RD Sharma Solutions Class 8 Rational Numbers Exercise 1.3

RD Sharma Class 8 Solutions Chapter 1 Ex 1.3 PDF Free Download

RD Sharma Solutions Class 8 Chapter 1 Exercise 1.3

Exercise 1.3

Q-1. Subtract the first rational number from the second in each of the following:

(i) \(\frac{ 3 }{ 8 } , \frac{ 5 }{ 8 }\)

(ii) \(\frac{ -7 }{ 9 } , \frac{ 4 }{ 9 }\)

(iii) \(\frac{ -2 }{ 11 } , \frac{ -9 }{ 11 }\)

(iv) \(\frac{ 11 }{ 13 } , \frac{ -4 }{ 13 }\)

(v) \(\frac{ 1 }{ 4 } , \frac{ -3 }{ 8 }\)

(vi) \(\frac{ -2 }{ 3 } , \frac{ 5 }{ 6 }\)

(vii) \(\frac{ -6 }{ 7 } , \frac{ -13 }{ 14 }\)

(viii) \(\frac{ -8 }{ 33 } , \frac{ -7 }{22 }\)

Solution:

(i) \(\frac{ 3 }{ 8 }, \frac{ 5 }{ 8 }\)

= \(\frac{ 5 }{ 8 } – \frac{ 3 }{ 8 }\)

= \( \frac{ 5 – 3 }{ 8 }\) = \( \frac{ 2 }{ 8 }\) = \( \frac{ 1 }{ 4 }\)

(ii) \(\frac{ -7 }{ 9 }, \frac{ 4 }{ 9 }\)

= \(\frac{ 4 }{ 9 } – \frac{ -7}{ 9 }\)

= \( \frac{ 4 + 7 }{ 9 }\) = \( \frac{ 11 }{ 9 }\)

(iii) \(\frac{ -2 }{ 11 },\frac{ -9 }{ 11 }\)

= \(\frac{ -9 }{ 11 } – \frac{ -2 }{ 11 }\) = \( \frac{ -9 + 2 }{ 11 }\) = \( \frac{ -7 }{ 11 }\)

(iv) \(\frac{ 11 }{ 13 }, \frac{ -4 }{ 13 }\)

= \(\frac{ -4 }{ 13 } – \frac{ 11 }{ 13 }\) = \( \frac{ -4 – 11 }{ 13 }\) = \( \frac{ -15 }{ 13 }\)

(v) \(\frac{ 1 }{ 4 }, \frac{ -3 }{ 8 }\)

= \(\frac{ -3 }{ 8 } – \frac{ 1 }{ 4 }\)

= \(\frac{ -3 }{ 8 } – \frac{ 1 \times 2 }{ 4 \times 2 }\)

= \(\frac{ -3 }{ 8 } – \frac{ 2 }{ 8 }\) = \( \frac{ -3 – 2 }{ 8 }\) = \( \frac{ -5 }{ 8 }\)

(vi) \(\frac{ -2 }{ 3 }, \frac{ 5 }{ 6 }\)

= \(\frac{ 5 }{ 6 } – \frac{ -2 }{ 3 }\)

= \(\frac{ 5 }{ 6 } – \frac{ -2 \times 2 }{ 3 \times 2 }\)

= \(\frac{ 5 }{ 6 } – \frac{ -4 }{ 6 }\) = \( \frac{ 5 + 4 }{ 6 }\) = \( \frac{ 9 }{ 6 }\) = \( \frac{ 3 }{ 2 }\)

(vii) \(\frac{ -6 }{ 7 }, \frac{ -13 }{ 14 }\)

= \(\frac{ -13 }{ 14 } – \frac{ -6 }{ 7 }\)

= \(\frac{ -13 }{ 14 } – \frac{ -6 \times 2 }{ 7 \times 2 }\)

= \(\frac{ -13 }{ 14 } – \frac{ -12 }{ 14 }\) = \( \frac{ -13 + 12 }{ 14 }\) = \( \frac{ -1 }{ 14 }\)

(viii) \(\frac{ -8 }{ 33 }, \frac{ -7 }{ 22 }\)

= \(\frac{ -7 }{ 22 } – \frac{ -8 }{ 33 }\)

