractions.

1

÷

12

then press

Estimation can be used to find approximate answers to addition and subtraction of mixed numbers. This allows you to easily check whether your answer to a calculation looks reasonable.

WORKED Example 18

Find 3 1--3- + 2 1--2- and check the answer by estimation. THINK

WRITE

1

Write the question.

3 1--3- + 2 1--2-

2

Group the whole numbers and group the fractions.

= (3 + 2) + ( 1--3- + 1--2- )

3

Add the whole numbers and write the fractions with the same denominator. Simplify the answer.

= 5 + ( 2--6- + 3--6- )

4

=5+

5 --6

= 5 5--65

Check the answer by adding the whole numbers to find an approximation.

3 1--3- + 2 1--2- > 3 + 2 so 3 1--3- + 2 1--2- > 5

remember remember 1. Fractions can be added and subtracted if they have the same denominator. 2. If fractions do not have the same denominator, use equivalent fractions to make the denominators the same. 3. Mixed numbers can be added or subtracted by first changing them to improper fractions. 4. Mixed numbers can also be added or subtracted by working with the whole numbers, then the fractional parts. If the second fractional part is bigger than the first fractional part, make the second fraction equal to a whole number by adding the same fraction to both parts.

103

Chapter 3 Fractions

3D WORKED

Example

14

Adding and subtracting fractions

1 Find: a

1 --5

d

20 -----50

g

5 --8

+

3 --5

+ −

11 -----50

+

1 --8

3 --8

b

9 -----11

−

3 -----11

c

3 --6

e

15 -----25

−

6 -----25

f

12 -----12

h

4 --7

i

33 --------100

+

5 --7

−

3 --7

+

2 --6

−

Adding and subtracting fractions

1 -----12

+

−

7 --------100

11 --------100

Adding fractions

2 Find the following, expressing the answers in simplest form.

WORKED

Example

15

+

a

5 -----12

d

37 --------100

g

13 -----16

1 -----12

+ +

13 --------100 9 -----16

−

10 -----16

b

7 -----16

e

9 --4

h

11 -----28

− − +

3 -----16 7 --4 10 -----28

c

9 -----15

f

4 --7

i

21 -----81

− + +

4 -----15

−

5 --7

21 -----81

2 --7

+

30 -----81

Subtracting fractions

3 Find the following, expressing the answers in simplest form. a

1 --2

+

1 --4

b

1 --2

−

d

1 --8

+

20 -----32

e

9 -----18

+

g

7 --9

−

8 -----27

h

31 -----35

−

+

2 -----21

c

1 --3

1 --6

f

17 -----40

+

1 --2

3 --7

i

8 -----10

−

28 -----70

1 --4

4 Find the lowest common multiple of each of the following pairs of numbers. a 2 and 3 b 3 and 5 c 5 and 10 d 4 and 6 e 5 and 6 f 9 and 6 g 4 and 8 h 6 and 8 i 7 and 5 5 Using the lowest common multiples that were found in question 4 as denominators, add or subtract these fractions. a

1 --2

−

1 --3

b

2 --3

−

2 --5

c

3 --5

+

3 -----10

d

3 --4

−

1 --6

e

4 --5

−

1 --6

f

1 --9

+

5 --6

g

1 --4

+

3 --8

h

5 --6

−

3 --8

i

5 --7

−

2 --5

6 Answer the following questions by first finding the lowest common denominator. Simplify your answer if necessary. +

b

5 --7

−

3 --8

c

4 --5

−

3 --4

2 -----15

e

5 --8

−

5 -----12

f

7 --9

−

1 --2

2 --9

h

7 -----11

+

1 --4

i

4 --5

−

2 -----11

+

7 -----10

k

3 -----10

−

1 -----20

l

7 --8

−

3 --5

2 --3

+

4 --5

n

5 --7

o

5 --9

−

2 --7

2 --3

−

1 -----13

q

11 -----12

r

4 --7

+

5 --6

a

2 --3

d

7 -----10

+

g

5 -----12

+

j

5 --6

m p

1 --4

+ −

3 --4 7 -----10

3.3

104

Maths Quest 7 for Victoria

7 Angela was a little hungry the day before her birthday, and spied the cake to be eaten 1 - of it, her Mum might not notice. the following day. She decided that if she ate only ----1 10 However, she was still hungry and took another --6- . a How much of the cake did Angela eat before her birthday? b How much of the cake was left for her birthday? 8 multiple choice a The lowest common denominator of 1--3- , A 16

B 24

and

1 --6

and

1 --5

C 18

B 10

is: D 72

1 1 --- , --2 4

b The lowest common denominator of A 8

1 --4

C 20

E 12

is: D 30

E 40

9 Answer the following: Adding and subtracting fractions

a

1 --2

d

11 -----12

g

1 --2

+ − +

1 --3

−

8 -----15 2 --3

+

1 --4

− 3 --4

3 -----20

b

3 -----10

+

4 -----15

e

1 -----16

+

1 --8

h

3 --4

+

1 --3

− +

+

10 A cake needed to be cooked at 200°C for is the total cooking time in hours?

1 --6 1 --4

c

7 --8

−

3 --4

f

5 --6

−

7 -----18

+ −

4 --6 1 --9

5 --6 3 --4

hour then a further

1 --2

hour at 150°C. What

11 Robert, Jason and Luke play football in the same team. Last Saturday, Robert kicked 1 1 5 --- of the team’s goals, Jason kicked --- and Luke ------ . What fraction of the team’s goals 4 3 12 were kicked by: a Jason and Luke? b Robert and Jason? c the three boys together? 12 Jacinta and her friend decided to climb the tree in her Nan’s front yard. It took her 10 minutes to climb halfway up, 10 more minutes to climb another sixth of the way and yet another 10 minutes to reach a further tenth of the way. Here she sat admiring the view for 5 minutes before beginning her descent. a What fraction of the height of the tree had Jacinta climbed after 20 minutes? b What fraction of the tree did Jacinta climb in total?

WORKED

Example

16

13 Find: a 1 1--5- + 3 1--2-

b 3 2--3- − 2 1--2-

c

5 2--3- − 2 1--4-

d 9 3--8- + 4

e 2 3--5- − 1 1--2-

f

3 1--2- + 2 3--4-

Chapter 3 Fractions

WORKED

Example

105

14 Find: a 7 − 5 2--3-

17

b 2 1--3- −

c 3 1--2- − 1 7--8-

5 --6

9 1 7 3 5 - e 5 --- − 1 ------ f 5 --- − 3 --d 4 3--5- − 2 ----10 4 12 4 6

15 Find: a WORKED

Example

1 --5

+ 2 2--7-

b

5 --8

+ 3 3--4- −

1 --2

c

2 2--9- + 1 1--3- +

d 4 2--3- + 1 3--8- − 3

1 --6

16 Find the following and check the answers by estimation. a 5 1--3- + 2 1--4-

18

b 1 1--6- + 2 1--2-

c 3 1--3- + 1 1--6-

d 6 1--2- − 2 1--4-

3 5 1 - f 7 --- − 3 --e 4 3--5- − 1 ----10 6 3

17 One-third of a litre of cordial is mixed with 1 1--2- litres of water. How many litres of drink has been made? 18 At a school dance-a-thon, the number of hours each student danced was recorded. Matthew 1 3--4- hours,

Leigh 2 2--3- hours,

Daniel 1 5--6- hours

How many hours did the boys dance in total? 19 The lengths of bolts and nails are often measured in inches. For the bolt shown, find the length of the threaded section if the total length of the bolt is 3 1--4- inches.

? 18–5" 5 — 16 "

20 To make the casing for a simple collapsible telescope you need 2 cylinders, one of which fits inside the other. One cylinder has measurements as shown below left.

? cm

7 3— 16 cm

? cm 1– 4

1– 4 cm

cm 1– 4

cm

QUEST

GE

S

EN

MAT H

a What is the width of a cylinder (see middle diagram above) which would slide over the one shown above left? b What is the width of a cylinder (see diagram above right) which would slide into the one shown above left?

CH

L AL

1 Find the fraction that is equivalent to

3 --5

with the product of its numer-

ator and denominator being equal to 375. 2 The sum of two fractions is 1 1--2- . Their difference is fractions?

5 ------ . 12

What are the

3 If a clock takes 5 seconds to strike 6 o’clock, how many seconds will it take to strike 12 o’clock?

106

Maths Quest 7 for Victoria

Egyptian fractions

The ancient Egyptians (from 3100 BC to 332 BC) used different symbols or hieroglyphics to show numbers. Using these symbols, the number 23 462 would be shown as:

1

10

1. Work out each number using the description given above. (a)

(b)

(c)

100

1000 10 000

23 462

(d)

As well as expressing words and whole numbers with hieroglyphics, the ancient Egyptians devised a way of expressing fractions as well. They placed a ‘mouth’ (pronounced ‘er’) over their symbols for whole numbers. For example:

1– 3

1– 5

Apart from some special fractions, all numerators were of the value 1, so to write a fraction like 3--5- , they would write 2. What do these symbols represent? (a)

(c)

1– 6

1 — 10

1– 5

(b)

(d)

+

1 — 100

1– 5

+

1– 5

or

1– 2

+

1 — 10

Chapter 3 Fractions

107

3. Perform the following calculations. Give your answer in Egyptian hieroglyphics and in Hindu–Arabic numerals. (a) Add these 2 numbers. (b) Subtract the second number from the first.

4. Write 4 of your own questions using Egyptian hieroglyphics. Swap these with a partner. Calculate the answers to your partner’s set of questions, showing clearly how you were able to work them out.

Multiplying fractions To multiply fractions, multiply the numerators and multiply the denominators.

WORKED Example 19 Find

1 --5

× 3--4- .

THINK

WRITE

1

Write the question.

2

Write as one fraction.

3

Multiply the numerators and the denominators and simplify your answer if appropriate.

1 --5

×

3 --4

1×3 = -----------5×4 3 = ----20

Fractions can be simplified by dividing both numerator and denominator by the same number before being multiplied. This is easier than working with large numbers. Dividing both numerator and denominator by the same number is often called cancelling.

WORKED Example 20 Find

2 --3

×

9 ------ . 10

THINK 1

Write the question.

2

Write the 2 fractions as 1 keeping the multiplication signs. Divide numerator and denominator by the same numbers or cancel. Multiply to find the answer.

3 4

WRITE 2 --3

×

9 -----10

1 2 × 93 = 1--------------- 5 3 × 10

=

3 --5

108

Maths Quest 7 for Victoria

WORKED Example 21 Find

1 --3

× 36.

THINK

WRITE

1

Write the question.

2 3

Express the whole number as a fraction and cancel if possible. Multiply the numerators and denominators.

4

Simplify your answer.

1 --3

× 36

1 36 12 = 1 --- × -----3 1 =

12 -----1

= 12

Mixed numbers must be changed to improper fractions before multiplying.

WORKED Example 22

Find 1 1--2- × 3 3--5- . THINK

WRITE

1

Write the question.

2 3

Change mixed numbers to improper fractions and cancel if possible. Multiply numerators and denominators.

4

Simplify your answer.

1 1--2- × 3 3--53 18 9 = 1--- × -----2 5 =

27 -----5

= 5 2--5-

Using the word ‘of ’ Sometimes the word of can mean the same as ‘multiply’. Antonio might take 1--3- of an hour (60 minutes) to cook his pizzas. A third of an hour or 1--3- of 60 minutes is the same ------ = 20 minutes. So it takes Antonio 20 minutes to cook a pizza. as 1--3- × 60 1 A customer wants to buy 1--4- of the 8 remaining pizzas for the family’s evening meal. A quarter of 8 is the same as 1--4- × 8--1- = 2. So the customer buys 2 pizzas.

WORKED Example 23 Find

2 --3

of 1--4- .

THINK

WRITE

1

Write the question.

2 --3

2

Change ‘of’ to ‘×’ and cancel if appropriate.

21 1 = --- × --- 2 3 4

3

Perform the multiplication.

=

of

1 --6

1 --4

109

Chapter 3 Fractions

WORKED Example 24 Mum put a 2 litre carton of flavoured milk in the refrigerator. When the children came home from school, Joanna drank one quarter of the milk, Kieren drank one half of it and Daisy drank one sixth of it. What fraction of the milk, in litres, did each person drink? THINK 1

WRITE

Write the fraction of milk that Joanna drank and simplify the answer.

