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8/14/2019 BUKSIS-2.2 REVISI AKHIR

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Estimating with Fractions Suppose you need to go somewhere or to buy

something. Do you need a tool to calculate the time

needed to go there? Do you need to know the exact

time? If not, what will you do?

As in integers, the computational results of fractionscan be estimated by rounding off or using

compatible numbers.

a. Rounding to 0,21 , or 1

22 ..22

at are you go ng to learn? To est ima te

f r a c t i o n s To a d d f ra c t i o n s To sub tra c t

fra c tions . To multiply fra c tions To d ivide fra c tions.

Key Terms:Ad d , subt rac t , multip ly,divide , est ima te,prod uc t, quo tient , likefra c tion, un likefra c tion, p rop er,imp rop er fra c tion

We can estimate the result of addition and

subtraction by rounding each fraction to 0,21 , or 1.

If I have87 of a bar and you give me

94 more of a

bar, estimate how long I have altogether.

EXAMPLE 1

Solution:

Since87

is close to 1 and94

is close to21

, then87

+94

approximates 1 +21 = 1

21 . So, I have approximately

121 bars.

Problem 1 Find two fractions which their sum is close but not

Mathematics for Junior High School - Year 7 / 61


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equal to 1.

4

1 3

2 8

7 8

5

21

101

43

105

83

81

125

109

127

103

31

107

b. Rounding to the nearest integer

If an addition, subtraction, multiplication, or

division involves a mixed number, we can estimate a

logical result by rounding each operand to the

nearest integer.

Estimate the result of 861

- 421

EXAMPLE 2

Solution :

861

can be rounded to 8 and 421

can be rounded to 5.

Therefore, 8 5 = 3 approximates 861

- 421

Write a story problem based on the expression of

Example 2.

c. Using compatible numbers

To estimate the quotient of mixed numbers,

compatible numbers are frequently used, especially

Problem 2

62 / Students Book Fractionss


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for mental calculation.

Estimate the result of 434

1 : 58

7

Well, it willbe easier if Idivide 42 by

EXAMPLE 3

Solution:

4341

: 587

can be estimated by using compatible

numbers. Round 4341 to 42 and 5

87

to 6.

Then 42 : 6 = 7 approximates 4341 : 5

87 .

Adding and Subtracting Fractions

Erna and Wati bought sliced bread, which

consists of 8 equal pieces. While they sat down in

the backyard, Erna ate81 part of the bread, and

Wati ate 83 of it. How many parts of the bread did

they eat?

To answer this question, let us carry out the Mini

Lab on the next page.

In this Mini Lab, we know that

81 +

83 =

84 or

21 .

Therefore, Wati and Erna have eaten21 of the

bread. Do the activities in the Mini Lab to find

61 + 6

3 and 51 + 5

3 , and so on.



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Based on the examples, we have the procedure to

add like fractions.

Apply the procedure to solve the following

example.

Add 53 to 5

4 .

53

+ 54

= 543 +

Work in PairsMaterials and Tools: millimeter blocks, ruler,two colors)

Add

drawing pencils (at least

81 to

83 .

Draw bar on the millimeter blocks as shown in theof this bar illustrates the one-eight.

following figure. Each part

Color one part of the bar to represent81 . Using di

parts of the bar to illustrate

fferent color, shade three

83 .

Discuss the following questions.

a. How many parts of the bar that are colored?b. What fraction represents the number of uncoloredc. If you color the two other parts of the bar, what

number of uncolored parts?

parts?fraction illustrates the

Mini-Lab

Adding Like FractionsTo add like fractions, add the numerators, whilethe denominators remain unchanged



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Subtracting Like Fractions

Tini found 85 parts of a cake on the table. Then she

ate81 part of the cake. What part of the cake

remained?

815

81

85 = =

84

or21

Therefore, the cake that was left was 21 .

Based on the example, write the procedure to

subtract like fractions, then use the procedure to

solve the problem below.

There were three-fourths of a watermelon on the

table. Then you ate one-fourth. How many parts of

the watermelon remained?

