Buksis-2.1 Revisi Akhir

2.1 What are you going to learn? To define

fractions

To simplify fractions To change a mixed number into fractions and its inverse To compare fractions To write a fraction in a decimal form

The Meaning of Fraction In our everyday life, sometimes we have to divide something into some equal parts. For instances: • A slice of bread divided into three equal parts, • A piece of paper cut into two equal parts, • An orange sliced into some equal parts,

To write a fraction in a percent form

All situations above are related to fractions.

Key Terms : •

fraction

•

numerator

•

denumerator

•

decimal

•

percent

Look at the picture below.

Tools and Material : •

paper

•

pencil

• ruler

Firstly, an orange is sliced into two equal parts. Each part is called one over two or one-half or half and it is written as

1 . 2

Then, each of the two equal parts is divided into two

Mathematics for Junior High School – Year 7 / 39

equal parts so that we obtain four equal parts. Each part of four equal parts is called one over four or one-fourth, written as

1 . 4

The numbers

1 1 3 , , and are examples of fraction 2 4 2

numbers, or fractions for short.

A fraction is a number that can be written in the form of a , where a and b are integers and b ≠ 0, and b is not a b

factor of a. The number a is called the numerator, b is called the denominator. Why b cannot be zero?

For the fraction

1 , 1 is the numerator, and 4 is the 4

denominator. For the fraction

3 , 3 is the numerator, and 2 is the 2

denominator.

Simplifying Fractions Look at the shadowed parts of the bars on the left. How many parts are in each bar?

How many parts are

shadowed? Which fractions represent the shadowed parts? Simple Fractions Fractions

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1 2 3 4 , , , and have equal values. Of these 2 4 6 8

fractions,

1 is the simplest form. A fraction is called the 2

simplest form (simple fraction) if the Greatest Common Factor (GCF) of the numerator and the denominator is 1. You can write the simplest form of a fraction by dividing the numerator and the denominator with the GCF of the numerator and the denominator.

EXAMPLE 1

Write

20 in the simplest form. 28

The GCF of 20 and 28 is 4. Divide the numerator and the denominator by 4.

Therefore, the simplest form of fraction

20 28

is 5 .

7

Converting a Mixed Number into a Fraction Two men rode horses. One rode for 1 1 km., and the 2

other covered a distance of 1 1 km. 4

The numbers 1 1 and 3 2

1 are called mixed numbers. 4

A mixed number can also be written as a fraction form. Work out the following Mini Lab.

Mathematics for Junior High School – Year 7 / 41

Mini-Lab Work in pairs. Material and equipment: paper, pencil, and ruler. Draw a model for a mixed number 1 1 in the following steps.

4

Draw a quadrilateral as shown below. Shade the quadrilateral to get 1.

Draw an identical quadrilateral beside the first one. Divide the quadrilateral on the right side into four equal parts to get one-fourth. Shade in one part to show 1 . Thus, we will get the model for 1 1 .

4

4

Divide the model of a quadrilateral into four equal parts (one-fourth).

The shadowed areas on the last picture show the mixed numbers 1 1 .

4

Discuss: b. How many shadowed one-fourth areas are there in the above picture? c. How many unshadowed one-fourth areas are there in the above drawing? d. What fraction has the same value as 1 1 ?

4

Based on the Mini-Lab, you may conclude that a mixed number can be written as a fraction that is called an improper fraction.

You can also use a different way to convert the form of a mixed number.

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Converting an Improper Fraction into a Mixed Number Suppose you have 28 liters of gasoline. You are asked to fill it in 8 containers. Each container will be filled with equal volume of gasoline. How many liters should be filled into each container? Solution:

28 ← Write down the division in the form of 8 fraction 3 8 28 24

Divide 28 by 8

4 −

34 = 31 8 2

Write down the remainder as a fraction, and then simplify it.

Therefore, each container is filled with 3

1 liters of 2

gasoline.

Comparing and Ordering Fractions Suppose there was a vote for chairperson of the Student Organization at your school. The result of the vote was as follows : • 1 of the students voted for Candidate I. 3 Mathematics for Junior High School – Year 7 / 43

• 2 of the students voted for Candidate II 7 Based on the above result, which candidate got more votes, Candidate I or Candidate II?

To answer the

question, you need to know how to compare fractions.

There are two aspects you should know to compare fractions: (1)

Comparing like fractions

Examine the length of the shaded parts of the two models of fraction below.

