Bombay Cambridge Gurukul FRACTIONS FRACTIONS FRACTIONS Bombay Cambridge Gurukul MATHEMATICS

Standard III Standard IV Standard V Choose the level Standard III Standard IV Standard V

III What are fractions? How to read fractions? More about fractions… Parts of a collection Revision More about fractions… …numerator and denominator Back

IV Equivalent fractions Types of fractions Fraction as division Mixed numbers Comparison of fractions Addition of like fractions Subtraction of like fractions Back

V Reduced form of fractions Factors and Multiples Addition of… …unlike fractions, mixed numbers V Subtraction of… …unlike fractions, mixed numbers Multiplication of fractions Reciprocal of a fraction Division of fractions Back

Standard III

What are fractions?

Look at the figure given below. It is a whole figure. We can divide it into 2 equal parts by drawing a line. 1 2 1 2 Shade only one part of the figure. Each part is called one half of the whole. 1 2 We write it as

Look at the figure given below. It is a whole figure. We can divide it into 2 equal parts by drawing a line. 1 2 1 2 Shade only one part of the figure. Each part is called one half of the whole. 1 2 We write it as Back

How to read fractions?

How to read a fraction? 1 is read as 1 upon 2 or 1 by 2. 2 3 7 is read as 3 upon 7 or 3 by 7. 2 5 is read as 2 upon 5 or 2 by 5. 7 9 is read as 7 upon 9 or 7 by 9. Back

More about fractions…

The following figures are divided into two equal parts. 1 2 1 2 1 2 1 2 whole whole When a whole is divided into two equal parts, each part is called half of the whole. 1 2 One half is written as Two halves make a whole.

Each figure is divided into two parts. Are both parts equal? Yes Yes No Yes No No Yes No

      Which of the following figures are divided into two equal parts?      

The following figures are divided into three equal parts. 1 3 1 3 When a whole is divided into three equal parts, each part is called one third of the whole. 1 3 One third is written as

Each figure is divided into three parts. Are all the three parts equal ? No Yes Yes No Yes No No Yes

        Which of the following figures are divided into three equal parts?        

The following figures are divided into four equal parts. 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 When a whole is divided into four equal parts, each part is called one fourth of the whole. 1 4 One fourth is written as

Each figure is divided into four parts. Are all the four parts equal? No No Yes Yes Yes Yes No No

        Which of the following figures are divided into four equal parts?        

Draw a line or lines to divide each of the following shapes into: three equal parts two equal parts four equal parts

Shade half (1/2) of each shape Shade one third (1/3) of each shape Shade one fourth (1/4) of each shape 1 4 1 4 1 3 1 3 1 4 1 4 1 2 1 2 1 3 1 3 1 3 1 2 1 2 1 4 1 4 1 4 1 3 1 4

Look at the figure given below: It has 3 equal parts. 2 parts are shaded. 2 3 The fraction for the shaded part is It is read as two third. It has 4 equal parts. 3 parts are shaded. The fraction for the shaded part is 3 4 It is read as three fourth.

Match the following 1 One fourth 2 1 One third 4 3 One half 4 1 Two third 2 3 Three fourth Back

Parts of a collection

The box given below has 12 stars. They can be divided into 2 equal parts. 6 6 Each part has 6 stars. To find the number of objects in one half of a collection, we divide the total number of objects by 2.

The box given below has 12 stars. They can be divided into 3 equal parts. 4 4 4 Each part has 4 stars. To find the number of objects in one third of a collection, we divide the total number of objects by 3.

The box given below has 12 stars. They can be divided into 4 equal parts. 3 3 3 3 Each part has 3 stars. To find the number of objects in one fourth of a collection, we divide the total number of objects by 4.

4 6 3 Total number of insects shown below is 12. 12 3 ¸ 4 12 4 ¸ 3 12 How many insects are there in 1 4 of the collection? How many insects are there in 1 2 of the collection? How many insects are there in 1 3 of the collection? = 12 3 ¸ 4 = 12 4 ¸ 3 = 12 2 ¸ 6 Encircle one fourth( 1 4 )of each collection. Encircle one third( 1 3 )of each collection. 1 2 )of each collection. Encircle one half( 4 6 3 One half of 12 is 6 One third of 12 is 4 One fourth of 12 is 3

Colour one half of the collection. Colour one fourth of the collection. Colour one half of the collection. Colour one third of the collection. Back

Revision

How many equal parts is each rod divided into?

