RD Sharma Solutions Class 8 Rational Numbers Exercise 1.5

RD Sharma Class 8 Solutions Chapter 1 Ex 1.5 PDF Free Download

RD Sharma Solutions Class 8 Chapter 1 Exercise 1.5

Exercise 1.5

 Q-1. Multiply:

(i) \(\frac{ 7 }{ 11 }\;  by\; \frac{ 5 }{ 4 }\)

(ii) \(\frac{ 5}{ 7 } \;by\; \frac{ -3 }{ 4 }\)

(iii) \(\frac{ -2}{ 9 }\; by\; \frac{ 5 }{ 11 }\)

(iv) \(\frac{ -3 }{ 17 } \;by\; \frac{ -5 }{ -4 }\)

(v) \(\frac{ 9 }{ -7 }\; by\; \frac{ 36 }{ -11}\)

(vi) \(\frac{ -11 }{ 13 } \;by\;  \frac{ -21 }{ 7 }\)

(vii) \(\frac{ -3 }{ 5 } \;by\; \frac{ -4 }{ 7 }\)

(viii) \(\frac{ -15 }{ 11 } \;by\; 7 \)

Solution:

(i) \(\frac{7}{11} \times \frac{5}{4} = \frac{7 \times 5}{11 \times 4} = \frac{35}{44}\)

(ii) \(\frac{ 5 }{ 7 } \times \frac{ -3 }{ 4 } = \frac{5 \times -3}{7 \times 4} = \frac{ -15 }{ 28 }\)

(iii) \(\frac{-2}{9} \times \frac{5}{11} = \frac{-2 \times 5}{11 \times 9} = \frac{-10}{99}\)

(iv) \(\frac{ -3 }{ 17 } \times \frac{ -5 }{ -4 } = \frac{-3 \times -5}{17 \times -4} = \frac{ 15 }{ -68 } = \frac{ 15 } {-68 }\)

(v) \(\frac{ 9 }{ -7 } \times \frac{ 36 }{ -11 } = \frac{9 \times 36}{-7 \times -11} = \frac{ 324 }{ 77 }\)

(vi) \(\frac{ -11 }{ 13 } \times \frac{ -21 }{ 7 } = \frac{-11 \times -21}{13 \times 7} = \frac{ 33 }{ 13 }\)

(vii) \(\frac{ -3 }{ 5 } \times \frac{ -4 }{ 7 } = \frac{-3 \times -4}{5 \times 7} = \frac{ 12 }{ 35 }\)

(viii) \(\frac{ -15 }{ 11 } \times 7 = \frac{-15 \times 7}{ 11 } = \frac{ -105 }{ 11 }\)

Q-2. Multiply:

(i) \(\frac{ -5 }{ 17 }\; by\; \frac{ 51 }{ -60 }\)

(ii) \(\frac{ -6 }{ 11 } \;by\; \frac{ -55 }{ 36 }\)

(iii) \(\frac{ -8 }{ 25 }\; by\; \frac{ -5 }{ 16 }\)

(iv) \(\frac{ 6 }{ 7 }\; by\; \frac{ -49 }{ 36 }\)

(v) \(\frac{ 8 }{ -9 }\; by\; \frac{ -7 }{ -16}\)

(vi) \(\frac{ -8 }{ 9 }\; by\; \frac{ 3 }{ 64 }\)

Solution:

(i) \(\frac{ -5 }{ 17 } \times \frac{ 51 }{ -60 } = \frac{-5 \times 51}{17 \times -60} = \frac{ 1}{ 4 }\)

(ii) \(\frac{ -6 }{ 11 } \times \frac{ -55 }{ 36 } = \frac{-6 \times -55}{11 \times 36} = \frac{ 5 }{ 6 }\)

(iii) \(\frac{ -8 }{ 25 } \times \frac{ -5 }{ 16 } = \frac{-8 \times -5}{25 \times 16} = \frac{ 1 }{ 10 }\)