= \(\frac{ -7 \times 3 }{ 22 \times 3 } – \frac{ -8 \times 2 }{ 33 \times 2 }\)

= \(\frac{ -21 }{ 66 } – \frac{ -16 }{ 66 }\) = \( \frac{ -21 + 16 }{ 66 }\) = \( \frac{ -5 }{ 66 }\)

Q-2. Evaluate each of the following:

(i) \(\frac{ 2 }{ 3 } – \frac{ 3 }{ 5 }\)

(ii) \(\frac{ -4 }{ 7 } – \frac{ 2 }{ -3 }\)

(iii) \(\frac{ 4 }{ 7 } – \frac{ -5 }{-7 }\)

(iv) \(\frac{ -2 }{ 1 } – \frac{ 5 }{ 9 }\)

(v) \(\frac{ -3 }{ -8 } – \frac{ -2 }{ 7 }\)

(vi) \(\frac{ -4 }{ 13 } – \frac{ -5 }{ 26 }\)

(vii) \(\frac{ -5 }{ 14 } – \frac{ -2 }{ 7 }\)

(viii) \(\frac{ 13 }{ 15 } – \frac{ 12 }{ 25 }\)

(ix) \(\frac{ -6 }{ 13 } – \frac{ -7 }{ 13 }\)

(x) \(\frac{ 7 }{ 24 } – \frac{ 19 }{ 36 }\)

(xi) \(\frac{ 5 }{ 63 } – \frac{ -8 }{ 21 }\)

Solution:

(i) \(\frac{ 2 }{ 3 } – \frac{ 3 }{ 5 }\)

= \(\frac{ 2 \times 5 }{ 3 \times 5 } – \frac{ 3 \times 3 }{ 5 \times 3 }\)

= \(\frac{ 10 }{ 15 } – \frac{ 9 }{ 15 }\)

= \(\frac{ 10 – 9 }{ 15 }\) = \( \frac{ 1 }{ 15 }\)

(ii) \(\frac{ -4 }{ 7 } – \frac{ 2 }{ -3 }\)

= \(\frac{ -4 \times 3 }{ 7 \times 3 } – \frac{ -2 \times 7 }{ 3 \times 7 }\)

= \(\frac{ -12 }{ 21 } – \frac{ -14 }{ 21 }\)

= \(\frac{ -12 + 14 }{ 21 }\) = \( \frac{ 2 }{ 21 }\)

(iii) \(\frac{ 4 }{ 7 } – \frac{ -5 }{ -7 }\)

= \(\frac{ 4 }{ 7 } – \frac{ 5 }{ 7 }\)

= \(\frac{ 4 – 5 }{ 7 }\) = \( \frac{ -1 }{ 7 }\)

(iv) \(\frac{ -2 }{ 1 } – \frac{ 5 }{ 9 }\)

= \(\frac{ -2 \times 9 }{ 1 \times 9 } – \frac{ 5 }{ 9 }\)

= \(\frac{ -18 }{ 9 } – \frac{ 5 }{ 9 }\)

= \(\frac{ -18 – 5 }{ 9 }\) = \( \frac{ -23 }{ 9 }\)

(v) \(\frac{ -3 }{ -8 } – \frac{ -2 }{ 7 }\)

= \(\frac{ 3 \times 7 }{ 8 \times 7 } – \frac{ -2 \times 8 }{ 7 \times 8 }\)

= \(\frac{ 21 }{ 56 } – \frac{ -16 }{ 56 }\)

= \(\frac{ 21 + 16 }{ 56 }\) = \( \frac{ 37 }{ 56 }\)

(vi) \(\frac{ -4 }{ 13 } – \frac{ -5 }{ 26 }\)

= \(\frac{ -4 \times 2 }{ 13 \times 2 } – \frac{ -5 }{ 26 }\)

= \(\frac{ -8 }{ 26 } – \frac{ -5 }{ 26 }\)

= \(\frac{ -8 + 5 }{ 26 }\) = \( \frac{ -3 }{ 26 }\)

(vii) \(\frac{ -5 }{ 14 } – \frac{ -2 }{ 7 }\)