What fraction do the colored portions in each of the following show? 2 5 2 3 3 4 1 4

Match the following fractions to the figures. 5 9 5 9 2 8 2 8 1 5 1 5 6 7 6 7 2 6 2 6 4 6 4 6

1 WHOLE 1 HALF 2 QUARTER 1 (ONE FOURTH) 4 3 THREE QUARTERS 4 (THREE FOURTH) 1 3 ONE THIRD 2 3 TWO THIRD Back

More about fractions… …numerator and denominator

PARTS OF A WHOLE ARE CALLED FRACTIONS. e.g. 1 2 Parts considered NUMERATOR Total number of equal parts DENOMINATOR NUMERATOR FRACTION = DENOMINATOR

3 3 8 8 So, in the fraction , 3 is the numerator. 3 8 Remember : Letter‘u’ is in the word ‘numerator’ and the word ‘ up’ . So, in the fraction , 3 is the numerator. 3 8 3 8 3 8 Remember : Letter ‘d’ starts the word denominator’ and the word ‘down’ . So, in the fraction , 8 is the denominator. 3 8

Write the numerator and denominator for each of the following fractions. Fraction Numerator Denominator 2 2 3 3 3 3 4 4 1 1 5 5 5 7 5 7

Write the fraction for the numerator and denominator given below. 1 5 1 5 1 5 1 5 1 5 4 7 4 7 4 7 4 7 4 7 3 4 3 4 3 4 3 4 3 4 5 8 5 8 5 8 5 8 5 8

Write the fraction for the shaded part. Numerator 5 Numerator 5 (shaded parts) (shaded parts) Denominator 10 Denominator 8 (total parts) (total parts) 5 10 5 8 Fraction Fraction

The End Created by Department of Research BOMBAY CAMBRIDGE GURUKUL Back

Standard IV

Equivalent fractions

Is the shaded part in each pair of figures same? Yes Yes Yes

What is the fraction for the shaded part? Is the shaded part in both the figures same? Yes What is the fraction for the shaded part? So, we see that =

What is the fraction for the shaded part? Is the shaded part in both the figures same? Yes What is the fraction for the shaded part? So, we see that =

What is the fraction for the shaded part? Is the shaded part in both the figures same? Yes What is the fraction for the shaded part? So, we see that =

Fractions which are equal in value to each other are called equivalent fractions. e.g. is equivalent to 1 2 4 is equivalent to 3 4 6 8

Match the following equivalent fractions. 1 2 2 8 2 4 1 3 2 6 1 1 4 3 Back

Types of fractions

Fractions where the numerator is smaller than the denominator are called proper fractions. 1 4 3 5 2 7 4 9 e.g. etc.

Fractions where the numerator is greater than the denominator are called improper fractions. 9 4 7 2 4 3 8 7 e.g. etc.

Fractions which have same denominator are called like fractions. 5 9 3 9 2 9 4 9 e.g. etc.

Fractions which have different denominators are called unlike fractions. 5 9 3 4 2 3 4 5 e.g. etc.

Fractions which have numeral 1 as numerator are called unit fractions. 9 1 4 1 3 1 5 e.g. etc. Back

Fraction as division

We can write each division sum as a fraction. 4 12 4 ¸ 12 = 3 6 3 6 ¸ = 1 5 1 ¸ 5 = 7 10 7 ¸ 10 =

¸ ¸ ¸ ¸ We can write each fraction as a division sum. 1 8 1 8 6 9 6 9 = 6 9 6 ¸ 9 = 4 12 ¸ 4 12 = 2 9 2 ¸ 9 = Back

Mixed numbers

Mixed numbers include a whole number and a fraction. + = (whole number) + (fraction) = (mixed number)

Converting mixed numbers to improper fractions. 1 2 4 to a improper fraction. Step 1 : Multiply the denominator 2 with whole number 4. = ´ 2 4 8 = + 1 8 9 Step 2 : Add numerator 1 to 8 Step 3 : Write 9 as the numerator of the improper fraction. 9 Step 4 : Write denominator 2 as the denominator of the improper fraction. 9 2 1 2 4 = 9 (improper fraction) (mixed number)