Joanna:

1 --4

of 2 =

1 --4

1 21 = 2--- × --4 1 =

2

3

Write the fraction of milk that Kieren drank and simplify the answer.

Kieren:

Write the fraction of milk that Daisy drank and simplify the answer.

Daisy:

1 --2 1 --2

1 --2

of 2 =

1 --6

1 21 = 1--- × --2 1 =1 of 2 = 1--6- × 2

×2

1 21 = 3--- × --6 1 =

4

×2

Write a sentence giving the answer in litres.

1 --3

Joanna drank a half a litre, Kieren drank 1 litre and Daisy drank a third of a litre.

remember remember 1. 2. 3. 4. 5.

To multiply fractions, multiply the numerators and multiply the denominators. Fractions can be simplified (or cancelled) before being multiplied. A whole number can be written as a fraction with a denominator of 1. Change mixed numbers to improper fractions before multiplying. In a problem involving fractions, ‘of’ can be replaced with ‘multiply’, then the fractions can be multiplied in the usual way.

3E WORKED

Example

19

Multiplying fractions

1 Find: ×

a

1 --2

e

11 -----13

i

5 --8

×

m

7 --8

×

b

1 --3

×

2 --3

c

2 --5

f

6 --7

×

9 -----11

g

11 -----12

1 --3

j

2 --3

×

2 --3

k

5 --6

×

7 --8

n

2 --3

×

4 -----13

o

3 --5

×

1 --4

×

1 --2

× ×

d

5 --9

×

4 --3

h

8 --9

×

5 --9

11 -----12

l

5 -----11

3 -----10

p

4 --7

3 --5 11 -----12

× ×

5 -----12 2 --3

Multiplying fractions

Multiplying fractions

110 WORKED

Example

20

Maths Quest 7 for Victoria

2 Find: a

4 --8

×

3 --9

b

2 --5

×

d

3 --8

×

4 -----15

e

10 -----11

×

g

18 -----32

×

64 -----72

h

48 -----56

×

j

8 -----49

×

14 -----16

k

4 --9

m

42 -----54

×

12 -----49

n

5 -----25

× ×

1 --2

×

c

3 --7

22 -----25

f

24 -----27

×

9 --8

24 -----60

i

15 -----27

×

36 -----45

l

6 -----12

×

8 -----10

o

18 -----24

×

12 -----36

6 --8 12 -----48

5 --6

3 Mark’s uncle gave him a family size block of chocolate. He divided it into thirds to share with his 2 sisters. His friends Tom and Nick then arrived to visit, and Mark shared his portion of the chocolate with them. a Draw the block of chocolate, and shade Mark’s third of the block. Then draw lines to represent Tom’s share. b What fraction of the block of chocolate did Nick receive? WORKED

Example

21 WORKED

Example

22

4 Find: b

5 -----14

×7

c

15 -----22

b

3 --4

× 1 5--8-

c

1 1--2- × 1 1--2-

e 2 1--5- × 3 1--2-

f

9 - × 10 ----10

× 3 2--3-

j

4 1--3- ×

a

2 --3

×3

× 11

d

5 -----16

×4

5 Find: a 1 1--4- ×

i

9 -----10

2 --3

2 --3

3 --5

m 5 × 3 1--2-

n 8 × 3 1--4-

q 5 1--5- × 6 2--3-

r

d 2 5--8- × 1 3--4-

g 2 1--4- × 2 1--2-

h

4 --5

× 1 6--7-

k 1 1--2- × 12

l

2 3--4- × 6

1 o 4 3--5- × 2 ----10

p 3 5--6- × 9 5--7-

3 3--4- × 2 2--3-

6 Tim was looking for something to do one Saturday afternoon and his Dad suggested he cook something for afternoon tea. Tim found a recipe for peanut butter muffins. The recipe is given below. Ingredients: 1 --4

1 --4

cup sugar,

1 --2

cup peanut butter, 2 eggs,

cup margarine,

1 1--2- cups milk, 2 1--2- cups self-raising flour, 1 --4

teaspoon baking soda

Method: Blend the sugar, margarine and peanut butter. Beat in the eggs and milk. Add the selfraising flour and baking soda. Place the mixture in greased muffin pans and sprinkle with cinnamon sugar. Bake for 15–20 minutes in a 200°C oven. This recipe makes 10 muffins, but Tim wants enough mixture to make 15. He needs to multiply the quantities by 1 1--2- . Write the quantities Tim needs to make 15 muffins.

Chapter 3 Fractions

WORKED

Example

23

7 Find: a

1 --2

b

3 --4

d

9 -----10

2 --3

5 --7

e

8 --9

of

1 --4

g

4 --5

of 20

h

7 --8

j

2 --5

of 45

k

3 --5

m

3 --4

of 6

n

9 -----10

of 15

p

3 --7

of 12

q

1 --2

s

2 --3

of

t

5 --6

of

1 --4

3 --2

c

5 --6

f

5 --7

of 32

i

9 -----10

of 25

l

8 --9

o

1 -----12

of 10

of

1 1--2-

r

3 --4

of

8 2--3-

of

18 -----25

u

7 --8

of 3 1--3-

Example

24

E

of

5 --6

of

7 -----25

Fraction of a number

of 50 of 81 GAME time

of

of

8 A car’s petrol tank holds 48 litres of fuel. a If the tank was full at the start of a trip, what fraction of the tank has been used? b How many litres of petrol have been used? WORKED

111

Fractions 02

F

9 Jamie’s Mum made 6 litres of punch for his birthday party. Emily drank 1--3- of the 1 1 - and Jonathan drank ------ . What fraction of the punch, in litres, punch, Tracy drank ----18 12 did each person drink? 10 Zoe and Sarah play basketball with the Sharp Shooters. The games are played for 40 minutes. Zoe played 4--5- of last week’s game and Sarah played 7--8- . How many minutes of the game did: a Zoe play? b Sarah play? 11 Gustave’s monthly take-home pay is $2400. From this he spends a quarter on his home loan payments, one half on food and drink and one sixth on clothing. One half of the remainder goes into his savings account. How much money does Gustave put into his savings account each month? 12 To make a fruit loaf using an automatic bread maker, the fruit needs to be added after the second rise stage and the top of the bread needs to be glazed two-thirds of the way through the bake stage. The instruction booklet shows you how long each stage of the total cycle takes. Stage Length of time First knead 5 minutes Second knead 20 minutes First rise then punch down 40 minutes Second rise then punch down 25 minutes Third rise 50 minutes Bake 40 minutes a After what length of time should the fruit be added? b Express this as a fraction of the total time in the bread maker. c After what length of time should the top of the bread be glazed?

112

Maths Quest 7 for Victoria

13 a b c d

Draw a quarter of a pie. Shade half of this piece of pie. What fraction of the whole pie is shaded? Complete the mathematical sentence 1--4- × 1--2- = __ to show how the answer was found.

14 a Find how many half pizzas in 1 pizza. b Find how many half pizzas in 2 whole pizzas. c Find how many half pizzas in 5 whole pizzas.

Musical fractions In music, particular symbols tells us how long a note is to be held for. The time value of each note is shown below.

4 beats 2 beats 1 beat (semibreve) (minim) (crotchet)

1– 2

beat 1–4 beat (quaver) (semi-quaver)

More than 1 quaver is shown as More than 1 semi-quaver is shown as

1. What is the total number of beats indicated by these notes? (a) (b) (c)

(d)

When musicians decide they want a note held for a little longer, they put a dot after the note. The dot adds an extra half of the number of beats to the original time value. For example, a dotted minim has a time value of 2+

1 --2

× 2 = 3 beats.

2. What is the time value indicated by a dotted crotchet? 3. What is the time value indicated by a dotted quaver? 4. What is the total number of beats indicated by these notes? (a) (b) 5. Write one note which is equivalent in time value to each of the following. (a) (b) (c) (d) (e)

(f)

(g)

(h)

To make music more organised, time signatures were added to the beginning of a piece of music so that each section or bar was divided up into equal parts. For example, 44 tells us that we need 4 crotchet beats (or notes that will make up that amount) in a bar. The 4 on the top shows us the amount of beats per bar, while the 4 on the bottom shows us that the note values we use are the ones that belong to crotchet time. In 34 time, we need 3 beats to make up a bar and in 24 time we need 2 beats to make up a bar.

Chapter 3 Fractions

113

6. Add one note to complete each of the following bars. (a) 2 (b) 4 4 4 (c) 3 (d) 2

4 (e) 4

4 (f) 3

4

4

(g) 3

(h) 2

4

4

7. Add 2 notes to complete each of the following bars. (a) 4 (b) 3

4

4

(c) 2

(d) 3

4

4

(e) 4

(f) 2

4

4

(g) 3

(h) 3

4

4

8. Put time signatures to each of the following rhythm patterns. (They are crotchet beats.) (a) (b) 9. Write your own rhythm patterns 4 bars long, using a different pattern in each bar for each of the following time signatures. (a)

3 4

(b)

4 4

Dividing fractions The reciprocal of a fraction is found by tipping the fraction upside down. So reciprocal of 5--3- . A whole number can be written as a fraction by putting it over 1 so that reciprocal of 4--1- .

WORKED Example 25

Find the reciprocal of 2--3- . THINK

WRITE

Turn the fraction upside down and write a sentence.

The reciprocal of

2 --3

is 3--2- .

3 --5

is the

1 --4

is the

114

Maths Quest 7 for Victoria

To find the reciprocal of a mixed number, express it as an improper fraction first.

WORKED Example 26

Find the reciprocal of 1 2--3- . THINK

WRITE

1

Write 1 2--3- as an improper fraction.

1 2--3- =

2

Turn it upside down and write a sentence.

The reciprocal of 1 2--3- is 3--5- .

5 --3

Antonio had a half of one of his famous pizzas left. A small child asked him how many quarters there were in his half. Antonio cut the pizza as shown, so that it was clear that one half divided into quarters gave 2 pieces or 2 quarters. 1 --2

÷

1 --4

=2

The answer could also have been found by changing the division sign to a multiplication sign and tipping the second fraction upside down. 1 --2

÷

1 --4

=

1 --2

×

4 --1

=2 Division by a number is the same as multiplication by its reciprocal.

WORKED Example 27 Find

2 --3

÷ 4--9- .

THINK

WRITE

1

Write the question.

2 --3

÷

4 --9

2

Change the division sign to a multiplication sign and tip the second fraction.

=

2 --3

×

3

Cancel if possible.

4

Perform multiplication.

=

3 --2

5

Simplify your answer.

= 1 1--2-

9 --4

To divide mixed numbers, you can first change them to improper fractions.

Chapter 3 Fractions

115

WORKED Example 28

Find 1 1--2- ÷ 3 2--5- . THINK

WRITE

1

Write the question.

1 1--2- ÷ 3 2--5-

2

Change mixed numbers to improper fractions.

=

3 --2

÷

17 -----5

3

Change the ÷ to × and tip the second fraction.

=

3 --2

×

5 -----17

Multiply the numerators and the denominators and simplify your answer if necessary.

=

15 -----34

4

Graphics Calculator tip!

Dividing a fraction by a fraction

With the TI graphics calculator we use the ÷ key to enter a fraction as well as when we need to divide two numbers. Using brackets around each fraction helps to keep the operations clear, particularly with mixed numbers. The calculation for 1 1--2- ÷ 3 2--5- is shown.

remember remember 1. The reciprocal of a fraction is found by tipping the fraction upside down. 2. The reciprocal of a mixed number is found by expressing it as an improper fraction, then tipping it upside down. 3. A whole number can be written as a fraction by putting it over 1. 4. Division by a number is the same as multiplication by its reciprocal. 5. To divide mixed numbers, first change them to improper fractions.