= \(\frac{ -5 }{ 14 } – \frac{ -2 \times 2 }{ 7 \times 2 }\)

= \(\frac{ -5 }{ 14 } – \frac{ -4 }{ 14 }\)

= \(\frac{ -5 + 4 }{ 14 }\) = \( \frac{ -1 }{ 14 }\)

(viii) \(\frac{ 13 }{ 15 } – \frac{ 12 }{ 25 }\)

= \(\frac{ 13 \times 5 }{ 15 \times 5 } – \frac{ 12 \times 3 }{ 25 \times 3 }\)

= \(\frac{ 65 }{ 75 } – \frac{ 36 }{ 75 }\)

= \(\frac{ 65 – 36 }{ 75 }\) = \( \frac{ 29 }{ 75 }\)

(ix) \(\frac{ -6 }{ 13 } – \frac{ -7 }{ 13 }\)

= \(\frac{ -6 + 7 }{ 13 }\) = \( \frac{ 1 }{ 13 }\)

(x) \(\frac{ 7 }{ 24 } – \frac{ 19 }{ 36 }\)

= \(\frac{ 7 \times 3 }{ 24 \times 3 } – \frac{ 19 \times 2 }{ 36 \times 2 }\)

= \(\frac{ 21 }{ 72 } – \frac{ 38 }{ 72 }\)

= \(\frac{ 21 – 38 }{ 72 }\) = \( \frac{ -17 }{ 72 }\)

(xi) \(\frac{ 5 }{ 63 } – \frac{ -8 }{ 21 }\)

= \(\frac{ 5 }{ 63 } – \frac{ -8 \times 3 }{ 21 \times 3 }\)

= \(\frac{ 5 }{ 63 } – \frac{ -24 }{ 63 }\)

= \(\frac{ 5 + 24 }{ 63 }\) = \( \frac{ 29 }{ 63 }\)

Q-3. The sum of the two numbers is \(\frac{ 5 }{ 9 }\). If one of the numbers is \(\frac{ 1 }{ 3 }\), Find the other.

Solution:

Given: Sum of the two numbers is \( \frac{ 5 }{ 9 }\).

One of the numbers is \( \frac{ 1 }{ 3 }\)

Let us say, the other number be x.

\( x + \frac{ 1 }{ 3 } = \frac{ 5 }{ 9 }\)

\(\Rightarrow x = \frac{ 5 }{ 9 } – \frac{ 1 }{ 3 }\)

\(\Rightarrow x = \frac{ 5 – 3 }{ 9 }\)

\(\Rightarrow x = \frac{ 2 }{ 9 }\)

Hence, the other number is \( \frac{ 2 }{ 9 }\).

Q-4. The sum of the two numbers is \(\frac{ -1 }{ 3 }\). If one of the numbers is \(\frac{ -12 }{ 3 }\), Find the other.

Solution:

Given : Sum of the two numbers is \( \frac{ -1 }{ 3 }\).

One of the numbers is \( \frac{ -12 }{ 3 }\)

Let x be the other number

\( x + \frac{ -12 }{ 3 } =\frac{ -1 }{ 3 } \)

\(\Rightarrow x = \frac{ -1 }{ 3 } – \frac{ -12 }{ 3 } \)

\(\Rightarrow x = \frac{ -1 + 12 }{ 3 }\)

\(\Rightarrow x = \frac{ 11 }{ 3 }\)

Hence, the other number is \( \frac{ 11 }{ 3 }\).

Q-5. The sum of the two numbers is \(\frac{ -4 }{ 3 }\). If one of the numbers is -5, Find the other.

Solution:

Given:

Sum of the two numbers is \( \frac{ -4 }{ 3 }\).

One of the numbers is -5

Let x be the other number

\( x + \left( -5 \right ) = \frac{ -4 }{ 3 } \)

\(\Rightarrow x = 5 – \frac{ -4 }{ 3 }\)

\(\Rightarrow x = \frac{ -4 }{ 3 } – \frac{5 \times 3 }{ 1 \times 3 } \)

\(\Rightarrow x = \frac{ -4 + 15 }{ 3 }\)

\(\Rightarrow x = \frac{ 11 }{ 3 }\)

Hence, the other number is \( \frac{ 11 }{ 3 }\).

