RD Sharma Solutions Class 8 Rational Numbers Exercise 1.7

RD Sharma Class 8 Solutions Chapter 1 Ex 1.7 PDF Free Download

RD Sharma Solutions Class 8 Chapter 1 Exercise 1.7

Exercise 1.7

Q-1. Divide:

(i) \(1\;  by \; \frac{1}{2}\)

(ii) \(5 \; by\;  \frac{-5}{7}\)

(iii) \(\frac{ -3 }{ 4 } \; by \; \frac{1}{2}\)

(iv) \(\frac{ -7 }{ 8 } \; by \; \frac{-21}{ 16 }\)

(v) \(\frac{ 7 }{ -4 } \; by \; \frac{ 63 }{ 64 }\)

(vi) \(0 \; by \; \frac{ -7 }{ 5 }\)

(vii) \(\frac{ -3 }{ 4 } \; by \; 6 \)

(viii) \(\frac{ 2 }{ 3 } \; by \; \frac{ -7 }{ 12 }\)

(ix) \(-4 \; by \; \frac{ -3 }{ 5 }\)

(x) \(\frac{ -3 }{ 13 } \; by \; \frac{ -4 }{ 65 }\)

Solution: 

(i) \(1 \div \frac{ 1 }{ 2 } = 1 \times \frac{ 2 }{ 1 } = 2\)

(ii) \(5 \div \frac{ -5 }{ 7 } = 5 \times \frac{ 7 }{ -5 } = -7\)

(iii) \(\frac{ -3 }{ 4 } \div \frac{ 9 }{ -16} = \frac{ -3 }{ 4 } \times \frac{ -16 }{ 9 } = \frac{ 4 }{ 3 }\)

(iv) \(\frac{ -7 }{ 8 } \div \frac{ -21 }{ 16} = \frac{ -7 }{ 8 } \times \frac{ -16 }{ 21 } = \frac{ 2 }{ 3 }\)

(v) \(\frac{ -7 }{ 4 } \div \frac{ 63 }{ 64 } = \frac{ 7  }{ -4 } \times \frac{ 64 }{ 63 } = \frac{ -16 }{ 9 }\)

(vi) \(0 \div \frac{ -7 }{ 5 } = 0 \times \frac{ -5 }{ 7 } = 0\)

(vii) \( \frac{ -3 }{ 4 } \div -6 = \frac{ -3 }{ 4 } \times \frac{ 1 }{ -6 } = \frac{ 1 }{ 8 }\)

(viii) \(\frac{ 2 }{ 3 } \div \frac{ -7 }{ 12 } = \frac{ 2 }{ 3 } \times \frac{ 12 }{ -7 } = \frac{ -8 }{ 7 }\)

(ix) \(-4  \div \frac{ -3 }{ 5 } = -4 \times \frac{ 5 }{ -3 } = \frac{ 20 }{ 3 }\)

(x) \(\frac{ -3 }{ 13 } \div \frac{ -4 }{ 65 } = \frac{ -3 }{ 13 } \times \frac{ 65 }{ -4 } = \frac{ 15 }{ 4 }\)

Q-2. Find the value and express as a rational number in standard form:

(i) \(\frac{ 2 }{ 5 } \div \frac{ 26 }{ 15 }\)

(ii) \(\frac{ 10 }{ 3 } \div \frac{ -35 }{ 12 }\)

(iii) \( -6  \div \frac{ -8 }{ 17 }\)

(iv) \(\frac{ -40 }{ 99 } \div \left( -20 \right )\)

(v) \(\frac{ -22 }{ 27 } \div \frac{ -110 }{ 18 }\)

(vi) \(\frac{ -36 }{ 125 } \div \frac{ -3 }{ 75 }\)

Solution:

(i) \(\frac{ 2 }{ 5 } \div \frac{ 26 }{ 15 } = \frac{ 2 }{ 5 } \times \frac{ 15 }{ 26 } = \frac{ 3 }{ 13 }\)