Converting improper fractions to mixed numbers. 7 3 to a mixed number. Step 1 : Divide 7 by 3. Divisor : 3 Quotient: 2 Remainder : 1 Step 2 : Write the mixed number. * 2 The quotient becomes the whole number. * 3 2 The divisor becomes the denominator. The remainder becomes the numerator. 1 3 2 7 3 1 2 = (improper fraction) (mixed number)

Converting improper fractions to mixed numbers. can be changed to Improper fractions can be changed to Mixed numbers Back

Comparison of fractions

How to compare like fractions ? like fractions Look at the figures shown below. Each figure is divided into 4 equal parts. (A) (B) Which figure has more shaded parts? The first figure (A) has more shaded parts.

How to compare like fractions ? like fractions Look at the figures shown below. Write the fraction for both figures. 2 6 4 6 4 6 Which fractions has more shaded area? So, we can say that 4 6 2 >

How to compare like fractions ? like fractions Look at the figures shown below. Write the fraction for both figures. 2 7 6 7 2 7 Which fraction has less shaded area? So, we can say that 2 7 6 <

How to compare like fractions ? like fractions If there are two like fractions, then the fraction with greater numerator is greater in value. e.g. 4 7 3 7 > If there are two like fractions, then the fraction with smaller numerator is lesser in value. e.g. 2 9 8 9 <

Compare the following using < , > or = . 4 5 1 5 > 3 7 6 7 < 2 9 2 9 = 4 6 3 6 > Back

Addition of like fractions

Addition of like fractions In the circle given below only one part out of five is shaded. Two more parts of the circle are shaded. The circle has three shaded parts.

Addition of like fractions + = 1 4 2 4 3 4 1 4 2 4 3 4 + =

Addition of like fractions + = 2 6 3 6 5 6 2 6 3 6 5 6 + =

Addition of like fractions 1 3 + 1 3 2 3 1 3 1 3 2 3 + =

Addition of like fractions When two or more like fractions are added, then only the numerators are added together. The denominators are not added together.

+ + Addition of like fractions ¸ The answer should be written 4 + 4 + ¸ The answer should be written in the reduced form of fractions.

Addition of like fractions 2 5 2 5 4 5 2 + 2 5 + = = 1 7 2 7 3 7 1 + 2 7 + = = 5 8 2 8 5 + 2 8 7 8 + = = 3 9 3 9 3 + 3 9 6 9 + = = Back

Subtraction of like fractions

Subtraction of like fractions In the figure given below, three parts out of five parts are shaded. Two parts are taken away. One part out of five is left.

Subtraction of like fractions In the figure given below, three parts out of four are shaded. Two parts are taken away. 3 4 2 4 1 4 - = One part out of four is left.

Subtraction of like fractions When two like fractions are subtracted, then the smaller numerator is subtracted from the bigger numerator. The denominators are not subtracted.

Subtraction of like fractions ¸ 2 6 12 4 12 2 12 1 6 - = = 2 14 16 2 16 12 16 2 2 6 8 3 4 - = = = 2 2 The answer should be written in the reduced form of fractions.

Subtraction of like fractions 3 6 2 6 3 - 2 6 1 6 - = = 5 7 2 7 5 - 2 7 3 7 - = = 6 8 1 8 6 - 1 8 5 8 - = = 7 9 5 9 7 - 5 9 2 9 - = =

The End Created by Department of Research BOMBAY CAMBRIDGE GURUKUL Back

Standard V

Reduced form of fractions

Reduced form of fractions A fraction is said to be in the reduced form if its numerator and denominator cannot be divided by a common number.

Look at the fraction given below. 6 8 We can divide the numerator and denominator both by 2. 6 8 6 8 2 So, ¸ 2 3 4 = = 2 6 divided by 2 is 3. 8 divided by 2 is 4. Now we can not divide 3 and 4 both by any number. 3 4 6 8 So, we can say that is the reduced form of

Reduce the given fraction to its lowest form. 3 9 We can divide both, the numerator and the denominator by 3. 3 3 9 1 3 = 3 The reduced form of 3 9 1 is We can divide both, the numerator and the denominator by 2. 10 12 10 12 2 5 6 = 2 The reduced form of 10 12 5 6 is