Q-6. The sum of the two rational numbers is -8. If one of the numbers is \(\frac{ -15 }{ 7 }\), Find the other.

Solution: It is given that,

The sum of the two numbers is -8.

One of the number is \( \frac{ -15 }{ 7 }\)

Let, the other number be x.

\(∴ x + \frac{ -15 }{ 7 } = \left( -8 \right ) \)

\(\Rightarrow x = -8 – \frac{ -15 }{ 7 }\)

\(\Rightarrow x = \frac{ -8 \times 7 }{ 1 \times 7 } – \frac{ -15 }{ 7 } \)

\(\Rightarrow x = \frac{ -56 + 15 }{ 7 }\)

\(\Rightarrow x = \frac{ -41 }{ 7 }\)

Hence, the other number is \( \frac{ -41 }{ 7 }\).

Q-7. What should be added to \(\frac{ -7 }{ 8 }\) so as to get \(\frac{ 5 }{ 9 }\)?

Solution:

The sum of the two numbers is \( \frac{ 5 }{ 9 }\)

One of the numbers is \( \frac{ -7 }{ 8 }\)

Let, the other number be x.

\(∴ x + \frac{ -7 }{ 8 } = \frac{ 5 }{ 9 }\)

\(\Rightarrow x = \frac{ 5 }{ 9 } – \frac{ -7 }{ 8 } \)

\(\Rightarrow x = \frac{ 5 \times 8 }{ 9 \times 8 } – \frac{ -7 \times 9 }{ 8 \times 9 } \)

\(\Rightarrow x = \frac{ 40 }{ 72 } – \frac{ -63 }{ 72 } \)

\(\Rightarrow x = \frac{ 40 + 63 }{ 72 }\)

\(\Rightarrow x = \frac{ 103 }{ 72 }\)

Hence, the other number is \( \frac{ 103 }{ 72 }\).

Q-8. What number should be added to \(\frac{ -5 }{ 11 }\) so as to get \(\frac{ 26 }{ 33 }\)?

Solution:

It is given that:

The sum of the two numbers is \( \frac{ 26 }{ 33 }\)

One of the numbers is \( \frac{ -5 }{ 11 }\)

Let, the other number be x.

\(∴ x + \frac{ -5 }{ 11 } \) = \( \frac{ 26 }{ 33 }\)

\(\Rightarrow x = \frac{ 26 }{ 33 } – \frac{ -5 }{ 11 } \)

\(\Rightarrow x = \frac{ 26 }{ 33 } – \frac{ -5 \times 3 }{ 11 \times 3 } \)

\(\Rightarrow x = \frac{ 26 }{ 33 } – \frac{ -15 }{ 33 } \)

\(\Rightarrow x = \frac{ 26 + 15 }{ 33 }\)

\(\Rightarrow x = \frac{ 41 }{ 33 }\)

Hence, the other number is \( \frac{ 41 }{ 33 }\).

Q-9. What number should be added to \(\frac{ -5 }{ 7 }\) to get \(\frac{ -2 }{ 3 }\)?

Solution:

It is given that:

The sum of the two numbers is \( \frac{ -2 }{ 3 }\)

One of the numbers is \( \frac{ -5 }{ 7 }\)

Let, the other number be x.

\(∴ x + \frac{ -5 }{ 7 } = \frac{ -2 }{ 3 }\)

\(\Rightarrow x = \frac{ -2 }{ 3 } – \frac{ -5 }{ 7 } \)

\(\Rightarrow x = \frac{ -2 \times 7 }{ 3 \times 7 } – \frac{ -5 \times 3 }{ 7 \times 3 } \)

\(\Rightarrow x = \frac{ -14 }{ 21 } – \frac{ -15 }{ 21 } \)

\(\Rightarrow x = \frac{ -14 + 15 }{ 21 }\)

\(\Rightarrow x = \frac{ 1 }{ 21 }\)

Hence, the other number is \( \frac{ 1 }{ 21 }\).

Q-10. What number should be subtracted from \(\frac{ -5 }{ 3 }\) to get \(\frac{ 5 }{ 6 }\)?

Solution:

Let the number be x.