53 +

54 = 5

7

= 521

521

64748

Subtracting Like Fractions.....

Problem 3



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(Relation with the Real World)

Mr. Salmeto had65 drum of kerosene. His

neighbour bought this kerosene so that the amountof kerosene in the drum was now

63 drum. How

many drums of kerosene did Mr. Salmetos

neighbour buy?

You could do the problem in the following way.

65 - ..... =

63

Solution:

65 -

62 =

63 , so we know that Mr. Salmetos

neighbour had bought62 or

31 drum of kerosene.

Adding and Subtracting Unlike

Fractions

Ani read a novel. Two days ago, she finished41

of the book. Today she continued reading3

2of

the book. What parts of the novel has Ani read?

To answer this question, you need to add unlike

fractions and may use fraction bars.

How many parts of the novel has Ani read?

What numbershould be added

to63 to get

65 ?



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Add 41 to

32 .

41 +

32 =

1211 or 4

1 +32 =

123 +

128 =

1283 + =

1211

Thus, Ani has read1211 parts of the book.

Solve31

21 using fraction bars.

EXAMPLE 6

61

623

62

63

31

21 ===

Based on Example 6 and Example 7, it is clear

that to add or subtract unlike fractions, you have

to convert the fractions to like fractions first by

using the Least Common Multiple ( LC M ) of the

denominators.

ba r fo r 63 o r 2

1

ba r fo r 62

o r 31

.

Sub trac t: 62

63 .

Find the b ar forthe sum .

ba r fo r 41

. ba r fo r 3

2 .

EXAMPLE 7



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Find the sum of72

41 + .

Since the LCM of 4 and 7 is 28, then

28x4x1

41 7 ==

7

7

28x7x2

72 8==

4

4

and41 + =

72

28287 8

+ =28

15

Adding Mixed Numbers

Before you proceed to adding mixed numbers, do

the following activities.

One of the methods to add mixed numbers is to

calculate the whole parts and the fraction parts

separately. Sometimes the sum of the fraction

parts is an improper fraction. If so, first convertthe improper fraction to a mixed number.

EXAMPLE 8

Repeat step 1 for some other pairs of ropes. Check each of the expression by measuring the whole

lengths for each pair of ropes.

Take a pair of ropes. Estimate the sum of the length of thetwo ropes, and then add them. Write down an expression forthis addition.

Take some pieces of rope4

12 cm,2

11 cm,10

14 cm,8

13 cm, and

435 cm in length.

Put two ropes in such a way that the end of the first ropetouches the end of the second rope.



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Find the result of 213

4315 + .

4315 = 4

315

213 = 22

213

= 4

23

= 4518

= 18 + 411

= 4119

Subtracting Mixed Numbers

Sometimes, you need to change the form of

fractions before you subtract them.

Calculate 316 - 2

14

316 - 2

14 = 626 - 6

34

= 685 - 6

34

= 651

Thus, 316 - 2

14 = 651 .

EXAMPLE 9

The LC M is 4. Exp ress it a slike frac tions.

A d d th e w h o le p a r tsa nd the frac tion pa rts.

Co nvert the fra c tion.

41

141

44

45 =+=

Add the w hole p ar ts.

EXAMPLE 10

Expre ss it a s like f rac tio ns

Conver t the f rac t ion p ar t .

62 = 5 +6

621 =

685 .

Sub tra c t the who le p a rtsen the f rac t ion p ar tsth



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Multiplying and Dividing Fractions

Before operating fractions, do the following activity.

a. Multiplying a Fraction by a Fraction

You can use a model (fraction bars) to multiply a

fraction by a fraction.

Mr. Arif has a large garden. He wants to plant

one-half of his garden with medicinal herbs.

Wild gingers will be planted on one-third of it .

What part of the garden will be planted wildgingers?

Solution:

The shadowed part is 61 of the whole bar. This

part represents the part of the garden that will be

Activities1. Take a piece of paper and cut the paper into two equal

parts. What fraction represents each part?2. Take one of the parts. Then, cut it i nto two equal parts.

What fraction represents each part?3. Take one of the parts. What fraction represents each part?