(ii) \(\frac{ 10 }{ 3 } \div \frac{ -35 }{ 12 } = \frac{ 10 }{ 3 } \times \frac{ 12 }{ -35 } = \frac{ -8 }{ 7 }\)

(iii) \( -6 \div \frac{ -8 }{ 17 } = -6 \times \frac{ 17 }{ -8 } = \frac{ 51 }{ 4 }\)

(iv) \(\frac{ -40 }{ 99 } \div \left( -20 \right ) = \frac{ -40 }{ 99 } \times \frac{ 1 }{ -20 } = \frac{ 2 }{ 99 }\)

(v) \(\frac{ -22 }{ 27 } \div \frac{ -110 }{ 18 } = \frac{ -22 }{ 27 } \times \frac{ 18 }{ -110 } = \frac{ 2 }{ 15 }\)

(vi) \(\frac{ -36 }{ 125 } \div \frac{ -3 }{ 75 } = \frac{ -36 }{ 125 } \times \frac{ 75 }{ -3 } = \frac{ 36 }{ 5 }\)

Q-3. The product of two rational numbers is 15. If one  of the numbers  is -10. Find the other number.

Solution:

Let, the other number be x.

So, \(x \times \left ( -10 \right ) = 15\)

\(\Rightarrow x = \frac{ 15 }{ -10 } = \frac{ 3 }{ -2 }\)

So, the other number is \(\frac{ -3 }{ 2 }\).

Q-4. The product of two rational numbers is \(\frac{-8 }{ 9 }\). If one of the number is \(\frac{ -4 }{ 15 }\), Find the other number.

Solution: Let, the other number be x.

So, \(x \times \frac{ -4 }{ 15 } = \frac{ -8 }{ 9 }\)

\(\Rightarrow x = \frac{ -8 }{ 9 } \div \frac{ -4 }{ 15 }\)

\(\Rightarrow x = \frac{ -8 }{ 9 } \times \frac{ 15 }{ -4  }\)

\(\Rightarrow x = \frac{ 10 }{ 3 } \)

Thus, the other number is \(\frac{ 10 }{ 3 }\)

Q-5. By what number should we multiply \(\frac{ -1 }{ 6 }\) so that the product may be \(\frac{ -23 }{ 9 }\)?

Solution: 

Let, the number be x.

\(x \times \frac{ -1 }{ 6 } = \frac{ -23 }{ 9 }\)

\(\Rightarrow x = \frac{ -23 }{ 9 } \div \frac{ -1 }{ 6 }\)

\(\Rightarrow x = \frac{ -23 }{ 9 } \times \frac{ 6 }{ -1  }\)

\(\Rightarrow x = \frac{ 46 }{ 3 } \)

Thus, the other number is \(\frac{ 46 }{ 3 }\)

Q-6. By what number should we multiply \(\frac{ -15 }{ 28 }\) so that the product may be \(\frac{ -5 }{ 7 }\)?

Solution:

Let, the number be x.

\(x \times \frac{ -15 }{ 28 } = \frac{ -5 }{ 7 }\)

\(\Rightarrow x = \frac{ -5 }{ 7 } \div \frac{ -15 }{ 28 }\)

\(\Rightarrow x = \frac{ -5 }{ 7 } \times \frac{ 28 }{ -15  }\)

\(\Rightarrow x = \frac{ 4 }{ 3 } \)

Thus, the other number is \(\frac{ 4 }{ 3 }\)

Q-7. By what number should we multiply \(\frac{ -8 }{ 13 }\) so that the product may be 24?

Solution:

Let, the number be x.

\(x \times \frac{ -8 }{ 13 } = 24\)

\(\Rightarrow x = 24 \div \frac{ -8 }{ 13 }\)

\(\Rightarrow x = 24 \times \frac{ 13 }{ -8  }\)

\(\Rightarrow x = -39 \)

Thus, the other number is -39.

Q-8. By what number should \(\frac{ -3 }{ 4 }\) be multiplied in order to produce  \(\frac{ 2 }{ 3 }\)?