Based on these models, you can conclude that 5 6 4 6

5 > 4 . Why? 6 6

Also, check that one-sixth can be considered as a new unit. 5 means 5 sixths, and 4 means 4 sixths. 6 6

Which one is greater, 5 sixths or 4 sixths? Based on the explanation, it is clear that 5 > 4 . 6

6

Therefore, to compare some like fractions, you just need to compare the numerator. If the numerator of one fraction is greater, then the fraction is also greater than the other. (2) Comparing unlike fractions

44 / Student’s Book – Fractions

Let us begin by comparing 1 to 1 . 2

3

We know that 1 is equal to 3 and 1 is equal to 2 . 2 3 6 6 The four fractions above can be modeled as follows.

is equal to 1 2

1 3

3 6

is equal to

2 6

Which fraction is greater? It is clear that 1 > 1 and 2

3

3 > 2 , because 1 = 3 and 1 = 2 . 6 6 2 6 3 6

Note

Therefore, one way to compare fractions is to express the fractions to like fractions, and then compare the

In a measurement the sizes can be compared if they have the same units. Analogically, to compare fractions, we have to change the fractions so that they have the same denominators

numerators. To get the same numerator, we use the Least Common Multiple (LCM) of two numbers. For more explanation, look at the procedures to compare fractions 1 with 2 in Example 2 below. 3

7

Use the sign <, =, or > to compare 1 to 2 . 3

7

Step I: Find LCM of 3 and 7. Multiple of 3 are 3, 6, 9, 12, 15, 18, 21 ,24 Multiple of 7 are 7, 14, 21 ,28

Mathematics for Junior High School – Year 7 / 45

LCM of 3 and 7 is 21, because 21 is the least number in which 3 and 7 match for the first time. Step II: Find the fraction which is equivalent to 1 3

2 7

and

using the LCM in Step I, as its

denominator.

1 3

=

... , so that 1 = 7 21 3 21

2 7

=

... , so that 2 = 6 21 7 21

Step III: Compare the like fractions in Step II. 7 and 6 . 21 21

Compare the numerator of Since 7 > 6, then 7 > 6 . 21

21

We may conclude that 1 > 2 . Since 1 > 2 , the answer 3

7

3

7

to the question about the election of the chairperson of the Student Organization is that Candidate I got more votes than Candidate II.

Use the sign <, =, or > to compare 7 with 5 . 18 24

Remember 24 = 2 × 2 × 2 × 3 To write an equivalent fraction, multiply the numerator and the denominator with the same number, excluding zero.

18 = 2 × 3 × 3

Find the LCM of 18 and 24 then circle all different factors that ap pear most frequently.

Multiply all circled factors to get the LCM of 24 and 18.

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2 × 2 × 2 × 3 × 3 = 72

7 24

=

21 72

5 18

=

20 72

Compare the numerator.

21 > 20 Since

Write equivalent fractions using LCM as the denominator.

5 21 20 7 > , then > . 72 72 24 18

Ordering Fractions Look at the fraction bars on the left. 1. What fraction represents each model? 2. Which fraction is the greatest? The smallest? 3. Put them in sequence from the smallest to the greatest.

To put fractions in sequence is the same as to compare three or more fractions. If you want to put like fractions in sequence, then order them based on the numerator. However, if you want to put unlike fractions in sequence, you need to find the LCM of the denominator of the original fractions. The LCM will be their denominators.

EXAMPLE 4

Put the fractions

3 2 7 in sequence from the , , and 8 5 20

least to the greatest.

Mathematics for Junior High School – Year 7 / 47

8= 2× 2 × 2 5= 5

Find the LC M of 8, 5, and 20 by writing down all prime factors of each number, an d then circle all different factors th at appear most frequently.

20 = 2 × 2 × 5

Multiply all circled factors. The LCM of 8, 5, and 20 is 40.

2 × 2 × 2 × 5 = 40

3 8

=

15 40

16 > 15 > 14

2 5

=

16 40

7 20

=

14 40

Compare the numerator and put them in order.

Since 16 > 15 > 14 , then 2 > 3 > 7 . 5 8 20 40 40 40

Therefore, if we put the fractions in order, from the least to the greatest, we get: 7 , 3 , 2 . 20 8 5

Decimal A fraction or a mixed number can also be expressed as decimal. Similarly, a decimal can be expressed as a fraction or a mixed number.

Get a calculator and do the following activities.

48 / Student’s Book – Fractions

Find the value of Push the bottom:

9 using a calculator. 40

9

/

4

0

=

What number do you get?

Such a number as 0.225 is called decimal and read “zero point two two five”.

Likewise, a decimal can be converted to a simple fraction. For example, convert 0.225 to a simple fraction.

Solution: Write it as a simple fraction.