Circle the fractions which are in the reduced form. 5 6 3 3 9 2 8 2 1 4 3 2 4 12 4 5 7 3 5 9 18 9 4 9 6 14 2 3 8 4 9 8 12 2 2 2 Back

Factors and Multiples

A number that divides a given number completely (without leaving a remainder) is called its factor. e.g. 5 divides 20 exactly. So, 5 is a factor of 20. And 20 is a multiple of 5. Is 20 exactly divisible by 3? No Is 3 a factor of 20? No Is 20 a multiple of 3? No

List the numbers that divide 15 exactly. 3 5 15 15 So, we can say that factors of 15 are 1, 3, 5 and 15. List the numbers that divide 12 exactly. 1 1 2 3 4 6 12 12 So, we can say that factors of 12 are 1, 2, 3, 4, 6 and 12. Every number has at least 2 factors : 1 and the number itself.

Which of the following are factors of 16? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16            1 2 4 8 16 Try the following…. Is 4 a factor of 14 ? No Is 6 a factor of 24 ? Yes Is 4 a factor of 32 ? Yes Is 3 a factor of 17 ? No

Which of the following are multiples of 4 ? 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30             8 12 16 20 24 28 Try the following…. Is 15 a multiple of 6 ? No Is 28 a multiple of 7 ? Yes Is 24 a multiple of 8 ? Yes Is 21 a multiple of 9 ? No

Common factors Common factors of 24 and 30 are : The factors of 24 are : 1, 2, 3, 4, 6, 8, 12 and 24. The factors of 30 are : 1, 2, 3, 5, 6, 10, 15 and 30. Common factors of 24 and 30 are : 1, 2, 3, 6 Highest common factor (H.C.F.) of 24 and 30 is : 6

Common multiples Common multiples of 3 and 4 are : The multiples of 3 are : 3, 6, 9, 12, 15, 18, 21, 24 … The multiples of 4 are : 4, 8, 12, 16, 20, 24, 28 ... Common multiples of 3 and 4 are : 12 , 24 … Least common multiple (L.C.M.) of 3 and 4 is : 12 Back

Addition of… …unlike fractions, mixed numbers

When we add two unlike fractions (with different denominators), we need to find the least common multiple ( L.C.M.) of the two denominators.

Addition of unlike fractions 1 2 1 6 + To change both fractions to like fractions, we find the L.C.M. of 2 and 6. Multiples of 2 are : 2, 4, 6, 8, 10, 12… Multiples of 6 are : 6, 12, 18, 24, 30… Common multiples of 2 and 6 are : 6, 12… Least common multiple (L.C.M.) of 2 and 6 is : 6

¸ ¸ ________ 6 ´ 1 2 6 + L.C.M. of 2 and 6 is 6. Now we can add The denominator of both the fractions is the same as the L.C.M. Step 1 : ________ 6 = Step 2 : Divide the common denominator with the denominator of the first fraction. 6 2 = 3 ¸ Step 3 : Multiply 3 with the numerator of the first fraction. 3 1 = 3 ´ Step 4 : Write 3 in place of the first numerator. = 3 + 6

¸ ´ Divide the common denominator with the denominator of the second fraction. Step 5 : 6 ¸ 6 = 1 Step 6 : Multiply 1 with the numerator of the second fraction. 1 = 1 ´ 1 Step 7: Write 1 in place of the second numerator. = 3 + 1 6 = 4 6 Step 8 : Add the numerators 1 2 6 + So, = 4

The denominators are different, Addition of unlike fractions The denominators are different, so, we find the L.C.M. of 2 and 4. L.C.M. of 2 and 4 is 4. = Then, numerators are added.

Addition of mixed numbers 1 5 4 2 5 + Step1: Change the mixed number to an improper fraction. 1 5 4 ´ 4 5 + 1 21 5 = = Step 2: (21 + 2) 5 5 Add both the fractions. 2 5 + 21 21 + 2 5 23 5 = = So, 1 5 4 2 + = 23 Back

Subtraction of… …unlike fractions, mixed numbers

When we subtract two unlike fractions (with different denominators), we need to find the least common multiple ( L.C.M.) of the two denominators.