3F WORKED

Example

25

Dividing fractions

1 Find the reciprocals of each of the following. a

3 --4

b

2 --7

e

6 --5

i

1 --5

m 5

c

5 --3

f

9 --2

j

1 -----12

n 20

d

5 -----12

g

5 --3

h

10 -----3

k

3 -----15

l

2 -----22

o 13

p 1

Reciprocals

116 WORKED

Example

26

Maths Quest 7 for Victoria

2 Find the reciprocals of these mixed numbers. a 1 2--3-

b 4 3--7-

c

3 6--7-

g 3 3--8-

h 5 1--3-

f

7 1--2-

d 4 2--5i

e 9 2--7-

10 2--9-

9 6 ----10

j

3 Multiply each of these numbers by its reciprocal. a

5 --7

b

2 --9

3 --5

c

d 3 2--7-

e 2 1--8-

f 3 3--4-

4 Copy and complete the following statement: When any number is multiplied by its . WORKED

Example

Dividing fractions

27

Dividing fractions

WORKED

Example

28

, the answer is always

5 Find: ÷

a

1 --4

d

1 -----12

÷

g

10 -----3

÷

j

4 --9

÷

m

4 --3

2 --3

÷

c

9 -----10

÷

1 --2

5 --6

f

7 -----12

÷

9 -----11

20 -----21

i

8 -----15

÷

16 -----15

4 --7

÷ 10

l

5 --8

÷ 20

15 -----11

÷5

o

7 --9

÷ 30

b 9 ÷ 4 1--2-

c

9 --4

÷ 3 1--2-

e 5 ÷ 2 3--4-

f

7 4--9- ÷

h 1 1--5- ÷ 2 1--3-

i

5 6--7- ÷ 2 3--4-

9 - ÷ 4 k 10 ----10

l

4 1--2- ÷ 3 6--7-

n 4 2--3- ÷

o 3 5--8- ÷ 1 3--4-

b

8 --9

2 --3

e

4 -----11

15 -----21

h

2 --7

÷

4 --9

k

÷ 12

n

7 --6

÷

6 Find: a 1 1--4- ÷

3 --4

÷ 3 1--3-

d

10 -----3

g

1 --8

j

1 1--2- ÷

÷ 2 1--21 --2

m 3 2--7- ÷ 2 1--7-

7 --6

7 Felicity is dividing 1 1--2- kilograms of mince steak into 8 hamburger portions. How many kilograms of mince will be in each hamburger portion? 8 Ned spends his school holidays helping his father with the shearing. a It takes Ned 1--4- hour to shear one sheep. How many sheep will Ned shear in 5 1--2- hours? 1 - of b Ned’s father, Wesley, shears a sheep every ----12 an hour. How many sheep would Wesley shear if he worked continuously for 8 hours?

3.2

9 Michelle can run one lap of a cross-country course in 1--5- hour. a How many laps could she complete in 1 1--3- hours? (Assume she can keep up the same pace.) b How many minutes does it take Michelle to run one lap?

1 --3

Chapter 3 Fractions

This part part of settled in 1804. 5 8

M

÷ 163

A

=

was first Answer these fraction questions to discover the puzzle answer code.

÷ 31

5 6

1

2 7

=

÷ 37

S How many are there in

D

5

1 2

L

1

3 4

N

= 1 4 2 3

s

I

?

=

5÷

1 10 ÷ 7

5 8

2 5

V

3

= 3 4

3

÷2

=

÷ 87

E

1

22

÷ 32

Divide 4 41 by 1 21

D

= 3 7

1 2

÷ 1 47 =

= A

3 4

Divide by 85

N

=

E

1

117

÷2 2

N

=

1 2

1

1 5

1

2

2 3

44

5

÷ 1 52

5 6

8

34

2

3 14

3

2

3

33

1

4 7

3

1 2

118

Maths Quest 7 for Victoria

2 Calculate each of the following, expressing your answer in simplest form. −

4 -----15

2

5 --6

+

6 3 5--9- ×

15 -----20

7

2 --7

of

1

12 -----15

3 -----10 21 -----40

3 3 1--4- + 2 2--5÷

5 --9

8

4 11 5--8- − 3 1--29 8 ÷ 1 3--4-

2 --3

×

4 -----18

10 4 2--5- ÷

2 -----11

5

12 -----20

Mixed operations with fractions If more than one operation is involved in a question with fractions, the same set order is used as with whole numbers. Therefore, the order of operations is: 1. brackets 2. multiplication and division (from left to right) 3. addition and subtraction (from left to right)

WORKED Example 29 Find

6 --7

× 1 5--6- ÷ 1--4- .

THINK

WRITE

1

Write the question.

6 --7

× 1 5--6- ÷

2

Change the mixed number to an improper fraction.

=

6 --7

3

Change division to multiplication and tip the next fraction (times and tip). Cancel where appropriate. Multiply and simplify.

4

×

11 -----6

1 --4

÷

1 --4

1 6 11 4 = --- × 1------ × --7 6 1

=

44 -----7

= 6 2--7-

On a TI graphics calculator, the calculation in worked example 29 can be entered as (6 ÷ 7) × (1 + 5 ÷ 6) ÷ (1 ÷ 4), then pressing MATH selecting 1:Frac and pressing ENTER .

119

Chapter 3 Fractions

remember remember The order of operations is the same for fractions as it is for whole numbers: 1. brackets 2. multiplication and division (left to right) 3. addition and subtraction (left to right).

Mixed operations with fractions

3G

Mixed operations

1 Find the following.

WORKED

Example

a

5 --8

+

5 --6

b

d

4 --5

−

3 --8

e 1 3--5- + 9 4--7-

g

1 4--5-

×

9 --7

÷

3 --4

f

6 1--3- ÷ 4 1--2-

i

5 1--4-

b 1 3--4- × 5 ÷ 3 1--2-

c

1 9 - + 2 --- − 4 ----4 10

e 2 5--6- × 2 − 3 1--2-

f

1 --2

i

11 -----12

b (5 − 3 6--7- ) × 4 1--3-

c

(1 1--2- + 3--4- ) × (5 − 3 3--4- )

÷ (4 1--2- + 5--6- )

f

1 -----11

i

7 -) 6 ÷ (3 2--3- + 2 ----12

5 --2

−

21 -----32

×

c

h

7 --8

1 7--8-

9 -----10

−

Four operations with fractions

2 1--5-

2 Find the following. 3 --5

a

29

× 2 1--3- ÷

1 --2

d 5 1--3- + 7 3--8- − 8 1--24 --5

g

× 10 + 2 3--4-

h

1 --3

+

2 --9

×

3 --4

+ −

2 --3 1 --2

× ÷

3 --5

1 --5

Mixed operations

3 --4

3 Calculate these, remembering to remove the brackets first. a ( 5--4- + 2--3- ) ×

1 --2

d 6 − (7 5--6- ÷ 2) 1 1 - ) − (7 --- × g ( 2--3- × 4 ----12 2

h 4 × (2 − 1 3--7- )

1 ------ ) 10

× (8 2--9- − 5 1--6- )

QUEST

GE

S

3 --2

( 2--3- − 1--6- ) ÷ (3 1--8- − 2 5--6- )

EN

M ATH

j

e

CH

A

LL

1 Without using a calculator find three quarters of four ninths of 78. 1 1 1 --- + --- + --2 3 4 -. 2 Without using a calculator find the answer to -----------------1 1 --- + --4 9

3 The fraction

1 --4

is a third of what number?

4 According to the will, the $120 000 estate is to be left to three children. Sam is to receive 2--5- of the money, Georgia is to receive 1--3- of the money and Olivia is to receive the remainder of the money. How much money is Olivia to receive? 5 If a half, a quarter and an eighth of a number are added together, the total is 63. What is the number?

120

Maths Quest 7 for Victoria

You want more pizza? Do you remember the question posed at the start of this chapter? Martin eats 2 slices of a pepperoni pizza which has been cut into 6 equal pieces while Olivia eats 3 pieces of the sausage pizza which has been cut into 8 equal pieces. Who has eaten more?

1. 2. 3. 4.

Try this question again, showing your calculations. How many pieces of each pizza is left? What fraction of each pizza is left? What is the total amount of pizza left?

Daniella takes a slice of pepperoni pizza and Christopher takes a slice of sausage pizza. However, Christopher complains that his piece is smaller than Daniella’s. 5. What is the difference in size between the larger and the smaller slice? (Write your answer as a fraction of a whole pizza.) Daniella continues to eat the pepperoni pizza and Christopher continues to eat the sausage pizza. 6. How many slices of pizza should Daniella and Christopher each have so as to eat the same amount? 7. What fraction of each pizza is now left over?

Powers and square roots of fractions Fractions and mixed numbers can be squared (index or power of 2) or cubed (index or power of 3) by using index notation just as with whole numbers. For example,

2 2 --- 3

=

2 --3

×

2 --3

=

22 ----32

=

4 --9

WORKED Example 30

Find: a

1--- 2 4

b

3 2 --- 5

c (1 1--3- )2.

THINK

WRITE

a

a

1

Write the question.

2

Square the numerator and the denominator. Simplify.

3

2 1 --- 4

=

12 ----42

=

1 -----16

Chapter 3 Fractions

THINK

WRITE

b

b

c

3 2 --- 5

1

Write the question.

2

Cube the numerator and the denominator.

=

3

Simplify.

=

1

Write the question.

2

Change the mixed number to an improper fraction. Square both the numerator and the denominator.

=

2 4 --- 3

=

42 ----32 16 -----9

Change back to a mixed number.

= 1 7--9-

3

4

23 ----53

2×2×2 = --------------------5×5×5 8 --------125

c (1 1--3- )2

=

It is also possible to find the square roots of fractions and mixed numbers.

WORKED Example 31

Find: a

9 -----16

b

6 1--4- .

THINK

WRITE

a

a

1

Write the question.

2

Write the numerator and denominator as separate square roots. Calculate the square root of both the numerator and the denominator.

3

b

1

Write the question.

2

Change the mixed number to an improper fraction. Write the numerator and the denominator as separate square roots. Calculate the square root of both the numerator and the denominator. Change to a mixed number.

3

4 5

9 -----16

=

9 ---------16

=

3 --4

6 1--4-

b =

25 -----4

=

25 ---------4

=

5 --2

= 2 1--2-

121

122

Maths Quest 7 for Victoria

Graphics Calculator tip!

Powers and square roots

As with whole numbers, we can use the TI graphics calculator to find different powers and square roots of fractions. To enter the index or power we use the ^ key or if we are squaring a number we can also use the x2 key. Remember to press the appropriate keys to obtain your answer as a fraction. The screen at right shows the calculations for worked example 30. To obtain the square root of a fraction, first enter the square root sign by pressing 2nd [ ] (that is, press 2nd then x2 ) then follow with the fraction. The screen obtained for worked example 31 is shown opposite.

remember remember 1. To square a fraction, square both numerator and denominator (base is raised to an index of 2). 2. To cube a fraction, cube both numerator and denominator (base is raised to an index of 3). 3. To find the square root of a fraction, take the square root of both numerator and denominator. 4. To find the square root of a mixed number, change the mixed number to an improper fraction, then find the square root of both the numerator and the denominator.

3H WORKED

Example

30a

WORKED

Example

30b Squaring and cubing fractions

Powers and square roots of fractions

1 Find: a

1--- 2 2

b

2 1 --- 3

c

2 2 --- 5

d

5--- 2 6

e

2 3 --- 4

f

7 2 ---- 10

g

5 2 ---- 12

h

3 2 ---- 11

i

2 3 --- 8

2 Find: a

1--- 3 4

b

3 1 --- 2

c

3 2 --- 5

d

3 3 ---- 10

e

3 2 --- 3

f

3 1 --- 6

123

Chapter 3 Fractions

WORKED

Example

30c

WORKED

Example

31

3 Find: a (1 1--3- )2

b (2 1--4- )2

c

(1 1--2- )2

1 2 -) d (1 ----10

e (2 1--5- )2

f

(3 2--3- )2

4 Find: a

16 -----25

b

49 --------100

c

9 -----64

d

25 -----36

e

2 1--4-

f

1 7--9-

g

9 1 ----16

h

-----2 14 25

i

-----1 15 49

Square root of a fraction

2 1 --- 2

+

2 3 --- 4

b

2 2 --- 3

d

2 4 --- 7

÷

2 3 --- 7

e

4 --9

−

+

2 1 --- 2

2 2 --- 3

−

1 --2

c

2 3 --- 5

×

2 5 --- 9

f

16 -----25

×

2 1 --- 2

QUEST

GE

S

a

EN

MAT H

5 Calculate:

CH

AL

1 Find the missing power for this calculation:

3 --- 5

=

2187 ---------------78 125

.