0,225 = 225 1000

Remember You may read 1.32 as one-thirty two per one hundred”

225 1.000

=

9 40

Simplifiy by dividing the numerator and the denominator by the GCF. The GC F of 225 and 1,000 is 25.

Therefore, 0.225 = 9 . 40 When a decimal is greater than 1, it can be written in a mixed number.

Write 1.32 as a mixed number in the simplest form.

Mathematics for Junior High School – Year 7 / 49

Solution: 1.32 = 1 32 100

An integer is written separately from the fraction .

Simplify the fraction. The GCF of 100 and 32 is 4.

Therefore 1.32 = 1 8 . 25

In addition, to write a fraction in a decimal form, you may divide the numerator by the denominator. For instance, 3 can be calculated as 4 4

0.75 3 2.8 0.2 0.2 0

You could also use a calculator to divide the numerator by the denominator.

EXAMPLE 7

Relationwith the Real World A carpenter wants to make a hole with a diameter of not more than 0.6 inch. Could he use

5 inch drill? 8

You may also use a calculator to divide 5 by 8.

Since 0.625 > 0.6, the carpenter could not use a drill with

50 / Student’s Book – Fractions

a size of

5 inch because the hole is too big. 8

For a fraction, if you divide the numerator by the denominator and the remainder is zero, then the quotient is in the form of an irrecursive decimal number. On the other hand, if the quotient repeats a number or a set of certain numbers without ending, then such a decimal number is called a recursive decimal number. Example:

The bar over the number me ans that the number 4 is recursive.

0.4444 . . . . = 0.4

Rounding off If the decimal number is rounded off up to one decimal place, it can be written as 0.4. The number 4 cannot be changed because the number on its right (that is 4) is less than 5.

Write down the following fractions as decimal numbers. a.

4 15

Solution:

15

0.266 4 3

The number 6 is recursive.

1 0.9 0.1 0.09 0.1

Therefore, 4 = 0.26 15 Mathematics for Junior High School – Year 7 / 51

b.

8 11

Solution: By using a calculator, we know that 72 is recursive.

Th erefore, 8 = 0.72 11

Rounding off • If the number 0.266 is rounded off up to one decimal place, it becomes 0.3 ( because 6 is larger than 5) • If the number 0.266 is rounded off up to two decimal places, it becomes 0.27 (because 6 is larger than 5) • If the number 0.725 is rounded off up to one decimal place, it becomes 0.7 ( because 2 is less than 5) • If the number 0.725 is rounded off up to two decimal places, it becomes 0.73. Besides protein, meat contains fat, carbohydrate, vitamins, minerals and water. The amount of each depends on species, age and sex. Chicken, for example, has about 18% protein and 60-70% water.

Percent and Permil Consider the copy of an article on the left. In this article, it is written 18% and 60-70%. Do you know what percentage is? If you compare a number to 100, then you will have a percentage.

Percent means “per hundred”.

52 / Student’s Book – Fractions

75 12.5 You may write a ratio 15 as 15%, as 75%, as 100 100 100

12.5%, and so on.

You could make a model of percent with a scale paper of 10 x 10 as follows.

EXAMPLE 9

Modeling What is the percentage of the shadowed parts?

the number of shadowed parts 15 = the number of all parts 100

Write down the ratio as percentage. 15 = 15% 100

Therefore, the shadowed parts are 15% of all parts. What is the percentage of partitioned paper in Example 9 that is not shadowed? How do you answer it without counting the number of squares?

You may use what you already know about percent to express a percentage as a fraction.

EXAMPLE 10

Express 36% as a fraction in its simplest form.

36% = 36 100

Express the percent as a fraction having 100 as th e denominator.

Mathematics for Junior High School – Year 7 / 53

Express the fraction in its simplest form.

36 = 9 100 25

Sometimes, you need to express a fraction as a decimal

first,

before

writing

the

equivalent

percentage.

EXAMPLE 11

Natural Science About 7 of the earth surface is covered by water. 10 Express 7 in percent. 10 7 = 70 = 70% 10 100

0.7 = 70% You may use a calculator to convert a fraction to a percent as demonstrated in the following example.

EXAMPLE 12

Use a calculator to express the fraction 2 as a 3

percent.

Therefore, the fraction 2 is about 66.7%. 3

EXAMPLE 13

Express 14 as a percent. 1 = ..... 4 100 1 = 1 × 25 = 25 4 4 × 25 100

54 / Student’s Book – Fractions

Thus, 1 is equal to 25%. 4

Permil Percent mean per hundred, while permil means ‘per thousand’ symbolized o / oo . per thousand,

EXAMPLE 14

Express

23 can be read 23 1000

17.5 can be read 17.5 per thousand. 1000

13 in per thousand. 25

Solution: 13 13 × 40 520 = = . 25 25 × 40 1000

Therefore,

EXAMPLE 15

13 is equal to 520 per thousand. 25

Express 125 per thousand as a fraction in its simplest form.

