# RS Aggarwal Solutions Class 6 Ex 5A

## RS Aggarwal Class 6 Ex 5A Chapter 5

Q1) (i) $\frac{ 2 }{ 3 }$ (ii) $\frac{ 4 }{ 5 }$

(iii) $\frac{ 5 }{ 8 }$ (iv) $\frac{ 7 }{ 10 }$

(v) $\frac{ 3 }{ 7 }$ (vi) $\frac{ 6 }{ 11 }$

(vii) $\frac{ 7 }{ 9 }$ (viii) $\frac{ 5 }{ 12 }$

Ans. 1) (i)$\frac{ 2 }{ 3 } = \frac{ 2 \times 2 }{ 3 \times 2} = \frac{ 2 \times 3 }{ 3 \times 3 } = \frac{ 2 \times 4 }{ 3 \times 4} = \frac{ 2 \times 5}{ 3 \times 5 } = \frac{ 2 \times 6}{ 3 \times 6 }$

Therefore, $\frac{ 2 }{ 3 } = \frac{ 4 }{ 6 } = \frac{ 6 }{ 9 } = \frac{ 8 }{ 12 } = \frac{ 10 }{ 15 } = \frac{ 12 }{ 18 }$

Hence, the five fractions equivalent to $\frac{ 2 }{ 3 }$ are $\frac{ 4 }{ 6 }$ , $\frac{ 6 }{ 9 }$ , $\frac{ 8}{ 12 }$ , $\frac{ 10 }{ 15 }$ and $\frac{ 12 }{ 18 }$

(ii) $\frac{ 4 }{ 5 } = \frac{ 4 \times 2 }{ 5 \times 2} = \frac{ 4 \times 3 }{ 5 \times 3 } = \frac{ 4 \times 4 }{ 5 \times 4} = \frac{ 4 \times 5}{ 5 \times 5 } = \frac{ 4 \times 6}{ 5 \times 6 }$

Therefore, $\frac{ 4 }{ 5 } = \frac{ 8 }{ 10 } = \frac{ 12 }{ 15 } = \frac{ 16 }{ 20 } = \frac{ 20 }{ 25 } = \frac{ 24 }{ 30 }$

Hence, the five fractions equivalent to $\frac{ 4 }{ 5 }$ are $\frac{ 8 }{ 10 }$ , $\frac{ 12 }{ 15 }$ , $\frac{ 16}{ 20 }$ , $\frac{ 20 }{ 25 }$ and $\frac{ 24 }{ 30 }$

(iii) $\frac{ 5 }{ 8 } = \frac{ 5 \times 2 }{ 8 \times 2} = \frac{ 5 \times 3 }{ 8 \times 3 } = \frac{ 5 \times 4 }{ 8 \times 4} = \frac{ 5 \times 5}{ 8 \times 5 } = \frac{ 5 \times 6}{ 8 \times 6 }$

Therefore, $\frac{ 5 }{ 8 } = \frac{ 10 }{ 16 } = \frac{ 15 }{ 24 } = \frac{ 20 }{ 32 } = \frac{ 25 }{ 40 } = \frac{ 30 }{ 48 }$

Hence, the five fractions equivalent to $\frac{ 5 }{ 8 }$ are $\frac{ 10 }{ 16 }$ , $\frac{ 15 }{ 24 }$ , $\frac{ 20}{ 32 }$ , $\frac{ 25 }{ 40 }$ and $\frac{ 30 }{ 48 }$ .

(iv) $\frac{ 7 }{ 10 } = \frac{ 7 \times 2 }{ 10 \times 2} = \frac{ 7 \times 3 }{ 10 \times 3 } = \frac{ 7 \times 4 }{ 10 \times 4} = \frac{ 7 \times 5}{ 10 \times 5 } = \frac{ 7 \times 6}{ 10 \times 6 }$

Therefore, $\frac{ 7 }{ 10 } = \frac{ 14 }{ 20 } = \frac{ 21 }{ 30 } = \frac{ 28 }{ 40 } = \frac{ 35 }{ 50 } = \frac{ 42 }{ 60 }$

Hence, the five fractions equivalent to $\frac{ 7 }{ 10 }$ are $\frac{ 14 }{ 20 }$ , $\frac{ 21 }{ 30 }$ , $\frac{ 28}{ 40 }$ , $\frac{ 35 }{ 50 }$ and $\frac{ 42 }{ 60 }$