Solution:

Let, the other number that should be multiplied with \(\frac{ -3 }{ 4 }\) to produce \(\frac{ 2 }{ 3 }\) be x.

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

\(\Rightarrow x = \frac{ 2 }{ 3 } \div \frac{ -3 }{ 4 }\)

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

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

Thus, the other number is \(\frac{ -8 }{ 9 }\)

Q-9. Find \(\left ( x + y \right ) \div \left ( x – y \right )\), if

(i) \(x = \frac{ 2 }{ 3 } , y = \frac{ 3 }{ 2 }\)

 (ii) \(x = \frac{ 2 }{ 5 } , y = \frac{ 1 }{ 2 }\)

(iii) \(x = \frac{ 5 }{ 4 } , y = \frac{ -1 }{ 3 }\)

(iv) \(x = \frac{ 2 }{ 7 } , y = \frac{ 4 }{ 3 }\)

(v) \(x = \frac{ 1 }{ 4 } , y = \frac{ 3 }{ 2 }\)

Solution:

(i) \(\left ( x + y \right ) \div \left ( x – y \right )\)

= \(\left ( \frac{ 2 }{ 3 } + \frac{ 3 }{ 2 } \right ) \div \left ( \frac{ 2 }{ 3 } – \frac{ 3 }{ 2 } \right )\)

= \(\frac{13}{6} \times \frac{6}{-5}\) = \(\frac{-13}{5} \)

Thus, \(\left ( x + y \right ) \div \left ( x – y \right ) = \frac{ -13 }{ 5 }\)

(ii) \(\left ( x + y \right ) \div \left ( x – y \right )\)

= \(\left ( \frac{ 2 }{ 5 } + \frac{ 1 }{ 2 } \right ) \div \left ( \frac{ 2 }{ 5 } – \frac{ 1 }{ 2 } \right )\)

= \(\frac{9}{10} \times \frac{10}{-1}\) = -9

Thus, \(\left ( x + y \right ) \div \left ( x – y \right ) = -9 \)

(iii) \(\left ( x + y \right ) \div \left ( x – y \right )\)

= \(\left ( \frac{ 5 }{ 4 } + \frac{ -1 }{ 3 } \right ) \div \left ( \frac{ 5 }{ 4 } – \frac{ -1 }{ 3 } \right )\)

= \(\frac{11}{12} \times \frac{12}{11}\) = \(\frac{ 11 }{ 19 }\)

Thus, \(\left ( x + y \right ) \div \left ( x – y \right ) = \frac{11 }{ 19 } \)

(iv) \(\left ( x + y \right ) \div \left ( x – y \right )\)

= \(\left ( \frac{ 2 }{ 7 } + \frac{ 4 }{ 3 } \right ) \div \left ( \frac{ 2 }{ 7 } – \frac{ 4 }{ 3 } \right )\)

= \(\frac{34}{21} \times \frac{21}{-22}\) = \(\frac{-17}{11}\)

Thus, \(\left ( x + y \right ) \div \left ( x – y \right ) = \frac{ -17 }{ 11 } \)

(v) \(\left ( x + y \right ) \div \left ( x – y \right )\)

= \(\left ( \frac{ 1 }{ 4 } + \frac{ 3 }{ 2 } \right ) \div \left ( \frac{ 1 }{ 4 } – \frac{ 3 }{ 2 } \right )\)

= \(\frac{7}{4} \times \frac{4}{-5}\) = \(\frac{ -7 }{ 5 }\)

Thus, \(\left ( x + y \right ) \div \left ( x – y \right ) = \frac{ -7 }{ 5 } \)

Q-10: The cost of \(7\frac{2}{3}\) metres of rope is Rs \(12\frac{3}{4}\). Find its cost per metres.

Solution: The cost of \(7\frac{2}{3}\) metres of rope is Rs. \(7\frac{2}{3}\).