L 2 A length of rope is cut in half and one half is used. Then one third of the other half is cut off and used. If the remaining piece of rope is 10 cm long, how long was the original rope? 3 Find the answer to 3

( 1--2- ) ------------------------. 1 2 -) ( 1 + ----------1+3 4 Look for a pattern in this list of fractions and write down the next two in the sequence. 2 --3

→

9 --5

→

16 -----14

→

23 -----50

→

30 -----53

3.3 −

1 -----10

124

Maths Quest 7 for Victoria

summary 1 2 3 4 5 6

7 8

9

10

11

12 13

Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows. The is the top part of the fraction. The is the bottom part of the fraction. Different fractions can be compared or ordered by writing each one with the denominator. order is from smallest to largest. order is from largest to smallest. are found by multiplying or dividing both the numerator and the denominator by the same number as long as the number is not equal to zero. To fractions, find the largest number that is a factor of the numerator and the denominator and divide both by that number. fractions can be changed to mixed numbers by dividing the numerator by the denominator. The becomes the numerator and the denominator remains the same. Mixed can be changed to improper fractions by multiplying the whole number and the denominator then adding the numerator. The result becomes the numerator and the denominator remains the same. Fractions must have the same denominator to be added or subtracted. This is done by finding a denominator. Once the denominators are the same, the numerators can be added or subtracted. To multiply fractions, simply the numerators and multiply the denominators. It is often easier to or simplify numerators and denominators before multiplying. Remember to change mixed numbers to improper fractions first. To find the of a number, swap the numerator and denominator. To divide fractions, change the ÷ to × and the second fraction. Remember to change numbers to improper fractions first.

WORD denominator ascending mixed tip multiply improper

LIST common same equivalent fractions cancel numerator simplify

descending remainder numbers reciprocal

Chapter 3 Fractions

125

CHAPTER review 1 What fraction of the eggs in this carton are brown?

3A

2 Write 3 equivalent fractions for the following. a

3 --8

b

4 --9

5 -----11

c

d

3A

6 --7

3 Fill the gaps to make equivalent fractions. a

1 --2

=

--4

d

2 --5

=

-----10

= =

3 --16 ------

b

1 --3

=

3 ---

=

-----15

c

3 --4

e

1 --4

=

5 ---

=

--------100

f

3 -----10

= =

=

-----28

--------100

=

9 ---

3A 300 ---------

4 Simplify these fractions. a

42 -----49

b

81 --------108

c

18 -----20

d

21 -----28

e

36 -----72

f

50 --------150

g

75 --------125

h

2 -----80

-----4 15 20

20 d 1 -------100

3B

5 Simplify the following mixed numbers. a

b 3 6--9-

8 6 ----10

c

3B

6 What fraction of the months of the year begin with the letter J? Express your answer in simplest form.

3B

7 Convert these improper fractions to mixed numbers.

3C

a

16 -----3

b

21 -----5

c

12 -----7

d

11 -----2

e

5 --4

f

80 -----7

g

55 -----9

h

72 -----10

8 Express these mixed numbers as improper fractions. a

2 3--4-

b 9 7--8-

e

3 2--3-

f

3 5--7-

c

9 9 ----10

-----g 3 11 12

3C

d 5 5--6h 8 1--2-

9 Find: a

5 --6

+

1 --8

b

d

5 --4

+

11 -----2

e 1 7--8- + 2 5--6-

g

6 1--3- − 2 5--9-

7 --9

−

2 --5

1 1--5- + 3 1--3- −

12 -----5

4 --3

f

10 3--7- − 8 1--2-

3D

h 5 1--2- − 2 5--6-

10 Find: a

−

c

13 -----15

b 4 7--9- − 1 5--6- + 3 1--3-

3D

126 3D

Maths Quest 7 for Victoria

11 Tony caught 3 fish with masses: 7 1 4 - kg, 2 --- kg and 2 --- kg. 2 ----10 2 5

a Which fish is the heaviest? b What is the combined mass of the fish? c What is the difference in mass between the heaviest and the lightest fish?

3E 3F

3F 3G 3H

3H

12 Find: ×

a

4 --7

c

25 -----36

e

5 --9

×

b

2 --5

d

56 -----81

2 4 2--5- × 3 ----11

f

10 2--7- × 6 2--9-

g

4 --5

of 55

h

2 -----11

of 121

i

7 --9

of

j

1 -----12

of 200

×

18 -----20

18 -----84

×

3 --5 18 -----64

13 Find: a

3 --5

×

d

4 --9

× 2 1--4- ×

5 --6

×

2 --7

b

3 --4

×

16 -----21

×

7 -----20

5 3 1 - × 1 --- × 2 --e 3 ----12 5 2

3 --5

c

5 -----11

×

f

6 -----17

× 1 1--4- × 3 2--5-

c

7 --9

f

6 2--3- ÷ 4 1--6-

c

( 2--3- − 1--6- ) × ( 3--4- + 5--8- )

f

7 − 2 1--4- ÷ 1 1--2-

3 -----10

1 × 1 ----10

14 Write the reciprocals of: a

5 --2

b

2 --7

c

1 --9

e

124

f

1 --4

g 5 3--4-

h 8 1--2-

i

7 2--5-

j

2 4--9-

k 3 5--8-

l

d 12

4 1--5-

15 Find: a

4 --5

÷

d

3 --7

÷ 10

8 -----15

b

9 -----10

÷

27 -----40

e 1 1--2- ÷ 4 4--5-

÷4

16 Calculate: a

( 4--9- + 5--6- ) ÷ 3

d

3 --4

÷

9 -----10

+

5 --7

b ×

5 -----14

3 2 - + --- ) × ( ----10 5

e 2 + 4 1--2- ÷ 2 1--4-

14 -----15

17 Calculate: a

4--- 2 5

b

9 2 ---- 10

c

2 3 --- 4

d

7--- 2 8

e

3 1 --- 3

f

3 5 --- 6

g

(1 1--2- )2

h (2 1--4- )2

i

(1 3--5- )2

test yourself

18 Calculate: a

81 --------100

b

25 -----49

c

2 7--9-

d

9 2 ----36

CHAPTER

3E

3

Xem thêm...
3 A pepperoni pizza is cut into 6 equal pieces while a sausage pizza of the same size is cut into 8 equal pieces. If Martin eats 2 pieces of the pepperoni pizza and Olivia eats 3 pieces of the sausage pizza, who has eaten more? How can you prove this?

84

Maths Quest 7 for Victoria

Understanding fractions Antonio makes pizzas. Besides selling whole pizzas, he often sells part of a pizza to his customers. For example, this pizza has been cut into 3 equal pieces. If a customer wants only 2 of these slices, then Antonio is selling 2--3- of the pizza to the customer.

Fractions are used to describe parts of a whole. The numerator or top number of the fraction shows how many parts there are, while the denominator or bottom number shows the number of parts the whole has been divided into. So 2--3- means that the whole thing has been divided into 3 equal parts and that we are concerned with 2 parts. The horizontal bar separating the numerator from the denominator is called the vinculum. A proper fraction is a fraction with the numerator smaller than the denominator, like 2--3- . An improper fraction has a numerator that is larger than the denominator, like 5--4- . A mixed number is made up of a number and a fraction, like 1 3--5- .

WORKED Example 1 What fraction of the rectangle shown has been shaded?

THINK 1 2 3

Count how many equal parts the rectangle has been divided into. How many parts are shaded? Write the answer in a sentence.

WRITE Total number of parts = 8 3 parts are shaded. 3 --- of the rectangle has been shaded. 8

Chapter 3 Fractions

85

Fractions can be marked on a number line in the 0 1– 2– 1 11– 12– 2 2 1– 3 3 3 3 3 same way as counting numbers. The number line above can be used to count in fractional steps. The fractions marked in bold are found by adding 1 to the previous fraction or mixed number so that the following sequence of numbers is obtained: 1 --- , 3

1 1--3- , 2 1--3- , 3 1--3- , 4 1--3- , . . .

WORKED Example 2 Copy the number line shown, filling in the missing numbers. Use it to find the next 3 terms in the given sequence: 3--4- , 1, 1 1--4- , _, _, _. THINK 1 2 3 4

0

1– 4

1– 2

3– 4

1

13–4

WRITE

Copy the number line and fill in the missing numbers. Circle the 3 given numbers in the sequence on the number line. On the number line, find the next 3 terms which continue the sequence. Write the sequence with the missing numbers.

0

1– 4

1– 2

3– 4

1 11–4 11–2 13–4

2

0

1– 4

1– 2

3– 4

1 11–4 11–2 13–4

2

3 --- , 4

1, 1 1--4- , 1 1--2- , 1 3--4- , 2

Finding equivalent fractions Antonio cuts a pizza into 2 equal pieces as shown. Each piece is exactly half of the pizza. A lady with a small child comes into the shop and wants to buy some pizza. She decides to cut one of the halves into 2 smaller, equal pieces as shown.

Now the child eats 2 quarters of a pizza and the mother eats half a pizza. If you look carefully you will see that each person has the same amount of pizza. We can say that 1 2 1 2 --- is equivalent to --- or --- = --- . 2 4 2 4 Equivalent fractions can be found by multiplying or dividing both the numerator and the denominator by the same number as long as the number is not zero. Therefore, 1 1×2 2 --- = ------------ = --2 2×2 4 So

and

1 1×3 3 --- = ------------ = --- . 2 2×3 6

1 2 3 --- = --- = --- . 2 4 6

86

Maths Quest 7 for Victoria

WORKED Example 3

For the equivalent fractions 4--5- = numerator and denominator.

8 -----10

find the number which has been used to multiply both

THINK 1

2

What number is 4 multiplied by to equal 8? (4 × _ = 8) What number is 5 multiplied by to equal 10? (5 × _ = 10) Write the answer.

WRITE 4 4×2 8 --- = ------------ = -----5 5 × 2 10

The number used is 2.

WORKED Example 4

From the list, find those fractions which are equivalent to 1--2- : 3--9- , 3--6- , THINK 1

Multiply the numerator and denominator of 1--2- by the numerator of the first fraction (3) to check whether the new fraction is in the list.

2

Multiply the numerator and denominator of 1--2- by the next different numerator (9) and check whether the new fraction is in the list.

3

Continue until all fractions have been considered.

4

Write the equivalent fractions.

9 10 7 17 ------ , ------ , ------ , ------ . 18 15 14 34

WRITE 1×3 3 ------------ = --2×3 6

1×9 9 ------------ = -----2 × 9 18

1 × 10 10 --------------- = -----2 × 10 20 7 1×7 ------------ = -----2 × 7 14 1 × 17 17 --------------- = -----2 × 17 34 Equivalent fractions of

1 --2

are 3--6- ,

WORKED Example 5

Write 3 equivalent fractions for 2--3- . THINK 1

Multiply both the numerator and denominator by 2, 3 and 4.

2

Write the equivalent fractions.

WRITE 2 2×2 2×3 2×4 --- = ------------ = ------------ = -----------3 3×2 3×3 3×4 2 4 6 8 --- = --- = --- = -----3 6 9 12

9 7 17 ------ , ------ , ------ . 18 14 34

Chapter 3 Fractions

87

Note: There are other fractions which are equivalent to 2--3- . These can be found by multiplying the numerator and denominator by other numbers. Answers to exercises (at the back of the book) will always give the 3 smallest equivalent fractions. Equivalent fractions can also be shown using diagrams. Equivalent fractions for 2--3are shown below.

2– 3

2– 3

= 4–6

= 6–9

2– 3

8 =— 12

Notice that the same amount of the rectangle has been shaded in each case.

Comparing fractions If Antonio’s pizza is cut into quarters it would be easy to see that 1 piece is less than 2 pieces, or that 1--4- of the pizza is less than 2--4- . If Antonio has another pizza of the same size but cuts it into thirds, it can still be easily seen that 1--4- is less than 1--3- . This is because a pizza divided into 4 equal pieces will give smaller pieces than one divided into 3 equal pieces. If the numerators are the same, the smaller fraction is the one with the larger 3 3 - is less than --- . If the numerators denominator. For example, 1--6- is greater than 1--7- and ----10 4 are different, another method is required to compare fractions. Fractions which do not have the same denominator can be compared by using equivalent fractions. We need to rewrite each fraction so that it has the same denominator.