(v)$\frac{ 3 }{ 7 } = \frac{ 3 \times 2 }{ 7 \times 2} = \frac{ 3 \times 3 }{ 7 \times 3 } = \frac{ 3 \times 4 }{ 7 \times 4} = \frac{ 3 \times 5}{ 7 \times 5 } = \frac{ 3 \times 6}{ 7 \times 6 }$

Therefore, $\frac{ 3 }{ 7 } = \frac{ 6 }{ 14 } = \frac{ 9 }{ 14 } = \frac{ 12 }{ 28 } = \frac{ 15 }{ 35 } = \frac{ 18 }{ 42 }$

Hence, the five fractions equivalent to $\frac{ 3 }{ 7 }$ are $\frac{ 6 }{ 14 }$ , $\frac{ 9 }{ 21 }$ , $\frac{ 12}{ 28 }$ , $\frac{ 15 }{ 35 }$ and $\frac{ 18 }{ 42 }$

(vi) $\frac{ 6 }{ 11 } = \frac{ 6 \times 2 }{ 11 \times 2} = \frac{ 6 \times 3 }{ 11 \times 3 } = \frac{ 6 \times 4 }{ 11 \times 4} = \frac{ 6 \times 5}{ 11 \times 5 } = \frac{ 6 \times 6}{ 11 \times 6 }$

Therefore, $\frac{ 6 }{ 11 } = \frac{ 12 }{ 22 } = \frac{ 18 }{ 33 } = \frac{ 24 }{ 44 } = \frac{ 30 }{ 55 } = \frac{ 36 }{ 66 }$

Hence, the five fractions equivalent to $\frac{ 6 }{ 11 }$ are $\frac{ 12 }{ 22 }$ , $\frac{ 18 }{ 33 }$ , $\frac{ 24}{ 44 }$ , $\frac{ 30 }{ 55 }$ and $\frac{ 36 }{ 66 }$

(vii) $\frac{ 7 }{ 9 } = \frac{ 7 \times 2 }{ 9 \times 2} = \frac{ 7 \times 3 }{ 9 \times 3 } = \frac{ 7 \times 4 }{ 9 \times 4} = \frac{ 7 \times 5}{ 9 \times 5 } = \frac{ 7 \times 6}{ 9 \times 6 }$

Therefore, $\frac{ 7 }{ 9 } = \frac{ 14 }{ 18 } = \frac{ 21 }{ 27 } = \frac{ 28 }{ 36 } = \frac{ 35 }{ 45 } = \frac{ 42 }{ 54 }$

Hence, the five fractions equivalent to $\frac{ 7 }{ 9 }$ are $\frac{ 14 }{ 18 }$ , $\frac{ 21 }{ 27 }$ , $\frac{ 28}{ 36 }$ , $\frac{ 35 }{ 45 }$ and $\frac{ 42 }{ 54 }$

(viii) $\frac{ 5 }{ 12 } = \frac{ 5 \times 2 }{ 12 \times 2} = \frac{ 5 \times 3 }{ 12 \times 3 } = \frac{ 5 \times 4 }{ 12 \times 4} = \frac{ 5 \times 5}{ 12 \times 5 } = \frac{ 5 \times 6}{ 12 \times 6 }$

Therefore, $\frac{ 5 }{ 12 } = \frac{ 10 }{ 24 } = \frac{ 15 }{ 36 } = \frac{ 20 }{ 48 } = \frac{ 25 }{ 60 } = \frac{ 30 }{ 72 }$

Hence, the five fractions equivalent to $\frac{ 5 }{ 12 }$ are $\frac{ 10 }{ 24 }$ , $\frac{ 15 }{ 36 }$ , $\frac{ 15}{ 36 }$ , $\frac{ 20 }{ 48 }$ and $\frac{ 25 }{ 60 }$

Q2) (i) $\frac{ 5 }{ 6 }$ and $\frac{ 20 }{ 24 }$

(ii) $\frac{ 3 }{ 8 }$ and $\frac{ 15 }{ 40 }$

(iii) $\frac{ 4 }{ 7 }$ and $\frac{ 16 }{ 21 }$

(iv)$\frac{ 2 }{ 9 }$ and $\frac{ 14 }{ 63 }$

(v) $\frac{ 1 }{ 3 }$ and $\frac{ 9 }{ 24 }$

(vi) $\frac{ 2 }{ 3 }$ and $\frac{ 33 }{ 22 }$

Ans. 2) The pairs of equivalent fractions are as follows:

(i) $\frac{ 5 }{ 6 }$ and $\frac{ 20 }{ 24 }$

( $\frac{ 20 }{ 24 } = \frac{ 5\times 4}{ 6 \times 4}$ )

(ii) $\frac{ 3 }{ 8 }$ and $\frac{ 15 }{ 40 }$

( $\frac{ 15 }{ 40 } = \frac{ 3 \times 5}{ 8 \times 5}$ )

(iv) $\frac{ 2 }{ 9 }$ and $\frac{ 14 }{ 40 }$

( $\frac{ 14 }{ 63 } = \frac{ 2 \times 7}{ 9 \times 7}$ )

Q3) Find the equivalent fraction of $\frac{ 3 }{ 5 }$ having

(i) denominator 30 (ii) numerator 24

Ans. 3) (i) Let $\frac{ 3 }{ 5 } = \frac{ ?}{ 30 }$

Clearly, 30 = 5 * 6

So, we multiply the numerator by 6.

Therefore, $\frac{ 3 }{ 5 } = \frac{ 3 \times 6 }{ 5 \times 6 } = \frac{ 18 }{ 30 }$

Hence, the required fraction is $\frac{ 18 }{ 30 }$.

(ii) Let $\frac{ 3 }{ 5 } = \frac{ 24 }{ ? }$

Clearly, 24 = 3 * 8

So, we multiply the denominator by 8.

Therefore, $\frac{ 3 }{ 5 } = \frac{ 3 \times 8 }{ 5 \times 8 } = \frac{ 24 }{ 40 }$

Hence, the required fraction is $\frac{ 24 }{ 40 }$.

Q4) Find the equivalent fraction $\frac{ 5 }{ 9 }$ having

(i) denominator 54 (ii) numerator 35

Ans. 4) (i) Let $\frac{ 5 }{ 9 } = \frac{?}{ 54 }$

Clearly, 54 = 9 * 6

So, we multiply the numerator by 6.

Therefore, $\frac{ 5 }{ 9 } = \frac{ 5 \times 6 }{ 9 \times 6 } = \frac{ 30 }{ 54 }$

Hence, the required fraction is $\frac{ 30 }{ 54 }$

(ii) Let $\frac{ 5 }{ 9 } = \frac{ 35 }{ ?}$

Clearly, 35 = 5 * 7

So, we multiply the denominator by 7.

Therefore, $\frac{ 5 }{ 9 } = \frac{ 5 \times 7 }{ 9 \times 7 } = \frac{ 35 }{ 63 }$

Hence, the required fraction is $\frac{ 35 }{ 63 }$.

Q5) Find the equivalent fraction of $\frac{ 6 }{ 11 }$

(i) denominator 77 (ii) numerator 60

Ans. 5) (i)Let $\frac{ 6 }{ 11 } = \frac{ ? }{ 77 }$

Clearly, 77 = 11 * 7

So, we multiply the numerator by 7.

Therefore, $\frac{ 6 }{ 11 } = \frac{ 6 \times 7 }{ 11 \times 7 } = \frac{ 42 }{ 77 }$

Hence, the required fraction is $\frac{ 30 }{ 54 }$

(ii) Let $\frac{ 6 }{ 11 } = \frac{ 60 }{ ?}$

Clearly, 60 = 6 * 10

So, we multiply the denominator by 10.

Therefore, $\frac{ 6 }{ 11 } = \frac{ 6 \times 10 }{ 11 \times 10 } = \frac{ 60 }{ 110 }$

Hence, the required fraction is $\frac{ 60 }{ 110 }$.

Q6) Find the equivalent fraction of $\frac{ 24 }{ 30 }$ having numerator 4.

Ans. 6) Let $\frac{ 24 }{ 30 } = \frac{ 4 }{ ?}$

Clearly, 4 = 24 / 6

So, we divide the denominator by 6.

Therefore, $\frac{ 24 }{ 30 } = \frac{ 24 \div 6 }{ 30 \div 6 } = \frac{ 4 }{ 5 }$

Hence, the required fraction is $\frac{ 4 }{ 5 }$.