Therefore,

Cost per metre = \(7\frac{2}{3} \div 7\frac{2}{3} \)

\(= \frac{ 51 }{ 4 } \div \frac{ 23 }{ 3 }\)

\(= \frac{ 51 }{ 4 } \times \frac{ 3 }{ 23 }\)

\(= \frac{ 153 }{ 92 }\) = Rs. \(1 \frac{ 61 }{ 92 }\)

Hence, the cost of rope per metres = Rs. \(1 \frac{ 61 }{ 92 }\)

Q-11. The cost of \(2\frac{1}{3}\) metres of cloth is Rs. \(75\frac{1}{4}\). Find the cost of cloth per metres.

Solution: The cost of \(2\frac{1}{3}\) metres of cloth is Rs. \(75\frac{1}{4}\).

Therefore,

Cost per metre = \(75\frac{1}{4} \div 2\frac{1}{3} \)

\(= \frac{ 301 }{ 4 } \div \frac{ 7 }{ 3 }\)

\(= \frac{ 301 }{ 4 } \times \frac{ 3 }{ 7 }\)

\(= \frac{ 129 }{ 4 }\) = Rs. \(32 \frac{ 1 }{ 4 }\)

Thus, Rs. \(32 \frac{ 1 }{ 4 }\) or Rs. 32.25 is the cost of cloth per metre.

Q-12. By what number should \(\frac{ -33 }{ 16 }\) be divided to get \(\frac{ -11 }{ 4 }\)?

Solution:

Let, the other number  be x.

\( \frac{ -33 }{ 16 } \div x = \frac{ -11 }{ 4 }\)

\(\Rightarrow \frac{ -33 }{ 16 } \times  \frac{ 1 }{ x } = \frac{ -11 }{ 4 }\)

\(\Rightarrow \frac{1} {x} = \frac{ -11 }{ 4 } \times \frac{ 16 }{ -33  }\)

\(\Rightarrow \frac{ 1 }{ x } = \frac{ 4 }{ 3 } \)

\(\Rightarrow x = \frac{ 3 }{ 4 }\)

Thus, the other number is \(\frac{ 3 }{ 4 }\)

Q-13. Divide the sum of \(\frac{ -13 }{ 5 }\) and \(\frac{ 12 }{ 7 }\) by the product of \(\frac{ -31 }{ 7 }\) and \(\frac{ -1 }{ 2 }\)?

Solution:

\(\left ( \frac{ -13 }{ 5 } + \frac{ 12 }{ 7 } \right ) \div \left ( \frac{ -31 }{ 7 } \times \frac{ -1 }{ 2 } \right )\)

= \(\frac{ -13 \times 7 + 12 \times 5 }{ 35 } \div \frac{ 31 }{ 14 }\)

= \(\frac{ -91 + 60 }{ 35 } \div \frac{31 }{ 14 }\)

= \(\frac{ -31 }{ 35 } \times \frac{14 }{ 31 }\) = \(\frac{ -2 }{ 5 }\)

Q-14. Divide the sum of \(\frac{ 65 }{ 12 }\) and \(\frac{ 12 }{ 7 }\) by their differences.

Solution:

\(\left ( \frac{ 65 }{ 12} + \frac{ 12 }{ 7 } \right ) \div \left ( \frac{ 65 }{ 12 } – \frac{ 12 }{ 7 } \right )\)

= \(\frac{65 \times 7 + 12 \times 12 }{ 84 } \div \frac{ 65 \times 7 – 12 \times 12 }{ 84 }\)

= \(\frac{455 + 144 }{ 84 } \div \frac{ 455 – 144 }{ 84 }\)

= \(\frac{ 599 }{ 84 } \div \frac{ 311 }{ 84 }\)

= \(\frac{ 599 }{ 84 } \times \frac{ 84 }{ 311 }\) = \(\frac{ 599 }{ 311 } \)

Q-15. If 24 trousers of equal size can be prepared in 54 meters of cloth, what length of cloth is required for each trouser?

Solution:

Cloth needed to prepare 24 trousers = 54 m

So,

Length of the cloth required for each trousers = \(54 \div 24\) = \(\frac{ 54 }{ 24 }\) = \(\frac{ 9 }{ 4 }\) m = \(2 \frac{ 1 }{ 4 }\) metres.

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