As per statement,

\( \frac{ -5 }{ 3 } – x = \frac{ 5 }{ 6 }\)

Or x = -5/3 – 5/6

x = (-10-5)/6

x = -15/6

The number is -15/6.

Q-11. What number must be subtracted from \(\frac{ 3 }{ 7 }\) to get \(\frac{ 5 }{ 4 }\)?

Solution:

Let x be the unknown number.

As per statement, we have

\(\frac{ 3 }{ 7 } – x = \frac{ 5 }{ 4 }\)

\(\Rightarrow x = \frac{ 3 }{ 7 } – \frac{ 5 }{ 4 } \)

\(\Rightarrow x = \frac{ 3 \times 4 }{ 7 \times 4 } – \frac{ 5 \times 7 }{ 4 \times 7 } \)

\(\Rightarrow x = \frac{ 12 }{ 28 } – \frac{ 35 }{ 28 } \)

\(\Rightarrow x = \frac{ 12 – 35 }{ 28 }\)

\(\Rightarrow x = \frac{ -23 }{ 28 }\)

Hence, the other number is \( \frac{ -23 }{ 28 }\).

Q-12. What should be added to \(\left (\frac{ 2 }{ 3 } + \frac{ 3 }{ 5 } \right )\) to get \(\frac{ -2 }{ 15 }\)?

Solution:

It is given that:

The sum of the numbers is \( \frac{ -2 }{ 15 }\)

One of the numbers is \(\left (\frac{ 2 }{ 3 } + \frac{ 3 }{ 5 } \right )\)

Let, the other number be x.

\(∴ \frac{ 2 }{ 3 } + \frac{ 3 }{ 5 } + x = \frac{ -2 }{ 15 }\)

\(\Rightarrow x = \frac{ -2 }{ 15 } – \frac{ 2 }{ 3 } – \frac{ 3 }{ 5 } \)

\(\Rightarrow x = \frac{ -2 }{ 15 } – \frac{ 2 \times 5 }{ 3 \times 5 } – \frac{ 3 \times 3 }{ 5 \times 3 } \)

\(\Rightarrow x = \frac{ -2 }{ 15 } – \frac{ 10 }{ 15 } – \frac{ 9 }{ 15 } \)

\(\Rightarrow x = \frac{ -2 – 10 – 9 }{ 15 }\)

\(\Rightarrow x = \frac{ -21 }{ 15 }\)

\(\Rightarrow x = \frac{ -7 }{ 5 }\)

Hence, the other number is \( \frac{ -7 }{ 5 }\).

Q-13. What should be added to \(\left ( \frac{ 1 }{ 2 } + \frac{ 1 }{ 3 } + \frac{ 1 }{ 5 } \right )\) to get 3?

Solution:

Given:

The sum of the numbers is 3

One of the numbers is \(\left (\frac{ 1 }{ 2 } + \frac{ 1 }{ 3 } + \frac{ 1}{ 5 } \right ) \)

Now, Let x be the other number.

\(\frac{ 1 }{ 2 } + \frac{ 1 }{ 3 } + \frac{ 1}{ 5 } + x = 3 \)

\(\Rightarrow x = 3 – \frac{ 1 }{ 2 } – \frac{ 1 }{ 3 } – \frac{ 1 }{ 5 } \)

\(\Rightarrow x = \frac{ 3 \times 30 }{ 1 \times 30 } – \frac{ 1 \times 15 }{ 2 \times 15 } – \frac{ 1 \times 10 }{ 3 \times 10 } – \frac{ 1 \times 6 }{ 5 \times 6 } \)

\(\Rightarrow x = \frac{ 90 }{ 30 } – \frac{ 15 }{ 30 } – \frac{ 10 }{ 30 } – \frac{ 6 }{ 30 } \)

\(\Rightarrow x = \frac{ 90 – 15 – 10 – 6 }{ 30 }\)

\(\Rightarrow x = \frac{ 59 }{ 30 }\)

Hence, the other number is \( \frac{ 59 }{ 30 }\).

Q-14. What should be subtracted from \(\left ( \frac{ 3 }{ 4 } – \frac{ 2 }{ 3 } \right )\) to get \(\frac{ -1 }{ 6 }\)?