Q7) Find the equivalent fraction $\frac{ 36 }{ 48 }$ with

(i) numerator 9 (ii) denominator 4

Ans. 7) (i) Let $\frac{ 36 }{ 48 } = \frac{ 9 }{ ?}$

Clearly, 9 = 36 / 4

So, we divide the denominator by 4.

Therefore, $\frac{ 36 }{ 48 } = \frac{ 36 \div 4 }{ 48 \div 4 } = \frac{ 9 }{ 12 }$

Hence, the required fraction is $\frac{ 9 }{ 12 }$.

(ii) Let $\frac{ 36 }{ 48 } = \frac{ ? }{ 4 }$

Clearly, 4 = 48 / 12

So, we divide the numerator by 12.

Therefore, $\frac{ 36 }{ 48 } = \frac{ 36 \div 7 }{ 48 \div 12 } = \frac{ 3 }{ 4 }$

Hence, the required fraction is $\frac{ 3 }{ 4 }$ .

Q8) Find the equivalent fraction of $\frac{ 56 }{ 70 }$ with

(i) numerator 4 (ii) denominator 10

Ans. 8) (i) Let $\frac{ 56 }{ 70 } = \frac{ 4 }{ ? }$

Clearly, 4 = 56 / 14

So, we divide the denominator by 14.

Therefore, $\frac{ 56 }{ 70 } = \frac{ 56 \div 14 }{ 70 \div 14 } = \frac{ 4 }{ 5 }$

Hence, the required fraction is $\frac{ 4 }{ 5 }$.

(ii) Let $\frac{ 56 }{ 70 } = \frac{ ?}{ 10 }$

Clearly, 10 = 70 / 7

So, we divide the numerator by 7.

Therefore, $\frac{ 56 }{ 70 } = \frac{ 56 \times 7 }{ 70 \times 7 } = \frac{ 8 }{ 10 }$

Hence, the required fraction is $\frac{ 8 }{ 10 }$ .

Q9) Reduce each of the following fractions into its simplest form:

(i) $\frac{ 9 }{ 15 }$ (ii) $\frac{ 48 }{ 60 }$

(iii) $\frac{ 84 }{ 98 }$ (iv) $\frac{ 150 }{ 60 }$

(v) $\frac{ 72 }{ 90 }$

Ans. 9) (i) Here , numerator = 9 and denominator = 15

Factors of 9 are 1, 3 and 9.

Factors of 15 are 1, 3, 5 and 15.

Common factors of 9 and 15 are 1 and 3.

H.C.F. of 9 and 15 is 3.

Therefore, $\frac{ 9 }{ 15 }$ = $\frac{ 9 \div 3 }{ 15 \div 3 } = \frac{ 3 }{ 5 }$

Hence, the simplest form of $\frac{ 9 }{ 15 }$ is $\frac{ 3 }{ 5 }$ .

(ii) Here , numerator = 48 and denominator = 60

Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.

Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Common factors of 48 and 60 are 1, 2, 3, 4, 6 and 12.

H.C.F. of 48 and 60 is 12.

Therefore, $\frac{ 48 }{ 60 }$ = $\frac{ 48 \div 12 }{ 60 \div 12 } = \frac{ 4 }{ 5 }$

Hence, the simplest form of $\frac{ 48 }{ 60 }$ is $\frac{ 4 }{ 5 }$ .

(iii) Here, numerator = 84 and denominator = 98

Factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 42 and 84.

Factors of 98 are 1, 2, 3, 4, 7, 14, 15, 49 and 98.

Common factors of 84 and 98 are 1, 2, 7 and 14.

H.C.F. of 84 and 98 is 14.

Therefore, $\frac{ 84 }{ 98 }$ = $\frac{ 84 \div 14 }{ 98 \div 14 } = \frac{ 6 }{ 7 }$

Hence, the simplest form of $\frac{ 84 }{ 98 }$ is $\frac{ 6 }{ 7 }$ .

(iv) Here, numerator = 150 and denominator = 60

Factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 75 and 150.

Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Common factors of 150 and 60 are 1, 2, 3, 5, 6, 10, 15 and 30.

H.C.F. of 150 and 60 is 30.

Therefore, $\frac{ 150 }{ 60 }$ = $\frac{ 150 \div 30 }{ 60 \div 30 } = \frac{ 5 }{ 2 }$

Hence, the simplest form of $\frac{ 150 }{ 60 }$ is $\frac{ 5 }{ 2 }$ .

(v) Here , numerator = 72 and denominator = 90

Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.

Factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90.

Common factors of 72 and 90 are 1, 2, 3, 3, 6, 9 and 18.

H.C.F. of 72 and 90 is 18.

Therefore, $\frac{ 72 }{ 90 }$ = $\frac{ 72 \div 18 }{ 90 \div 18 } = \frac{ 4 }{ 5 }$

Hence, the simplest form of $\frac{ 72 }{ 90 }$ is $\frac{ 4 }{ 5 }$ .

Q10) Show that each of the following fractions is in the simplest form:

(i) $\frac{ 8 }{ 11 }$ (ii) $\frac{ 9 }{ 14 }$

(iii) $\frac{ 25 }{ 36 }$ (iv) $\frac{ 8 }{ 15 }$

(v) $\frac{ 21 }{ 10 }$

Ans. 10) (i) Here , numerator = 8 and denominator = 11

Factors of 8 are 1, 2, 4 and 8.

Factors of 11 are 1 and 11.

Common factors of 8 and 11 is 1.

Thus, H.C.F. of 8 and 11 is 1.

Hence, $\frac{ 8 }{ 11 }$ is the simplest form .

(ii) Here , numerator = 9 and denominator = 14

Factors of 9 are 1, 3 and 9.

Factors of 14 are 1, 2, 7, and 14.

Common factors of 9 and 14 is 1.

Thus, H.C.F. of 9 and 14 is 1.

Hence, $\frac{ 9 }{ 14 }$ is the simplest form .

(iii) Here , numerator = 25 and denominator = 36

Factors of 25 are 1, 5 and 25.

Factors of 36 are 1, 2, 3, 4, 6, 9,12, 18 and 36.

Common factors of 25 and 36 is 1.

Thus, H.C.F. of 25 and 36 is 1.

Hence, $\frac{ 25 }{ 36 }$ is the simplest form .

(iv) Here , numerator = 8 and denominator = 15

Factors of 8 are 1, 2, 4, and 8.

Factors of 15 are 1, 3, 5, and 15.

Common factors of 8 and 15 is 1.

Thus, H.C.F. of 8 and 15 is 1.

Hence, $\frac{ 8 }{ 15 }$ is the simplest form .

(v) Here , numerator = 21 and denominator = 10

Factors of 21 are 1, 3, 7, and 21.

Factors of 10 are 1, 2, 5, and 10.

Common factors of 21 and 10 is 1.

Thus, H.C.F. of 21 and 10 is 1.

Hence, $\frac{ 21 }{ 10 }$ is the simplest form .

Q11) Replace $?$ by the correct number in each of the following:

(i) $\frac{ 2 }{ 7 } = \frac{ 8 }{ ? }$

(ii) $\frac{ 3 }{ 5 } = \frac{ ? }{ 35 }$

(iii) $\frac{ 5 }{ 8 } = \frac{ 20 }{ ? }$

(iv) $\frac{ 45 }{ 60 } = \frac{ 9 }{ ?}$

(v) $\frac{ 40 }{ 56 } = \frac{ ? }{ 7 }$

(vi) $\frac{ 42 }{ 54 } = \frac{ 7 }{ ?}$

Ans. 11) (i) 28 ( $\frac{ 2 }{ 7 } = \frac{ 2 \times 4 }{ 7 \times 4 } = \frac{ 8 }{ 28 }$ )

(ii) 21 ( $\frac{ 3 }{ 5 } = \frac{ 3 \times 7 }{ 5 \times 7 } = \frac{ 21 }{ 35 }$ )

(iii) 32 ( $\frac{ 5 }{ 8 } = \frac{ 5 \times 4 }{ 8 \times 4 } = \frac{ 20 }{ 32 }$ )

(iv) 12 ( $\frac{ 45 }{ 60 } = \frac{ 45 \div 5 }{ 60 \div 5 } = \frac{ 9 }{ 12 }$ )

(v) 5 ( $\frac{ 40 }{ 56 } = \frac{ 40 \div 8 }{ 56 \div 8 } = \frac{ 5 }{ 7 }$ )

(vi) 9 ( $\frac{ 42 }{ 54 } = \frac{ 42 \div 6 }{ 54 \div 6 } = \frac{ 7 }{ 9 }$ )

#### Practise This Question

The perimeter and breadth of a rectangle is 84 cm and 8 cm respectively. Find the length of the rectangle.