125 per thousand=

1 125 = 8 1000

125 1000

Express per thousand as a fraction of which 1,000 is the denominator.

Express th e fraction in its simples t form.

Mathematics for Junior High School – Year 7 / 55

1. Thirty-five percent of the members of a club play football as their hobbies. What is the percentage of the members that do not play football as their hobbies?

2. Write down the following percentages as fractions in the simplest forms. a. 15%

b. 75%

c. 88%

d. 18%

3. Express each of the following fractions in per thousand. a.

3 20

b.

4. Biology.

34 50

c.

18 150

d.

23 250

The exhaled air consists of about 80%

nitrogen and 20% oxygen. Write down each percentage as a fraction in its simplest form.

5. Write down each fraction below as a percentage.

3 200

19 20

b. 7 50

c. 1 4

d. 1 8

e.

f. 9 50

g. 8 20

h. 3 10

i. 12 30

j. 2 25

a.

6. Express each of the following decimal numbers as a fraction or a mixed number in its simplest form. a. 0.3

b. 0.004

c. 2.625

d. 1.35

e. 5.500

7. Round off the following decimal numbers up to one

56 / Student’s Book – Fractions

and two decimal places. Give your reason for rounding them off. a. 0.075

b. 1.627

c. 0.155

d. 0.074

e. 10.023

8. Ali has 1-meter length of rope. This rope is cut into two parts, and one part has 0.55 meter length. Express the length of each part of the rope as a simple fraction.

9. Writin. Describe the steps to convert 0.8 to a fraction in its simplest form.

10. Express the following fractions as a decimal number. a.

3 20

b.

9 50

c.

7 32

d.

5 6

e.

11 16

11. Arrange the following numbers from the least to the greatest. a.

7 9 ; 0.8; ; 0.87 8 11

c. 3

2 3 b. 1.65;1 ; 1 ; 1.7 3 5

1 1 1 ; 3.1 ; 3 ; 3 ; 3.01 12 5 20

12. Ali ran as far as 1

3 7 km, and Budi ran as far as 1 4 10

km. Who ran farther than the other? 13. Arrange the following fractions in sequence, from the least to the greatest. a. 2 , 2 , 2 3 5 7

b. 4 , 5 , 7 8 6 9

d. 3 , 2 , 3 5 7 8

e. 2 8 , 2 17 , 2 5 9 18 6

c. 1 2 ,1 3 , 1 5 3 4 6 f. 11 , 5 , 5 24 8 12

Mathematics for Junior High School – Year 7 / 57

h. 1 8 , 2 1 , 1 3 11 4 4

g. 7 , 1 , 7 15 3 12

14. Critical Thinking. I am a fraction in my simplest form. My numerator and denominator are prime numbers having 2 as their differences. The sum of my numerator and denominator are the same as 12. What number am I?

15.

Writing. When you are given two different and

unequal fractions, write in your own words how you determine the greater fraction.

16.

Open Question. Write down three fractions and

arrange them in order, from the least to the greatest. Describe the steps that you apply to put the fractions in order.

17.

Describe in your own words how you find out

that a fraction is less than, the same as, or greater than 1.

18. Choose A, B, C, or D.

Which

number

mixed

describes

the

shaded parts? A. 4 3 4

58 / Student’s Book – Fractions

B. 3 3 4

C. 3 15 16

D. 3 1 4

19.

Critical Thinking. Express 100 as a fraction

using four similar numerals. Can it be expressed using six similar numerals? 20.

Express each of the following fractions as a

mixed number. a. 17 5

b. 13 7

21. Writing.

c. 27 5

d. 37 12

e. 21 f. 16 4 5

Think of two different situations in your

everyday life in which you use a mixed number. 22. Physics. The formula to convert the degree in the Celsius scale into the Fahrenheit scale is Express the fraction 9 5

9 o C + 32 = o F . 5

in this formula as a mixed

number. 23. Research.

Measure the height of your classmate or

family in a centimeter unit. When the height is over 100 cm, express such a measurement in a meter unit by using a mixed number.

24. Write down two fractions that are equal to each of the following fractions. a. 1 4

b. 10

25.

Open Question. Use the numbers 2, 3, 4, 6,

20

c. 4 5

d. 15

45

e. 6

8

12, 18, and 24 to write 3 pairs of equal fractions.

Mathematics for Junior High School – Year 7 / 59

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