WORKED Example 6 Which is the bigger fraction,

2 --3

or 3--5- ?

THINK 1

Find the lowest common multiple of the denominators. That is, find multiples of the larger denominator. Continue until there is one which is divisible by the other denominator.

2

Write each fraction as an equivalent fraction using this number (15) as the denominator. Decide which is bigger by comparing the numerators of the equivalent 9 ------ and ------ ). fractions ( 10 15 15

3

WRITE Multiples of 5 are 5, 10, 15 . . . 15 is divisible by 3 so 15 is the lowest common multiple.

2 2 × 5 10 3 3×3 9 --- = ------------ = ------ and --- = ------------ = -----5 5 × 3 15 3 3 × 5 15 2 --3

is bigger than 3--5- .

88

Maths Quest 7 for Victoria

The answer to worked example 6 can be written as 2--3- > 3--5- . The symbol ‘>’ means bigger than or greater than. The symbol ‘’ means greater than, the symbol ‘ or or < sign between each pair of fractions to make a true statement. a

8 --5

d

15 -----2

1 2--5-

19 -----2

8 1--2-

c

33 -----32

3 1 ----32

5 1--2-

e 3 3--4-

19 -----4

f

7 1--7-

65 -----7

8 --3

-----h 1 16 17

g 2 1--3-

b

35 -----17

1 1 Write down the equivalent fractions from this list: 123 31 10 16 19 8 6 3 1 2 --- , --- , ------ , ------ , ------ , ------ , ------ , ------ , --------- , --------- . 4 6 12 25 32 41 64 76 124 490

2 Write 3 equivalent fractions for 2--5- . 3 Write

12 -----20

in simplest form.

4 On a recent quiz, Simone answered 6 out of the 50 questions incorrectly. What fraction of the test did Simone answer correctly? Express your answer in simplest form. 5 Write

26 -----3

as a mixed number.

6 Change 1 1--4- to an improper fraction. 7 Write form.

12 -----9

as a mixed number in simplest

8 Change 3 6--7- to an improper fraction. 9 Rewrite these fractions in ascending (from smallest to largest) order: 2 14 --- , ------ , 5 5

1 3--5- , 3 3--5- ,

15 ------ , 5

4 2--5- ,

28 ------ , 5

6 1--5- .

10 A catering company provided 12 quiches which were each cut into 8 pieces for a luncheon. At the end of the function there were 12 pieces of quiche left over. How many whole quiches were eaten?

3.1

100

Maths Quest 7 for Victoria

Adding and subtracting fractions Fractions can be added and subtracted if they have the same denominator. Remember to simplify your answer if possible.

WORKED Example 14 Find

5 --9

+ 1--9- .

THINK

WRITE 5 --9

1

Write the question.

2

If the denominators are the same, add or subtract the numerators. Simplify the answer if possible.

3

+

1 --9

=

6 --9

=

2 --3

If fractions do not have the same denominator, use equivalent fractions to make the denominators the same.

WORKED Example 15 Find

5 --6

−

1 ------ , 12

expressing the answer in simplest form.

THINK

WRITE

1

Find multiples of each denominator.

2

Find the lowest common multiple (LCM) of 6 and 12 (12).

3

Rewrite the fractions if necessary so that they both have the LCM as the new denominator.

4

Add or subtract the numerators.

5

Simplify your answer if necessary by cancelling by the highest common factor (9 ÷ 3 = 3, 12 ÷ 3 = 4).

Multiples of 6 are 6, 12 . . . Multiples of 12 are 12 . . . LCM is 12. 5 1 --- – -----6 12 5×2 1 = ------------ – -----6 × 2 12 10 1 = ------ – -----12 12 9 = -----12 93 = ------ 4 12 =

3 --4

With practice many of these steps can be done in your head and the solution written as follows. 5 --6

−

1 -----12

= = =

10 -----12 9 -----12 3 --4

−

1 -----12

Chapter 3 Fractions

101

Adding and subtracting mixed numbers There are a number of ways of adding and subtracting mixed numbers. Use the method which you prefer. All mixed numbers can be added and subtracted by first changing them to improper fractions. Remember to simplify the answer if possible.

WORKED Example 16

Find 2 3--4- + 1 1--2- . THINK

WRITE

1

Write the question.

2 3--4- + 1 1--2-

2

Change to improper fractions.

=

11 -----4

+

3 --2

3

Write both fractions with the same denominator using equivalent fractions.

=

11 -----4

+

6 --4

4

Add the fractions.

=

17 -----4

5

Write the answer as a mixed number if appropriate.

= 4 1--4-

Alternatively, you can add the whole numbers first and then work out the fractional parts. So,

2 3--4- + 1 1--2- = (2 + 1) + ( 3--4- + 1--2- ) = 3 + ( 3--4- + 2--4- ) =3+

5 --4

= 3 + 1 1--4= 4 1--4This alternative method also works for subtracting fractions, unless the second fraction part is bigger than the first fraction part. Then use the method of equal addition. (Also see the section on subtraction in chapter 1.)

WORKED Example 17

Find 3 1--5- − 1 3--4- . THINK

WRITE

1

Write the question.

3 1--5- − 1 3--4-

2

Write both fraction parts with the same denominator (20). Add the same number of parts to both fractions to make 5 - . the second fraction a whole number ----20 Subtract and simplify if necessary.

4 15 - − 1 -----= 3 ----20 20

3 4

9 - − 2 = 3 ----20 9 = 1 ----20

102

Maths Quest 7 for Victoria

Graphics Calculator tip!

Adding and subtracting fractions

As with any calculation involving fractions, if you wish to have an answer expressed as a fraction then each calculation needs to end by pressing MATH , selecting 1:Frac and pressing ENTER . The calculation for worked ÷ 6 − example 15 would be entered as 5 MATH , select 1:Frac and press ENTER . When entering mixed numbers, it is a good idea to use brackets. This allows the correct order of operations to occur, particularly when subtracting. The calculations for worked examples 16 and 17 can be viewed in the screen shown. Note that the answers are given as improper fractions.

1

÷

12

then press

Estimation can be used to find approximate answers to addition and subtraction of mixed numbers. This allows you to easily check whether your answer to a calculation looks reasonable.

WORKED Example 18

Find 3 1--3- + 2 1--2- and check the answer by estimation. THINK

WRITE

1

Write the question.

3 1--3- + 2 1--2-

2

Group the whole numbers and group the fractions.

= (3 + 2) + ( 1--3- + 1--2- )

3

Add the whole numbers and write the fractions with the same denominator. Simplify the answer.

= 5 + ( 2--6- + 3--6- )

4

=5+

5 --6

= 5 5--65

Check the answer by adding the whole numbers to find an approximation.

3 1--3- + 2 1--2- > 3 + 2 so 3 1--3- + 2 1--2- > 5

remember remember 1. Fractions can be added and subtracted if they have the same denominator. 2. If fractions do not have the same denominator, use equivalent fractions to make the denominators the same. 3. Mixed numbers can be added or subtracted by first changing them to improper fractions. 4. Mixed numbers can also be added or subtracted by working with the whole numbers, then the fractional parts. If the second fractional part is bigger than the first fractional part, make the second fraction equal to a whole number by adding the same fraction to both parts.

103

Chapter 3 Fractions

3D WORKED

Example

14

Adding and subtracting fractions

1 Find: a

1 --5

d

20 -----50

g

5 --8

+

3 --5

+ −

11 -----50

+

1 --8

3 --8

b

9 -----11

−

3 -----11

c

3 --6

e

15 -----25

−

6 -----25

f

12 -----12

h

4 --7

i

33 --------100

+

5 --7

−

3 --7

+

2 --6

−

Adding and subtracting fractions

1 -----12

+

−

7 --------100

11 --------100

Adding fractions

2 Find the following, expressing the answers in simplest form.

WORKED

Example

15

+

a

5 -----12

d

37 --------100

g

13 -----16

1 -----12

+ +

13 --------100 9 -----16

−

10 -----16

b

7 -----16

e

9 --4

h

11 -----28

− − +

3 -----16 7 --4 10 -----28

c

9 -----15

f

4 --7

i

21 -----81

− + +

4 -----15

−

5 --7

21 -----81

2 --7

+

30 -----81

Subtracting fractions

3 Find the following, expressing the answers in simplest form. a

1 --2

+

1 --4

b

1 --2

−

d

1 --8

+

20 -----32

e

9 -----18

+

g

7 --9

−

8 -----27

h

31 -----35

−

+

2 -----21

c

1 --3

1 --6

f

17 -----40

+

1 --2

3 --7

i

8 -----10

−

28 -----70

1 --4

4 Find the lowest common multiple of each of the following pairs of numbers. a 2 and 3 b 3 and 5 c 5 and 10 d 4 and 6 e 5 and 6 f 9 and 6 g 4 and 8 h 6 and 8 i 7 and 5 5 Using the lowest common multiples that were found in question 4 as denominators, add or subtract these fractions. a

1 --2

−

1 --3

b

2 --3

−

2 --5

c

3 --5

+

3 -----10

d

3 --4

−

1 --6

e

4 --5

−

1 --6

f

1 --9

+

5 --6

g

1 --4

+

3 --8

h

5 --6

−

3 --8

i

5 --7

−

2 --5

6 Answer the following questions by first finding the lowest common denominator. Simplify your answer if necessary. +

b

5 --7

−

3 --8

c

4 --5

−

3 --4

2 -----15

e

5 --8

−

5 -----12

f

7 --9

−

1 --2

2 --9

h

7 -----11

+

1 --4

i

4 --5

−

2 -----11

+

7 -----10

k

3 -----10

−

1 -----20

l

7 --8

−

3 --5

2 --3

+

4 --5

n

5 --7

o

5 --9

−

2 --7

2 --3

−

1 -----13

q

11 -----12

r

4 --7

+

5 --6

a

2 --3

d

7 -----10

+

g

5 -----12

+

j

5 --6

m p

1 --4

+ −

3 --4 7 -----10

3.3

104

Maths Quest 7 for Victoria

7 Angela was a little hungry the day before her birthday, and spied the cake to be eaten 1 - of it, her Mum might not notice. the following day. She decided that if she ate only ----1 10 However, she was still hungry and took another --6- . a How much of the cake did Angela eat before her birthday? b How much of the cake was left for her birthday? 8 multiple choice a The lowest common denominator of 1--3- , A 16

B 24

and

1 --6

and

1 --5

C 18

B 10

is: D 72

1 1 --- , --2 4

b The lowest common denominator of A 8

1 --4

C 20

E 12

is: D 30

E 40

9 Answer the following: Adding and subtracting fractions

a

1 --2

d

11 -----12

g

1 --2

+ − +

1 --3

−

8 -----15 2 --3

+

1 --4

− 3 --4

3 -----20

b

3 -----10

+

4 -----15

e

1 -----16

+

1 --8

h

3 --4

+

1 --3

− +

+

10 A cake needed to be cooked at 200°C for is the total cooking time in hours?

1 --6 1 --4

c

7 --8

−

3 --4

f

5 --6

−

7 -----18

+ −

4 --6 1 --9

5 --6 3 --4

hour then a further

1 --2

hour at 150°C. What

11 Robert, Jason and Luke play football in the same team. Last Saturday, Robert kicked 1 1 5 --- of the team’s goals, Jason kicked --- and Luke ------ . What fraction of the team’s goals 4 3 12 were kicked by: a Jason and Luke? b Robert and Jason? c the three boys together? 12 Jacinta and her friend decided to climb the tree in her Nan’s front yard. It took her 10 minutes to climb halfway up, 10 more minutes to climb another sixth of the way and yet another 10 minutes to reach a further tenth of the way. Here she sat admiring the view for 5 minutes before beginning her descent. a What fraction of the height of the tree had Jacinta climbed after 20 minutes? b What fraction of the tree did Jacinta climb in total?