EXAMPLE 11

The g arde n p lant ed me d ic inal herbs is21 o f

Co lour one-half of the

for me d ic inal herbs into

Sha d e

the whole ga rde n ;b a r.

Div ide the g arden

three eq ual pa rts.

p a r t .31 of the co loured



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Do you still remember the meaning of 4 x 2? This

meaning also applies in multiplying whole

numbers and a fraction.

Consider the following illustration:

When multiplying a whole number with a

fraction, you may convert the whole number to a

fraction having 1 as the denominator, then

multiply the fractions.

Relation with Real World

Suppose you have 16 marbles. Three-eighths of

the whole marbles are green. How many marbles

are green?

To answer this question, you have to calculate 83

x 16.

83 x 16 = 8

3 x 116

= 1x816x3

1

2

= 1123

=16 = 6

322

38

32

32

32

32

32

4

=

=

+++=

EXAMPLE 14

Exp ress 16 a s 1

16 .

u lt ip ly the num erato ran d th e d en o m in a to r.Simplify.

M

Dith

v ide the numerato r ande d enom inato r by 8 .



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c. Multiplying Mixed Numbers

Rini has a big photo album. Eight and one third

pages of the album are still empty. Rini wants tofill one-half of the empty pages with her favorite

artists photographs. How many pages of the

album will be filled with artists photographs?

To answer this question, you need to calculate 21

of 318 or

21 x 3

18 .

Before multiplying a mixed number, you have to

express the mixed number as an improper

fraction.

318 =

325

31)38(

=+

.

Then multiply the fractions.

625

325x2

1 = = 614

Therefore, as many as 614 pages of the photo

album will be filled with the artists photographs.

You may also illustrate the above case by using

the following model.

318 p a g e s



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21 of 8 is 4, and 2

1 of 31 is 6

1 .

Hence, 3182

1 is 4 + 61 or 6

14 .

You may also use the model of an area to multiply

mixed numbers.

Draw a quadrilateral with a length of 211 cm and a

width of 412 cm as shown in the following figure.

4122

11 = 1 + 1 + 41 + 2

1 + 21 + 8

1 = 833 .

The area of the quadrilateral is 833 cm 2.

Find the result of 4133

22 .

21 o f

318 p a g e s

EXAMPLE 15



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Solution:

4133

22 = 413

38

=1

2

413

38

= 13132

= 3283

26 =

Solving Problems Related to the Real World

Suppose you can read one-page full of text in41

hours. At least, how many hours do you need to

read 421

pages?

Solution:

To read 421

pages of the text needs 421

41 hours =

89

hours. So, you need at least 181 hours to read 4

21

pages of the text.

d. Dividing Fractions

To understand the meaning of division of

fractions, follow the situations below.

Direction

You could determinethe result of

Simplify.

Multiply.

Simp lify t hthe G CF of

em b y using8 an d 4

Write d oa s a fra c

w n eac h m ixed n u m b ertion.

2128

by applying thefollowing method.

8 2 = 168 = 416 + 4 = 20

Thus2128 = 20

EXAMPLE 16



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1. You have 6 candies that will be given to your

friends. Each of your friends will get 2 candies.

How many people will get candies?

Answer: 3

Written as 6 : 2 = 3.

2. Suppose the 6 candies are given to your

friends. Each of them will get 1 candy. How

many people will get the candies?

Answer: 6

Written as 6 : 1 = 6.

3. Suppose the 6 candies are divided so that each of

your friends gets 21 candy. How many people will

get the candies?

Answer: 12

Written as 6 : 21 = 12.

Notice that:

6 : 21 = 12 relates to 6 1

2 = 12.

The numbers 21 and 2 have a specific relation, that

is, the product is 1. 12x21 = .

One half or 21 is called the multiplicative inverse of

2.



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Notice that any two numbers whose product is 1

are called the multiplicative inverse to each other.

Find other numbers that are the multiplicative

inverses to each other.

e. Dividing Mixed Numbers

To divide mixed numbers, firstly, express every

mixed number as a fraction and then multiply

them.