Solution:

It is given that:

The numbers is \( \frac{ -1 }{ 6 }\)

One of the numbers is \(\left (\frac{ 3 }{ 4 } – \frac{ 2 }{ 3 } \right )\)

Let, the other number be x.

\(∴ \frac{ 3 }{ 4 } – \frac{ 2 }{ 3 } – x = \frac{ -1 }{ 6 }\)

\(\Rightarrow x = \frac{ 3 }{ 4 } – \frac{ 2 }{ 3 } – \frac{ -1 }{ 6 }\)

\(\Rightarrow x = \frac{ 3 \times 3 }{ 4 \times 3 } – \frac{ 2 \times 4 }{ 3 \times 4 } – \frac{ -1 \times 2 }{ 6 \times 2 } \)

\(\Rightarrow x = \frac{ 9 }{ 12 } – \frac{ 8 }{ 12 } + \frac{ 2 }{ 12 } \)

\(\Rightarrow x = \frac{ 2 – 8 + 9 }{ 12 }\)

\(\Rightarrow x = \frac{ 3 }{ 12 }\)

\(\Rightarrow x = \frac{ 1 }{ 4 }\)

Hence, the other number is \( \frac{ 1 }{ 4 }\).

Q-15. Fill in the blanks:

(i) \(\frac{ -4 }{ 13 } – \frac{ -3 }{ 26 } = …….\)

(ii) \(\frac{ -9 }{ 14 } + ……. = -1\)

(iii) \(\frac{ -7 }{ 9 } + ……. = 3\)

(iv) \(……. + \frac{ 15}{ 23 } = 4\)

Solution:

(i) \(\frac{ -4 }{ 13 } – \frac{ -3 }{ 26 } = ……….. \)

Let the required number be x.

\(\frac{ -4 }{ 13 } – \frac{-3 }{ 26 } = x\)

\(\Rightarrow \frac{ -4 \times 2 }{ 13 \times 2 } – \frac{-3 }{ 26 } = x\)

\(\Rightarrow \frac{ -8 }{ 26 } – \frac{-3 }{ 26 } = x\)

\(\Rightarrow x = \frac{ -8 + 3 }{ 26 }\)

\(\Rightarrow x = \frac{ -5 }{ 26 }\)

\(\frac{ -4 }{ 13 } – \frac{ -3 }{ 26 } = \frac{ -5 }{ 26 } \)

(ii) \(\frac{-9}{14} + ……. = -1\)

Let, the required number be x.

\(\frac{-9}{14} + x = -1\)

\(\Rightarrow x = -1 – \frac{-9}{14}\)

\(\Rightarrow x = \frac{-1 \times 14 }{ 1 \times 14 } – \frac{-9}{14}\)

\(\Rightarrow x = \frac{ -14 + 9 }{ 14 }\)

\(\Rightarrow x = \frac{ -5 }{ 14 }\)

\(\frac{-9}{14} + \frac{ -5 }{ 14 }= -1\)

(iii) \(\frac{ -7 }{ 9 } + …… = 3\)

\(\Rightarrow \frac{ -7 }{ 9 } + x = 3\)

\(\Rightarrow x = 3 – \frac{ -7 }{ 9 }\)

\(\Rightarrow x = \frac{ 3 \times 9 }{1 \times 9 } – \frac{ -7 }{ 9 }\)

\(\Rightarrow x = \frac{ 27 }{ 9 } – \frac{ -7 }{ 9 }\)

\(\Rightarrow x = \frac{ 27 + 7 }{ 9 }\)

\(\Rightarrow x = \frac{ 34 }{ 9 }\)

Hence, \(\frac{ -7 }{ 9 } + \frac{ 34 }{ 9 } = 3\)

(iv) \(…… + \frac{ 15 }{ 23 } = 4\)

Let the required number be x.

\(\Rightarrow x + \frac{ 15 }{ 23 } = 4\)

\(\Rightarrow x = 4 – \frac{ 15 }{ 23 }\)

\(\Rightarrow x = \frac{ 92 – 15 }{ 23 }\)

\(\Rightarrow x = \frac{ 77 }{ 23 }\)

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