WORKED

Example

16

13 Find: a 1 1--5- + 3 1--2-

b 3 2--3- − 2 1--2-

c

5 2--3- − 2 1--4-

d 9 3--8- + 4

e 2 3--5- − 1 1--2-

f

3 1--2- + 2 3--4-

Chapter 3 Fractions

WORKED

Example

105

14 Find: a 7 − 5 2--3-

17

b 2 1--3- −

c 3 1--2- − 1 7--8-

5 --6

9 1 7 3 5 - e 5 --- − 1 ------ f 5 --- − 3 --d 4 3--5- − 2 ----10 4 12 4 6

15 Find: a WORKED

Example

1 --5

+ 2 2--7-

b

5 --8

+ 3 3--4- −

1 --2

c

2 2--9- + 1 1--3- +

d 4 2--3- + 1 3--8- − 3

1 --6

16 Find the following and check the answers by estimation. a 5 1--3- + 2 1--4-

18

b 1 1--6- + 2 1--2-

c 3 1--3- + 1 1--6-

d 6 1--2- − 2 1--4-

3 5 1 - f 7 --- − 3 --e 4 3--5- − 1 ----10 6 3

17 One-third of a litre of cordial is mixed with 1 1--2- litres of water. How many litres of drink has been made? 18 At a school dance-a-thon, the number of hours each student danced was recorded. Matthew 1 3--4- hours,

Leigh 2 2--3- hours,

Daniel 1 5--6- hours

How many hours did the boys dance in total? 19 The lengths of bolts and nails are often measured in inches. For the bolt shown, find the length of the threaded section if the total length of the bolt is 3 1--4- inches.

? 18–5" 5 — 16 "

20 To make the casing for a simple collapsible telescope you need 2 cylinders, one of which fits inside the other. One cylinder has measurements as shown below left.

? cm

7 3— 16 cm

? cm 1– 4

1– 4 cm

cm 1– 4

cm

QUEST

GE

S

EN

MAT H

a What is the width of a cylinder (see middle diagram above) which would slide over the one shown above left? b What is the width of a cylinder (see diagram above right) which would slide into the one shown above left?

CH

L AL

1 Find the fraction that is equivalent to

3 --5

with the product of its numer-

ator and denominator being equal to 375. 2 The sum of two fractions is 1 1--2- . Their difference is fractions?

5 ------ . 12

What are the

3 If a clock takes 5 seconds to strike 6 o’clock, how many seconds will it take to strike 12 o’clock?

106

Maths Quest 7 for Victoria

Egyptian fractions

The ancient Egyptians (from 3100 BC to 332 BC) used different symbols or hieroglyphics to show numbers. Using these symbols, the number 23 462 would be shown as:

1

10

1. Work out each number using the description given above. (a)

(b)

(c)

100

1000 10 000

23 462

(d)

As well as expressing words and whole numbers with hieroglyphics, the ancient Egyptians devised a way of expressing fractions as well. They placed a ‘mouth’ (pronounced ‘er’) over their symbols for whole numbers. For example:

1– 3

1– 5

Apart from some special fractions, all numerators were of the value 1, so to write a fraction like 3--5- , they would write 2. What do these symbols represent? (a)

(c)

1– 6

1 — 10

1– 5

(b)

(d)

+

1 — 100

1– 5

+

1– 5

or

1– 2

+

1 — 10

Chapter 3 Fractions

107

3. Perform the following calculations. Give your answer in Egyptian hieroglyphics and in Hindu–Arabic numerals. (a) Add these 2 numbers. (b) Subtract the second number from the first.

4. Write 4 of your own questions using Egyptian hieroglyphics. Swap these with a partner. Calculate the answers to your partner’s set of questions, showing clearly how you were able to work them out.

Multiplying fractions To multiply fractions, multiply the numerators and multiply the denominators.

WORKED Example 19 Find

1 --5

× 3--4- .

THINK

WRITE

1

Write the question.

2

Write as one fraction.

3

Multiply the numerators and the denominators and simplify your answer if appropriate.

1 --5

×

3 --4

1×3 = -----------5×4 3 = ----20

Fractions can be simplified by dividing both numerator and denominator by the same number before being multiplied. This is easier than working with large numbers. Dividing both numerator and denominator by the same number is often called cancelling.

WORKED Example 20 Find

2 --3

×

9 ------ . 10

THINK 1

Write the question.

2

Write the 2 fractions as 1 keeping the multiplication signs. Divide numerator and denominator by the same numbers or cancel. Multiply to find the answer.

3 4

WRITE 2 --3

×

9 -----10

1 2 × 93 = 1--------------- 5 3 × 10

=

3 --5

108

Maths Quest 7 for Victoria

WORKED Example 21 Find

1 --3

× 36.

THINK

WRITE

1

Write the question.

2 3

Express the whole number as a fraction and cancel if possible. Multiply the numerators and denominators.

4

Simplify your answer.

1 --3

× 36

1 36 12 = 1 --- × -----3 1 =

12 -----1

= 12

Mixed numbers must be changed to improper fractions before multiplying.

WORKED Example 22

Find 1 1--2- × 3 3--5- . THINK

WRITE

1

Write the question.

2 3

Change mixed numbers to improper fractions and cancel if possible. Multiply numerators and denominators.

4

Simplify your answer.

1 1--2- × 3 3--53 18 9 = 1--- × -----2 5 =

27 -----5

= 5 2--5-

Using the word ‘of ’ Sometimes the word of can mean the same as ‘multiply’. Antonio might take 1--3- of an hour (60 minutes) to cook his pizzas. A third of an hour or 1--3- of 60 minutes is the same ------ = 20 minutes. So it takes Antonio 20 minutes to cook a pizza. as 1--3- × 60 1 A customer wants to buy 1--4- of the 8 remaining pizzas for the family’s evening meal. A quarter of 8 is the same as 1--4- × 8--1- = 2. So the customer buys 2 pizzas.

WORKED Example 23 Find

2 --3

of 1--4- .

THINK

WRITE

1

Write the question.

2 --3

2

Change ‘of’ to ‘×’ and cancel if appropriate.

21 1 = --- × --- 2 3 4

3

Perform the multiplication.

=

of

1 --6

1 --4

109

Chapter 3 Fractions

WORKED Example 24 Mum put a 2 litre carton of flavoured milk in the refrigerator. When the children came home from school, Joanna drank one quarter of the milk, Kieren drank one half of it and Daisy drank one sixth of it. What fraction of the milk, in litres, did each person drink? THINK 1

WRITE

Write the fraction of milk that Joanna drank and simplify the answer.

Joanna:

1 --4

of 2 =

1 --4

1 21 = 2--- × --4 1 =

2

3

Write the fraction of milk that Kieren drank and simplify the answer.

Kieren:

Write the fraction of milk that Daisy drank and simplify the answer.

Daisy:

1 --2 1 --2

1 --2

of 2 =

1 --6

1 21 = 1--- × --2 1 =1 of 2 = 1--6- × 2

×2

1 21 = 3--- × --6 1 =

4

×2

Write a sentence giving the answer in litres.

1 --3

Joanna drank a half a litre, Kieren drank 1 litre and Daisy drank a third of a litre.

remember remember 1. 2. 3. 4. 5.

To multiply fractions, multiply the numerators and multiply the denominators. Fractions can be simplified (or cancelled) before being multiplied. A whole number can be written as a fraction with a denominator of 1. Change mixed numbers to improper fractions before multiplying. In a problem involving fractions, ‘of’ can be replaced with ‘multiply’, then the fractions can be multiplied in the usual way.

3E WORKED

Example

19

Multiplying fractions

1 Find: ×

a

1 --2

e

11 -----13

i

5 --8

×

m

7 --8

×

b

1 --3

×

2 --3

c

2 --5

f

6 --7

×

9 -----11

g

11 -----12

1 --3

j

2 --3

×

2 --3

k

5 --6

×

7 --8

n

2 --3

×

4 -----13

o

3 --5

×

1 --4

×

1 --2

× ×

d

5 --9

×

4 --3

h

8 --9

×

5 --9

11 -----12

l

5 -----11

3 -----10

p

4 --7

3 --5 11 -----12

× ×

5 -----12 2 --3

Multiplying fractions

Multiplying fractions

110 WORKED

Example

20

Maths Quest 7 for Victoria

2 Find: a

4 --8

×

3 --9

b

2 --5

×

d

3 --8

×

4 -----15

e

10 -----11

×

g

18 -----32

×

64 -----72

h

48 -----56

×

j

8 -----49

×

14 -----16

k

4 --9

m

42 -----54

×

12 -----49

n

5 -----25

× ×

1 --2

×

c

3 --7

22 -----25

f

24 -----27

×

9 --8

24 -----60

i

15 -----27

×

36 -----45

l

6 -----12

×

8 -----10

o

18 -----24

×

12 -----36

6 --8 12 -----48

5 --6

3 Mark’s uncle gave him a family size block of chocolate. He divided it into thirds to share with his 2 sisters. His friends Tom and Nick then arrived to visit, and Mark shared his portion of the chocolate with them. a Draw the block of chocolate, and shade Mark’s third of the block. Then draw lines to represent Tom’s share. b What fraction of the block of chocolate did Nick receive? WORKED

Example

21 WORKED

Example

22

4 Find: b

5 -----14

×7

c

15 -----22

b

3 --4

× 1 5--8-

c

1 1--2- × 1 1--2-

e 2 1--5- × 3 1--2-

f

9 - × 10 ----10

× 3 2--3-

j

4 1--3- ×

a

2 --3

×3

× 11

d

5 -----16

×4

5 Find: a 1 1--4- ×

i

9 -----10

2 --3

2 --3

3 --5

m 5 × 3 1--2-

n 8 × 3 1--4-

q 5 1--5- × 6 2--3-

r

d 2 5--8- × 1 3--4-

g 2 1--4- × 2 1--2-

h

4 --5

× 1 6--7-

k 1 1--2- × 12

l

2 3--4- × 6

1 o 4 3--5- × 2 ----10

p 3 5--6- × 9 5--7-

3 3--4- × 2 2--3-

6 Tim was looking for something to do one Saturday afternoon and his Dad suggested he cook something for afternoon tea. Tim found a recipe for peanut butter muffins. The recipe is given below. Ingredients: 1 --4

1 --4

cup sugar,

1 --2

cup peanut butter, 2 eggs,

cup margarine,

1 1--2- cups milk, 2 1--2- cups self-raising flour, 1 --4

teaspoon baking soda

Method: Blend the sugar, margarine and peanut butter. Beat in the eggs and milk. Add the selfraising flour and baking soda. Place the mixture in greased muffin pans and sprinkle with cinnamon sugar. Bake for 15–20 minutes in a 200°C oven. This recipe makes 10 muffins, but Tim wants enough mixture to make 15. He needs to multiply the quantities by 1 1--2- . Write the quantities Tim needs to make 15 muffins.

Chapter 3 Fractions

WORKED

Example

23

7 Find: a

1 --2

b

3 --4

d

9 -----10

2 --3

5 --7

e

8 --9

of

1 --4

g

4 --5

of 20

h

7 --8

j

2 --5

of 45

k

3 --5

m

3 --4

of 6

n

9 -----10

of 15

p

3 --7

of 12

q

1 --2

s

2 --3

of

t

5 --6

of

1 --4

3 --2

c

5 --6

f

5 --7

of 32

i

9 -----10

of 25

l

8 --9

o

1 -----12

of 10

of

1 1--2-

r

3 --4

of

8 2--3-

of

18 -----25

u

7 --8

of 3 1--3-

Example

24

E

of

5 --6

of

7 -----25

Fraction of a number

of 50 of 81 GAME time

of

of

8 A car’s petrol tank holds 48 litres of fuel. a If the tank was full at the start of a trip, what fraction of the tank has been used? b How many litres of petrol have been used? WORKED

111

Fractions 02

F

9 Jamie’s Mum made 6 litres of punch for his birthday party. Emily drank 1--3- of the 1 1 - and Jonathan drank ------ . What fraction of the punch, in litres, punch, Tracy drank ----18 12 did each person drink? 10 Zoe and Sarah play basketball with the Sharp Shooters. The games are played for 40 minutes. Zoe played 4--5- of last week’s game and Sarah played 7--8- . How many minutes of the game did: a Zoe play? b Sarah play? 11 Gustave’s monthly take-home pay is $2400. From this he spends a quarter on his home loan payments, one half on food and drink and one sixth on clothing. One half of the remainder goes into his savings account. How much money does Gustave put into his savings account each month? 12 To make a fruit loaf using an automatic bread maker, the fruit needs to be added after the second rise stage and the top of the bread needs to be glazed two-thirds of the way through the bake stage. The instruction booklet shows you how long each stage of the total cycle takes. Stage Length of time First knead 5 minutes Second knead 20 minutes First rise then punch down 40 minutes Second rise then punch down 25 minutes Third rise 50 minutes Bake 40 minutes a After what length of time should the fruit be added? b Express this as a fraction of the total time in the bread maker. c After what length of time should the top of the bread be glazed?