(iv) \(\frac{ 6 }{ 7 } \times \frac{ -49 }{ 36 } = \frac{6 \times -49}{7 \times 36} = \frac{ -7 }{ 6 }\)

(v) \(\frac{ 8 }{ -9 } \times \frac{ -7 }{ -16 } = \frac{8 \times -7}{-9 \times -16} = \frac{ -7 }{ 18 }\)

(vi) \(\frac{ -8 }{ 9 } \times \frac{ 3 }{ 64 } = \frac{-8 \times 3}{9 \times 64} = \frac{ -1 }{ 24 }\)

Q-3. Simplify each of the following and express the result as a rational number in standard form:

(i) \(\frac{ -16 }{ 21 } \times \frac{ 14 }{ 5 }\)

(ii) \(\frac{ 7 }{ 6 } \times \frac{ -3 }{ 28 }\)

(iii) \(\frac{ -19 }{ 36 } \times 16 \)

(iv) \(\frac{ -13 }{ 9 } \times \frac{ 27}{ -26 }\)

(v) \(\frac{ -9 }{ 16 } \times \frac{ -64 }{ -27 }\)

(vi) \(\frac{ -50 }{ 7 } \times \frac{ 14 }{ 3 }\)

(vii) \(\frac{ -11}{ 9 } \times \frac{ -81 }{ -88 }\)

(viii) \(\frac{ -5}{ 9 } \times \frac{ 72 }{ -25 }\)

Solution:

(i) \(\frac{-16}{21}\times \frac{14}{5} = \frac{-2 \times 2 \times 2 \times 2}{3 \times 7} \times \frac{2 \times 7}{5} = \frac{-32}{15}\)

(ii) \(\frac{7}{6}\times \frac{-3}{28} = \frac{7 }{2 \times 3} \times \frac{-3 }{2 \times 2\times 7} = \frac{-1}{8}\)

(iii) \(\frac{-19}{36}\times 16 = \frac{-19}{2 \times 2\times 3\times 3} \times 2 \times 2 \times 2\times 2 = \frac{-76}{9}\)

(iv) \(\frac{-13}{9}\times \frac{27}{-26} = \frac{-13}{3 \times 3} \times \frac{3 \times 3 \times 3}{-2 \times 13} = \frac{-3}{2}\)

(v) \(\frac{-9}{16}\times \frac{-64}{27} = \frac{-9}{16} \times \frac{-4 \times 16}{-3 \times 9} = \frac{-4}{3}\)

(vi) \(\frac{-50}{7}\times \frac{14}{3} = \frac{-50}{ 7} \times \frac{2 \times 7}{3} = \frac{-100}{3}\)

(vii) \(\frac{-11}{9}\times \frac{-81}{-88} = \frac{-11}{3 \times 3} \times \frac{-9 \times 9}{-8 \times 11} = \frac{-9 }{8}\)

(viii) \(\frac{-5}{9}\times \frac{72}{-25} = \frac{-5}{3 \times 3} \times \frac{8 \times 9}{-5 \times 5} = \frac{8}{5}\)

Q-4. Simplify:

(i) \(\left ( \frac{ 25 }{ 8 } \times \frac{ 2 }{ 5 } \right ) – \left ( \frac{ 3 }{ 5 } \times \frac{ -10 }{ 9 } \right )\)

(ii) \(\left ( \frac{ 1 }{ 2 } \times \frac{ 1 }{ 4 } \right ) + \left ( \frac{ 1 }{ 2 } \times 6 \right )\)

(iii) \(\left ( -5 \times \frac{ 2 }{ 15 } \right ) – \left ( -6  \times \frac{ 2 }{ 9 } \right )\)

(iv) \(\left ( \frac{ -9 }{ 4 } \times \frac{ 5 }{ 3 } \right ) + \left ( \frac{ 13 }{ 2 } \times \frac{ 5 }{ 6 } \right )\)