Calculate533:

412 .

Solution:

8524 51

185

49

518:

49

533:

412

2

1

=

=

=

=

Evaluate 3:3210 .

Solution:

953

932

3313231332

13:

3323:

3210

=

=

=

=

=

EXAMPLE 17

Chang e the m ixed numbe rs.

Divide the nume rato r and thede nominator by the GCF of 9 and 18,tha t is 9.

Find the p rod uc t.

EXAMPLE 18

Conver t the mixed numbe r.

Multiply b y the multiplic ative inverse o f 3.

Multiply thedenomina

nume rato rs a nd t hetors.

Write d ow n t

Exp ress a s a

h e p ro d u c t .

mixed number.



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1. Approximate the results of the following

operations.

a. 71 +

83

b. 9111

- 397

c. 1321

x31

d. 7 53

: 1 21

2. Estimate the product of51 and 248.

a. Use the result to approximate the product of53

and 248.

b. Explain how to estimate the result of85

x 55.

3. Open-ended . Write two numbers (either

fractions or mixed numbers) and then estimate the

computational result of the two numbers.

4. Which one of the following results is between 6

and 7?

a. 21

x 1421

b. 2 x 31615

c. 5 1211

+ 2524



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a. How many parts of Tinis after-school time are

used to sleep, eat, and study at school?

b. How many parts of Tinis after-school time are

used to do other activities?

10. Find the result of the following additions.

a. 3218 + b. 26

13 + c. 433

518 +

d.1612

8311 + e.

438

1219 +

11. Writing. Describe the method you use to find

the sum of .516

322

543

315 +++

12. Open Question . Write down two mixednumbers so that their sum is a whole number.

13. Explain why the result of addition of two mixed

numbers is not always a mixed number.

14. Cuisine . To make a kind of cake, we need

431 cups of sugar, while another kind of cake

needs 211 cups of sugar. Suppose you have about

3 cups of sugar. Is your sugar enough to make the

two kinds of cake? Explain .

15. Find the result of the following subtractions.

a. 83

343

7

b. 41

1165

2



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c. 534

549 d. 16

1118121 e. 2

111121115

16. Number Pattern . Write down the two

successive numbers in the following pattern.

319 , 6

18 , 7, 655 , 3

24 , ....., ......

17. Research . Determine the fraction representing

the number of female teachers at your school.What is the ratio of the female teachers to the

male teachers that teach at year I? What is the

ratio of the female teachers that teach at year I to

all teachers at your school?

18. Find the product of the following

multiplication.

a. 51 of 2

1 b. 21 of 5

1 c. 81

52

d. 21 of 3

1 e. 43

53

19. Evaluate ab if a =3

1 and b =7

6 .

20. Garden . Suppose two-thirds of a garden will

be planted grass. The remaining will be planted

plants. Three-fourths of the garden that are

planted plants, will be planted flowers. What is

the part of the garden that will be planted

flowers?



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21. Biology . About 107 of a human body consists

of liquid. If the weight of a man is 60 kg, how

many kilograms would the weight of the liquidbe ?

22. Find the results of the following

multiplication.

a.2

172

14 b.10

963

23 c.3

272

16

d. 2182

18

23. Writing . List the objects the area of which

you can find their area by the methods of

multiplication of mixed numbers.

24. Geometry Calculate the area of the

quadrilateral as shown on the left.

25. Application . Tine needs 433 metres of textile

to make a pair of dresses. If Tine wants to make 4

pairs of dresses, at least how many metres oftextile does Tine have to buy?

26. Geometry. Calculate the area of a rectangular

yard with a length of 215 m and a width of 3

24 m.

27. Critical Thinking . Is 2143

22 more or less

than 10? Explain your answer without calculating

the result.

21

4 cm

312 cm



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28. Measurement . How many parts of a rope of41

metre are there on a 2-metre rope?

29. Find the result of the following divisions.

a.83:5 b.

43:15 c.

53:

94 d.

31:12

30. Suppose you cut 3 apples. Each is cut into

eight equal parts. How many parts of apples will

you get?



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