112

Maths Quest 7 for Victoria

13 a b c d

Draw a quarter of a pie. Shade half of this piece of pie. What fraction of the whole pie is shaded? Complete the mathematical sentence 1--4- × 1--2- = __ to show how the answer was found.

14 a Find how many half pizzas in 1 pizza. b Find how many half pizzas in 2 whole pizzas. c Find how many half pizzas in 5 whole pizzas.

Musical fractions In music, particular symbols tells us how long a note is to be held for. The time value of each note is shown below.

4 beats 2 beats 1 beat (semibreve) (minim) (crotchet)

1– 2

beat 1–4 beat (quaver) (semi-quaver)

More than 1 quaver is shown as More than 1 semi-quaver is shown as

1. What is the total number of beats indicated by these notes? (a) (b) (c)

(d)

When musicians decide they want a note held for a little longer, they put a dot after the note. The dot adds an extra half of the number of beats to the original time value. For example, a dotted minim has a time value of 2+

1 --2

× 2 = 3 beats.

2. What is the time value indicated by a dotted crotchet? 3. What is the time value indicated by a dotted quaver? 4. What is the total number of beats indicated by these notes? (a) (b) 5. Write one note which is equivalent in time value to each of the following. (a) (b) (c) (d) (e)

(f)

(g)

(h)

To make music more organised, time signatures were added to the beginning of a piece of music so that each section or bar was divided up into equal parts. For example, 44 tells us that we need 4 crotchet beats (or notes that will make up that amount) in a bar. The 4 on the top shows us the amount of beats per bar, while the 4 on the bottom shows us that the note values we use are the ones that belong to crotchet time. In 34 time, we need 3 beats to make up a bar and in 24 time we need 2 beats to make up a bar.

Chapter 3 Fractions

113

6. Add one note to complete each of the following bars. (a) 2 (b) 4 4 4 (c) 3 (d) 2

4 (e) 4

4 (f) 3

4

4

(g) 3

(h) 2

4

4

7. Add 2 notes to complete each of the following bars. (a) 4 (b) 3

4

4

(c) 2

(d) 3

4

4

(e) 4

(f) 2

4

4

(g) 3

(h) 3

4

4

8. Put time signatures to each of the following rhythm patterns. (They are crotchet beats.) (a) (b) 9. Write your own rhythm patterns 4 bars long, using a different pattern in each bar for each of the following time signatures. (a)

3 4

(b)

4 4

Dividing fractions The reciprocal of a fraction is found by tipping the fraction upside down. So reciprocal of 5--3- . A whole number can be written as a fraction by putting it over 1 so that reciprocal of 4--1- .

WORKED Example 25

Find the reciprocal of 2--3- . THINK

WRITE

Turn the fraction upside down and write a sentence.

The reciprocal of

2 --3

is 3--2- .

3 --5

is the

1 --4

is the

114

Maths Quest 7 for Victoria

To find the reciprocal of a mixed number, express it as an improper fraction first.

WORKED Example 26

Find the reciprocal of 1 2--3- . THINK

WRITE

1

Write 1 2--3- as an improper fraction.

1 2--3- =

2

Turn it upside down and write a sentence.

The reciprocal of 1 2--3- is 3--5- .

5 --3

Antonio had a half of one of his famous pizzas left. A small child asked him how many quarters there were in his half. Antonio cut the pizza as shown, so that it was clear that one half divided into quarters gave 2 pieces or 2 quarters. 1 --2

÷

1 --4

=2

The answer could also have been found by changing the division sign to a multiplication sign and tipping the second fraction upside down. 1 --2

÷

1 --4

=

1 --2

×

4 --1

=2 Division by a number is the same as multiplication by its reciprocal.

WORKED Example 27 Find

2 --3

÷ 4--9- .

THINK

WRITE

1

Write the question.

2 --3

÷

4 --9

2

Change the division sign to a multiplication sign and tip the second fraction.

=

2 --3

×

3

Cancel if possible.

4

Perform multiplication.

=

3 --2

5

Simplify your answer.

= 1 1--2-

9 --4

To divide mixed numbers, you can first change them to improper fractions.

Chapter 3 Fractions

115

WORKED Example 28

Find 1 1--2- ÷ 3 2--5- . THINK

WRITE

1

Write the question.

1 1--2- ÷ 3 2--5-

2

Change mixed numbers to improper fractions.

=

3 --2

÷

17 -----5

3

Change the ÷ to × and tip the second fraction.

=

3 --2

×

5 -----17

Multiply the numerators and the denominators and simplify your answer if necessary.

=

15 -----34

4

Graphics Calculator tip!

Dividing a fraction by a fraction

With the TI graphics calculator we use the ÷ key to enter a fraction as well as when we need to divide two numbers. Using brackets around each fraction helps to keep the operations clear, particularly with mixed numbers. The calculation for 1 1--2- ÷ 3 2--5- is shown.

remember remember 1. The reciprocal of a fraction is found by tipping the fraction upside down. 2. The reciprocal of a mixed number is found by expressing it as an improper fraction, then tipping it upside down. 3. A whole number can be written as a fraction by putting it over 1. 4. Division by a number is the same as multiplication by its reciprocal. 5. To divide mixed numbers, first change them to improper fractions.

3F WORKED

Example

25

Dividing fractions

1 Find the reciprocals of each of the following. a

3 --4

b

2 --7

e

6 --5

i

1 --5

m 5

c

5 --3

f

9 --2

j

1 -----12

n 20

d

5 -----12

g

5 --3

h

10 -----3

k

3 -----15

l

2 -----22

o 13

p 1

Reciprocals

116 WORKED

Example

26

Maths Quest 7 for Victoria

2 Find the reciprocals of these mixed numbers. a 1 2--3-

b 4 3--7-

c

3 6--7-

g 3 3--8-

h 5 1--3-

f

7 1--2-

d 4 2--5i

e 9 2--7-

10 2--9-

9 6 ----10

j

3 Multiply each of these numbers by its reciprocal. a

5 --7

b

2 --9

3 --5

c

d 3 2--7-

e 2 1--8-

f 3 3--4-

4 Copy and complete the following statement: When any number is multiplied by its . WORKED

Example

Dividing fractions

27

Dividing fractions

WORKED

Example

28

, the answer is always

5 Find: ÷

a

1 --4

d

1 -----12

÷

g

10 -----3

÷

j

4 --9

÷

m

4 --3

2 --3

÷

c

9 -----10

÷

1 --2

5 --6

f

7 -----12

÷

9 -----11

20 -----21

i

8 -----15

÷

16 -----15

4 --7

÷ 10

l

5 --8

÷ 20

15 -----11

÷5

o

7 --9

÷ 30

b 9 ÷ 4 1--2-

c

9 --4

÷ 3 1--2-

e 5 ÷ 2 3--4-

f

7 4--9- ÷

h 1 1--5- ÷ 2 1--3-

i

5 6--7- ÷ 2 3--4-

9 - ÷ 4 k 10 ----10

l

4 1--2- ÷ 3 6--7-

n 4 2--3- ÷

o 3 5--8- ÷ 1 3--4-

b

8 --9

2 --3

e

4 -----11

15 -----21

h

2 --7

÷

4 --9

k

÷ 12

n

7 --6

÷

6 Find: a 1 1--4- ÷

3 --4

÷ 3 1--3-

d

10 -----3

g

1 --8

j

1 1--2- ÷

÷ 2 1--21 --2

m 3 2--7- ÷ 2 1--7-

7 --6

7 Felicity is dividing 1 1--2- kilograms of mince steak into 8 hamburger portions. How many kilograms of mince will be in each hamburger portion? 8 Ned spends his school holidays helping his father with the shearing. a It takes Ned 1--4- hour to shear one sheep. How many sheep will Ned shear in 5 1--2- hours? 1 - of b Ned’s father, Wesley, shears a sheep every ----12 an hour. How many sheep would Wesley shear if he worked continuously for 8 hours?

3.2

9 Michelle can run one lap of a cross-country course in 1--5- hour. a How many laps could she complete in 1 1--3- hours? (Assume she can keep up the same pace.) b How many minutes does it take Michelle to run one lap?

1 --3

Chapter 3 Fractions

This part part of settled in 1804. 5 8

M

÷ 163

A

=

was first Answer these fraction questions to discover the puzzle answer code.

÷ 31

5 6

1

2 7

=

÷ 37

S How many are there in

D

5

1 2

L

1

3 4

N

= 1 4 2 3

s

I

?

=

5÷

1 10 ÷ 7

5 8

2 5

V

3

= 3 4

3

÷2

=

÷ 87

E

1

22

÷ 32

Divide 4 41 by 1 21

D

= 3 7

1 2

÷ 1 47 =

= A

3 4

Divide by 85

N

=

E

1

117

÷2 2

N

=

1 2

1

1 5

1

2

2 3

44

5

÷ 1 52

5 6

8

34

2

3 14

3

2

3

33

1

4 7

3

1 2

118

Maths Quest 7 for Victoria

2 Calculate each of the following, expressing your answer in simplest form. −

4 -----15

2

5 --6

+

6 3 5--9- ×

15 -----20

7

2 --7

of

1

12 -----15

3 -----10 21 -----40

3 3 1--4- + 2 2--5÷

5 --9

8

4 11 5--8- − 3 1--29 8 ÷ 1 3--4-

2 --3

×

4 -----18

10 4 2--5- ÷

2 -----11

5

12 -----20

Mixed operations with fractions If more than one operation is involved in a question with fractions, the same set order is used as with whole numbers. Therefore, the order of operations is: 1. brackets 2. multiplication and division (from left to right) 3. addition and subtraction (from left to right)

WORKED Example 29 Find

6 --7

× 1 5--6- ÷ 1--4- .

THINK

WRITE

1

Write the question.

6 --7

× 1 5--6- ÷

2

Change the mixed number to an improper fraction.

=

6 --7

3

Change division to multiplication and tip the next fraction (times and tip). Cancel where appropriate. Multiply and simplify.

4

×

11 -----6

1 --4

÷

1 --4

1 6 11 4 = --- × 1------ × --7 6 1

=

44 -----7

= 6 2--7-

On a TI graphics calculator, the calculation in worked example 29 can be entered as (6 ÷ 7) × (1 + 5 ÷ 6) ÷ (1 ÷ 4), then pressing MATH selecting 1:Frac and pressing ENTER .

119

Chapter 3 Fractions

remember remember The order of operations is the same for fractions as it is for whole numbers: 1. brackets 2. multiplication and division (left to right) 3. addition and subtraction (left to right).

Mixed operations with fractions

3G

Mixed operations

1 Find the following.

WORKED

Example

a

5 --8

+

5 --6

b

d

4 --5

−

3 --8

e 1 3--5- + 9 4--7-

g

1 4--5-

×

9 --7

÷

3 --4

f

6 1--3- ÷ 4 1--2-

i

5 1--4-

b 1 3--4- × 5 ÷ 3 1--2-

c

1 9 - + 2 --- − 4 ----4 10

e 2 5--6- × 2 − 3 1--2-

f

1 --2

i

11 -----12

b (5 − 3 6--7- ) × 4 1--3-

c

(1 1--2- + 3--4- ) × (5 − 3 3--4- )

÷ (4 1--2- + 5--6- )

f

1 -----11

i

7 -) 6 ÷ (3 2--3- + 2 ----12

5 --2

−

21 -----32

×

c

h

7 --8

1 7--8-

9 -----10

−

Four operations with fractions

2 1--5-

2 Find the following. 3 --5

a

29

× 2 1--3- ÷

1 --2

d 5 1--3- + 7 3--8- − 8 1--24 --5

g

× 10 + 2 3--4-

h

1 --3

+

2 --9

×

3 --4

+ −

2 --3 1 --2

× ÷

3 --5

1 --5

Mixed operations

3 --4

3 Calculate these, remembering to remove the brackets first. a ( 5--4- + 2--3- ) ×

1 --2

d 6 − (7 5--6- ÷ 2) 1 1 - ) − (7 --- × g ( 2--3- × 4 ----12 2

h 4 × (2 − 1 3--7- )

1 ------ ) 10

× (8 2--9- − 5 1--6- )

QUEST

GE

S

3 --2

( 2--3- − 1--6- ) ÷ (3 1--8- − 2 5--6- )

EN

M ATH

j

e

CH

A

LL

1 Without using a calculator find three quarters of four ninths of 78. 1 1 1 --- + --- + --2 3 4 -. 2 Without using a calculator find the answer to -----------------1 1 --- + --4 9

3 The fraction

1 --4

is a third of what number?