(v) \(\left ( \frac{ -4 }{ 3 } \times \frac{ 12 }{ -5 } \right ) + \left ( \frac{ 3 }{ 7 } \times \frac{ 21 }{ 15 } \right )\)

(vi) \(\left ( \frac{ 13 }{ 5 } \times \frac{ 8 }{ 3 } \right ) – \left ( \frac{ -5 }{ 2 } \times \frac{ 11 }{ 3 } \right )\)

(vii) \(\left ( \frac{ 13 }{ 7 } \times \frac{ 11 }{ 26 } \right ) – \left ( \frac{ -4 }{ 3 } \times \frac{ 5 }{ 6 } \right )\)

(viii) \(\left ( \frac{ 8 }{ 5 } \times \frac{ -3 }{ 2 } \right ) + \left ( \frac{ -3 }{ 10 } \times \frac{ -11 }{ 16 } \right )\)

Solution:

(i) \(\left ( \frac{ 25 }{ 8 } \times \frac{ 2 }{ 5 } \right ) – \left ( \frac{ 3 }{ 5 } \times \frac{ -10 }{ 9 } \right )\)

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

= \(\frac{5 \times 3 + 2 \times 4}{12}\) = \(\frac{23}{12}\)

(ii) \(\left ( \frac{ 1 }{ 2 } \times \frac{ 1 }{ 4 } \right ) + \left ( \frac{ 1 }{ 2 } \times 6 \right )\)

= \(\frac{ 1 }{ 8 } + 3\)

= \(\frac{1 + 3 \times 8}{8}\) = \(\frac{25}{8}\)

(iii) \(\left ( -5 \times \frac{ 2 }{ 15 } \right ) – \left ( -6  \times \frac{ 2 }{ 9 } \right )\)

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

= \(\frac{ -2 + 4}{3}\) = \(\frac{2}{3}\)

(iv) \(\left ( \frac{ -9 }{ 4 } \times \frac{ 5 }{ 3 } \right ) + \left ( \frac{ 13 }{ 2 } \times \frac{ 5 }{ 6 } \right )\)

= \(\frac{ -3 \times 5 }{ 4 } + \frac{13 \times 5 }{ 12 }\)

= \(\frac{-15}{4} + \frac{65}{12}\)

= \(\frac{-15 \times 3 + 65}{12}\)

= \(\frac{20}{ 12}\) = \(\frac{5}{3}\)

(v) \(\left ( \frac{ -4 }{ 3 } \times \frac{ 12 }{- 5 } \right ) + \left ( \frac{ 3 }{ 7 } \times \frac{ 21 }{ 15 } \right )\)

= \(\frac{ 4 \times 4 }{ 5 } + \frac{1 \times 3 }{ 5 }\)

= \(\frac{16}{5} + \frac{3}{5}\)

= \(\frac{16 + 3}{5}\) = \(\frac{19}{ 5}\)

(vi) \(\left ( \frac{ 13 }{ 5 } \times \frac{ 8 }{ 3 } \right ) – \left ( \frac{ -5 }{ 2 } \times \frac{ 11 }{ 3 } \right )\)

= \(\frac{ 13 \times 8 }{ 15 } – \frac{-5 \times 11 }{ 6 }\)

= \(\frac{104}{15} – \frac{-55}{6}\)

= \(\frac{104 \times 2 + 55 \times 5}{30}\) = \(\frac{483}{30}\)

(vii) \(\left ( \frac{ 13 }{ 7 } \times \frac{ 11 }{ 26 } \right ) – \left ( \frac{ -4 }{ 3 } \times \frac{ 5 }{ 6 } \right )\)

= \(\frac{ 1\times 11 }{ 7 \times 2 } – \frac{-2\times 5 }{ 3 \times 3 }\)

= \(\frac{11}{14} – \frac{-10}{9}\)

= \(\frac{11 \times 9 + 10 \times 14}{126}\) = \(\frac{239}{126}\)