Subtraction of unlike fractions 2 3 _ 1 6 To change both fractions to like fractions, we find the L.C.M. of 3 and 6. Multiples of 3 are : 3, 6, 9, 12, 15, 18… Multiples of 6 are : 6, 12, 18, 24, 30… Common multiples of 8 and 4 are : 6, 12… Least common multiple (L.C.M.) of 8 and 4 is : 6

¸ ________ 6 ´ 2 3 1 6 - L.C.M. of 3 and 6 is 6. Now we can subtract The denominator of both the fractions is the same as the L.C.M. Step 1 : ________ 6 = Step 2 : Divide the common denominator with the denominator of the first fraction. 6 3 = 2 ¸ Step 3: Multiply 2 with the numerator of the first fraction. 2 2 = 4 ´ Step 4 : Write 4 in place of the first numerator. = 4 - 6

Subtract the numerators Divide the common denominator with the denominator of the second fraction. Step 5 : 6 6 = 1 ¸ Step 6 : Multiply 1 with the numerator of the second fraction. 1 1 = 1 ´ Step 7: Write 1 in place of the second numerator. = 4 - 1 6 Step 8 : = 3 6 Subtract the numerators 2 3 1 6 - So, =

The denominators are different, Subtraction of unlike fractions The denominators are different, so, we find the L.C.M. of 2 and 4. L.C.M. of 2 and 4 is 4. Then, numerators are subtracted.

Subtraction of mixed numbers 2 7 3 - Change the mixed number to an improper fraction. Step 1: 2 7 3 ´ 3 7 + 2 23 7 = = Step 2: Subtract both the fractions. (23 - 2) 7 7 2 7 - 23 23 - 2 7 21 7 = = So, 2 7 3 - = 21 Back

Multiplication of fractions

¸ ¸ ´ ´ ´ How to multiply a fraction by a whole number ? 5 8 4 We multiply only the numerator of the fraction with the whole number. The denominator remains the same. 4 5 ´ 8 20 8 = We should write the answer in the reduced form of fractions. 20 8 2 ¸ 10 4 2 5 2 = ¸ = 5 8 ´ 4 5 2 So, =

¸ ´ ´ ´ How to multiply a whole number by a fraction ? 6 9 5 We multiply the whole number only with the numerator of the fraction. The denominator remains the same. 6 5 ´ 9 30 9 = We should write the answer in the reduced form of fractions. 30 9 3 ¸ 10 3 = 6 9 ´ 10 3 5 So, =

¸ ´ ´ ´ How to multiply a fraction by a fraction ? 2 3 6 7 We multiply both the numerators. And we multiply both the denominators. 6 2 ´ 7 3 12 21 = We should write the answer in the reduced form of fractions. 12 21 3 ¸ 4 7 = 2 3 6 7 4 7 So, ´ = Back

Reciprocal of a fraction

How to write a reciprocal fraction ? The numerator becomes the denominator. And the denominator becomes the numerator. Fraction Reciprocal fraction 7 9 7 9

REMEMBER The reciprocal of 1 7 is 7 1 or 7 The reciprocal of a unit fraction is a whole number. The reciprocal of 7 or 7 1 is 1 7 The reciprocal of a whole number is a unit fraction. Back

Division of fractions

¸ ¸ ¸ ´ ´ How to divide a whole number by a fraction ? 6 4 8 We change the division sign to multiplication. 4 ¸ ´ Then we write the reciprocal of the second fraction. ´ 8 6 4 Multiply the numerators. 32 6 2 32 6 16 3 Reduce the fraction to its lowest form. = 2 ¸ 6 8 16 3 So, 4 =

¸ ¸ ¸ ´ ´ How to divide a fraction by a whole number ? 4 5 4 We change the division sign to multiplication. 4 5 ¸ ´ Then we write the reciprocal of the whole number. 4 5 ´ 1 4 Multiply the numerators. 4 20 2 2 4 20 2 10 1 5 Reduce the fraction to its lowest form. = = 2 2 4 5 1 5 So, ¸ 4 =

¸ ¸ ¸ ´ ´ How to divide a fraction by a fraction ? 4 8 2 3 We change the division sign to multiplication. 4 8 ´ ¸ Then we write the reciprocal of the second fraction. 4 8 ´ 3 2 Multiply the numerators and the denominators. 12 16 2 2 12 16 6 8 3 4 Reduce the fraction to its lowest form. = = 2 2 4 8 So, ¸ 2 3 3 4 =

The End Created by Department of Research BOMBAY CAMBRIDGE GURUKUL