4 According to the will, the $120 000 estate is to be left to three children. Sam is to receive 2--5- of the money, Georgia is to receive 1--3- of the money and Olivia is to receive the remainder of the money. How much money is Olivia to receive? 5 If a half, a quarter and an eighth of a number are added together, the total is 63. What is the number?

120

Maths Quest 7 for Victoria

You want more pizza? Do you remember the question posed at the start of this chapter? Martin eats 2 slices of a pepperoni pizza which has been cut into 6 equal pieces while Olivia eats 3 pieces of the sausage pizza which has been cut into 8 equal pieces. Who has eaten more?

1. 2. 3. 4.

Try this question again, showing your calculations. How many pieces of each pizza is left? What fraction of each pizza is left? What is the total amount of pizza left?

Daniella takes a slice of pepperoni pizza and Christopher takes a slice of sausage pizza. However, Christopher complains that his piece is smaller than Daniella’s. 5. What is the difference in size between the larger and the smaller slice? (Write your answer as a fraction of a whole pizza.) Daniella continues to eat the pepperoni pizza and Christopher continues to eat the sausage pizza. 6. How many slices of pizza should Daniella and Christopher each have so as to eat the same amount? 7. What fraction of each pizza is now left over?

Powers and square roots of fractions Fractions and mixed numbers can be squared (index or power of 2) or cubed (index or power of 3) by using index notation just as with whole numbers. For example,

2 2 --- 3

=

2 --3

×

2 --3

=

22 ----32

=

4 --9

WORKED Example 30

Find: a

1--- 2 4

b

3 2 --- 5

c (1 1--3- )2.

THINK

WRITE

a

a

1

Write the question.

2

Square the numerator and the denominator. Simplify.

3

2 1 --- 4

=

12 ----42

=

1 -----16

Chapter 3 Fractions

THINK

WRITE

b

b

c

3 2 --- 5

1

Write the question.

2

Cube the numerator and the denominator.

=

3

Simplify.

=

1

Write the question.

2

Change the mixed number to an improper fraction. Square both the numerator and the denominator.

=

2 4 --- 3

=

42 ----32 16 -----9

Change back to a mixed number.

= 1 7--9-

3

4

23 ----53

2×2×2 = --------------------5×5×5 8 --------125

c (1 1--3- )2

=

It is also possible to find the square roots of fractions and mixed numbers.

WORKED Example 31

Find: a

9 -----16

b

6 1--4- .

THINK

WRITE

a

a

1

Write the question.

2

Write the numerator and denominator as separate square roots. Calculate the square root of both the numerator and the denominator.

3

b

1

Write the question.

2

Change the mixed number to an improper fraction. Write the numerator and the denominator as separate square roots. Calculate the square root of both the numerator and the denominator. Change to a mixed number.

3

4 5

9 -----16

=

9 ---------16

=

3 --4

6 1--4-

b =

25 -----4

=

25 ---------4

=

5 --2

= 2 1--2-

121

122

Maths Quest 7 for Victoria

Graphics Calculator tip!

Powers and square roots

As with whole numbers, we can use the TI graphics calculator to find different powers and square roots of fractions. To enter the index or power we use the ^ key or if we are squaring a number we can also use the x2 key. Remember to press the appropriate keys to obtain your answer as a fraction. The screen at right shows the calculations for worked example 30. To obtain the square root of a fraction, first enter the square root sign by pressing 2nd [ ] (that is, press 2nd then x2 ) then follow with the fraction. The screen obtained for worked example 31 is shown opposite.

remember remember 1. To square a fraction, square both numerator and denominator (base is raised to an index of 2). 2. To cube a fraction, cube both numerator and denominator (base is raised to an index of 3). 3. To find the square root of a fraction, take the square root of both numerator and denominator. 4. To find the square root of a mixed number, change the mixed number to an improper fraction, then find the square root of both the numerator and the denominator.

3H WORKED

Example

30a

WORKED

Example

30b Squaring and cubing fractions

Powers and square roots of fractions

1 Find: a

1--- 2 2

b

2 1 --- 3

c

2 2 --- 5

d

5--- 2 6

e

2 3 --- 4

f

7 2 ---- 10

g

5 2 ---- 12

h

3 2 ---- 11

i

2 3 --- 8

2 Find: a

1--- 3 4

b

3 1 --- 2

c

3 2 --- 5

d

3 3 ---- 10

e

3 2 --- 3

f

3 1 --- 6

123

Chapter 3 Fractions

WORKED

Example

30c

WORKED

Example

31

3 Find: a (1 1--3- )2

b (2 1--4- )2

c

(1 1--2- )2

1 2 -) d (1 ----10

e (2 1--5- )2

f

(3 2--3- )2

4 Find: a

16 -----25

b

49 --------100

c

9 -----64

d

25 -----36

e

2 1--4-

f

1 7--9-

g

9 1 ----16

h

-----2 14 25

i

-----1 15 49

Square root of a fraction

2 1 --- 2

+

2 3 --- 4

b

2 2 --- 3

d

2 4 --- 7

÷

2 3 --- 7

e

4 --9

−

+

2 1 --- 2

2 2 --- 3

−

1 --2

c

2 3 --- 5

×

2 5 --- 9

f

16 -----25

×

2 1 --- 2

QUEST

GE

S

a

EN

MAT H

5 Calculate:

CH

AL

1 Find the missing power for this calculation:

3 --- 5

=

2187 ---------------78 125

.

L 2 A length of rope is cut in half and one half is used. Then one third of the other half is cut off and used. If the remaining piece of rope is 10 cm long, how long was the original rope? 3 Find the answer to 3

( 1--2- ) ------------------------. 1 2 -) ( 1 + ----------1+3 4 Look for a pattern in this list of fractions and write down the next two in the sequence. 2 --3

→

9 --5

→

16 -----14

→

23 -----50

→

30 -----53

3.3 −

1 -----10

124

Maths Quest 7 for Victoria

summary 1 2 3 4 5 6

7 8

9

10

11

12 13

Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows. The is the top part of the fraction. The is the bottom part of the fraction. Different fractions can be compared or ordered by writing each one with the denominator. order is from smallest to largest. order is from largest to smallest. are found by multiplying or dividing both the numerator and the denominator by the same number as long as the number is not equal to zero. To fractions, find the largest number that is a factor of the numerator and the denominator and divide both by that number. fractions can be changed to mixed numbers by dividing the numerator by the denominator. The becomes the numerator and the denominator remains the same. Mixed can be changed to improper fractions by multiplying the whole number and the denominator then adding the numerator. The result becomes the numerator and the denominator remains the same. Fractions must have the same denominator to be added or subtracted. This is done by finding a denominator. Once the denominators are the same, the numerators can be added or subtracted. To multiply fractions, simply the numerators and multiply the denominators. It is often easier to or simplify numerators and denominators before multiplying. Remember to change mixed numbers to improper fractions first. To find the of a number, swap the numerator and denominator. To divide fractions, change the ÷ to × and the second fraction. Remember to change numbers to improper fractions first.

WORD denominator ascending mixed tip multiply improper

LIST common same equivalent fractions cancel numerator simplify

descending remainder numbers reciprocal

Chapter 3 Fractions

125

CHAPTER review 1 What fraction of the eggs in this carton are brown?

3A

2 Write 3 equivalent fractions for the following. a

3 --8

b

4 --9

5 -----11

c

d

3A

6 --7

3 Fill the gaps to make equivalent fractions. a

1 --2

=

--4

d

2 --5

=

-----10

= =

3 --16 ------

b

1 --3

=

3 ---

=

-----15

c

3 --4

e

1 --4

=

5 ---

=

--------100

f

3 -----10

= =

=

-----28

--------100

=

9 ---

3A 300 ---------

4 Simplify these fractions. a

42 -----49

b

81 --------108

c

18 -----20

d

21 -----28

e

36 -----72

f

50 --------150

g

75 --------125

h

2 -----80

-----4 15 20

20 d 1 -------100

3B

5 Simplify the following mixed numbers. a

b 3 6--9-

8 6 ----10

c

3B

6 What fraction of the months of the year begin with the letter J? Express your answer in simplest form.

3B

7 Convert these improper fractions to mixed numbers.

3C

a

16 -----3

b

21 -----5

c

12 -----7

d

11 -----2

e

5 --4

f

80 -----7

g

55 -----9

h

72 -----10

8 Express these mixed numbers as improper fractions. a

2 3--4-

b 9 7--8-

e

3 2--3-

f

3 5--7-

c

9 9 ----10

-----g 3 11 12

3C

d 5 5--6h 8 1--2-

9 Find: a

5 --6

+

1 --8

b

d

5 --4

+

11 -----2

e 1 7--8- + 2 5--6-

g

6 1--3- − 2 5--9-

7 --9

−

2 --5

1 1--5- + 3 1--3- −

12 -----5

4 --3

f

10 3--7- − 8 1--2-

3D

h 5 1--2- − 2 5--6-

10 Find: a

−

c

13 -----15

b 4 7--9- − 1 5--6- + 3 1--3-

3D

126 3D

Maths Quest 7 for Victoria

11 Tony caught 3 fish with masses: 7 1 4 - kg, 2 --- kg and 2 --- kg. 2 ----10 2 5

a Which fish is the heaviest? b What is the combined mass of the fish? c What is the difference in mass between the heaviest and the lightest fish?

3E 3F

3F 3G 3H

3H

12 Find: ×

a

4 --7

c

25 -----36

e

5 --9

×

b

2 --5

d

56 -----81

2 4 2--5- × 3 ----11

f

10 2--7- × 6 2--9-

g

4 --5

of 55

h

2 -----11

of 121

i

7 --9

of

j

1 -----12

of 200

×

18 -----20

18 -----84

×

3 --5 18 -----64

13 Find: a

3 --5

×

d

4 --9

× 2 1--4- ×

5 --6

×

2 --7

b

3 --4

×

16 -----21

×

7 -----20

5 3 1 - × 1 --- × 2 --e 3 ----12 5 2

3 --5

c

5 -----11

×

f

6 -----17

× 1 1--4- × 3 2--5-

c

7 --9

f

6 2--3- ÷ 4 1--6-

c

( 2--3- − 1--6- ) × ( 3--4- + 5--8- )

f

7 − 2 1--4- ÷ 1 1--2-

3 -----10

1 × 1 ----10

14 Write the reciprocals of: a

5 --2

b

2 --7

c

1 --9

e

124

f

1 --4

g 5 3--4-

h 8 1--2-

i

7 2--5-

j

2 4--9-

k 3 5--8-

l

d 12

4 1--5-

15 Find: a

4 --5

÷

d

3 --7

÷ 10

8 -----15

b

9 -----10

÷

27 -----40

e 1 1--2- ÷ 4 4--5-

÷4

16 Calculate: a

( 4--9- + 5--6- ) ÷ 3

d

3 --4

÷

9 -----10

+

5 --7

b ×

5 -----14

3 2 - + --- ) × ( ----10 5

e 2 + 4 1--2- ÷ 2 1--4-

14 -----15

17 Calculate: a

4--- 2 5

b

9 2 ---- 10

c

2 3 --- 4

d

7--- 2 8

e

3 1 --- 3

f

3 5 --- 6

g

(1 1--2- )2

h (2 1--4- )2

i

(1 3--5- )2

test yourself

18 Calculate: a

81 --------100

b

25 -----49

c

2 7--9-

d

9 2 ----36

CHAPTER

3E

3

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