(viii) \(\left ( \frac{ 8 }{ 5 } \times \frac{ -3 }{ 2 } \right ) + \left ( \frac{ -3 }{ 10 } \times \frac{ 11 }{ 16 } \right )\)

= \(\frac{4 \times \left ( -3 \right )}{5} + \frac{-3 \times 11}{10 \times 16}\)

= \(\frac{ -12 }{ 5 } + \frac{-33 }{ 160 }\)

= \(\frac{-12 \times 32 – 33}{160}\) = \(\frac{-417}{160}\)

Q-5. Simplify:

(i) \(\left ( \frac{ 3 }{ 2 } \times \frac{ 1 }{ 6 } \right ) + \left ( \frac{ 5 }{ 3 } \times \frac{ 7 }{ 2 } \right ) – \left ( \frac{ 13 }{ 8 } \times \frac{ 4 }{ 3 } \right )\)

(ii) \(\left ( \frac{ 1 }{ 4 } \times \frac{ 2 }{ 7 } \right ) – \left ( \frac{ 5 }{ 14 } \times \frac{ -2 }{ 3 } \right ) + \left ( \frac{ 3 }{ 7 } \times \frac{ 9 }{ 2 } \right )\)

(iii)  \(\left ( \frac{ 13 }{ 9 } \times \frac{ -15 }{ 2 } \right ) + \left ( \frac{ 7 }{ 3 } \times \frac{ 8 }{ 5 } \right ) + \left ( \frac{ 3 }{ 5 } \times \frac{ 1 }{ 2 } \right )\)

(iv) \(\left ( \frac{ 3 }{ 11 } \times \frac{ 5 }{ 6 } \right ) – \left ( \frac{ 9 }{ 12 } \times \frac{ 4 }{ 3 } \right ) + \left ( \frac{ 5 }{ 13 } \times \frac{ 6 }{ 15 } \right )\)

Solution:

(i) \(\left ( \frac{3}{2} \times \frac{1}{6} \right ) + \left ( \frac{5}{3} \times \frac{7}{2} \right ) – \left ( \frac{13}{8} \times \frac{4}{3} \right )\)

= \(\frac{1}{4} + \frac{35}{6} – \frac{13}{6}\)

= \(\frac{1 \times 3 + 35 \times 2 – 13 \times 6}{12}\)

= \(\frac{3 + 70 – 26}{12}\) = \(\frac{47}{12}\)

(ii) \(\left ( \frac{1}{4} \times \frac{2}{7} \right ) – \left ( \frac{5}{14} \times \frac{-2}{3} \right ) + \left ( \frac{3}{7} \times \frac{9}{2} \right )\)

= \(\frac{1}{14} – \frac{-5}{21} + \frac{27}{14}\)

= \(\frac{1 \times 3 + 5 \times 2 + 27 \times 3}{21}\)

= \(\frac{3 + 10 + 81 }{21}\) = \(\frac{94}{21}\)

(iii) \(\left ( \frac{13}{9} \times \frac{-15}{2} \right ) + \left ( \frac{7}{3} \times \frac{8}{5} \right ) + \left ( \frac{3}{5} \times \frac{1}{2} \right )\)

= \(\frac{-13 \times 5}{6} + \frac{7 \times 8}{15} + \frac{3}{10}\)

= \(\frac{-65}{6} + \frac{56}{15} + \frac{3}{10}\)

= \(\frac{-65 \times 5 + 56 \times 2 + 3 \times 3}{30}\)

= \(\frac{-204}{30}\) = \(\frac{-34}{5}\)

(iv) \(\left ( \frac{3}{11} \times \frac{5}{6} \right ) – \left ( \frac{9}{12} \times \frac{4}{3} \right ) + \left ( \frac{5}{13} \times \frac{6}{15} \right )\)

= \(\frac{5}{22}  – 1 + \frac{2}{13}\)

= \(\frac{5 \times 13 – 286 + 2 \times22}{286}\)

= \(\frac{65 – 286 + 44}{286}\) = \(\frac{-177}{286}\)

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