Verify the following - i 3-7- 5-6 - 12-13 - 3-7-5-6 - 3-7-12-13 ii -15-4- 3-7 - -12-5 - -15-4-3-7 - -15-4-12-5 iii -8-3 - -13-12 -5-6 - -8-3-5-6 - -13-12-5-6 iv -16-7- -8-9 - -7-6 - -16-7-8-9 - -16-7-7-6

(i) L.H.S. = 3/7× (5/6 + 12/13) = 3/7× (65 + 72/78) = 3/7×137/78 = 411/546 R.H.S. = (3/7×5/6) + (3/7×12/13) = 3 × 5/7 × 6 + 3 × 12/7 × 13 = 15/42 + 36/91 = 195 ...

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Byju's Answer

Standard VIII

Mathematics

Distributive Property of Rational Numbers

Verify the fo...

Question

Verify the following :

(i) $\frac{3}{7} \times (\frac{5}{6} + \frac{12}{13}) = (\frac{3}{7} \times \frac{5}{6}) + (\frac{3}{7} \times \frac{12}{13})$
(ii) $\frac{- 15}{4} \times (\frac{3}{7} + \frac{- 12}{5}) = (\frac{- 15}{4} \times \frac{3}{7}) + (\frac{- 15}{4} \times \frac{- 12}{5})$
(iii) $(\frac{- 8}{3} + \frac{- 13}{12}) \times \frac{5}{6} = (\frac{- 8}{3} \times \frac{5}{6}) + (\frac{- 13}{12} \times \frac{5}{6})$
(iv) $\frac{- 16}{7} \times (\frac{- 8}{9} + \frac{- 7}{6}) = (\frac{- 16}{7} \times \frac{- 8}{9}) + (\frac{- 16}{7} \times \frac{- 7}{6})$

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Solution

(i) $L . H . S . = \frac{3}{7} \times (\frac{5}{6} + \frac{12}{13})$

$= \frac{3}{7} \times (\frac{65 + 72}{78}) = \frac{3}{7} \times \frac{137}{78} = \frac{411}{546}$

$R . H . S . = (\frac{3}{7} \times \frac{5}{6}) + (\frac{3}{7} \times \frac{12}{13})$

$= \frac{3 \times 5}{7 \times 6} + \frac{3 \times 12}{7 \times 13} = \frac{15}{42} + \frac{36}{91}$

$= \frac{195 + 216}{546} = \frac{411}{546}$

$∴ L . H . S . = R . H . S .$

(ii) $L . H . S . = \frac{- 15}{4} \times (\frac{3}{7} + \frac{- 12}{5})$

$= \frac{- 15}{4} \times (\frac{15 - 84}{35})$

$= \frac{- 15}{4} \times \frac{69}{35} = \frac{1035}{140}$

$R . H . S . = (\frac{- 15}{4} \times \frac{3}{7}) + (\frac{- 15}{4} \times \frac{- 12}{5})$

$= \frac{- 15 \times 3}{4 \times 7} + \frac{- 15 \times (- 12)}{4 \times 5}$

$= \frac{- 45}{28} + \frac{180}{20} = \frac{- 225 + 1260}{140} = \frac{1035}{140}$

$∴ L . H . S . = R . H . S .$

(iii) $L . H . S . = (\frac{- 8}{3} + \frac{- 13}{12}) \times \frac{5}{6}$

$= \frac{- 32 - 13}{12} \times \frac{5}{6} = \frac{- 45}{12} \times \frac{5}{6} = \frac{- 225}{72}$

$R . H . S . = (\frac{- 8}{3} \times \frac{5}{6}) + (\frac{- 13}{12} \times \frac{5}{6})$

$= \frac{- 8 \times 5}{3 \times 6} + \frac{- 13 \times 5}{12 \times 6} = \frac{- 40}{18} + \frac{- 65}{72}$

$= \frac{- 160 + (- 65)}{72} = \frac{- 160 - 65}{72}$

$= \frac{- 225}{72}$

$∴ L . H . S . = R . H . S .$

(iv) $L . H . S . = \frac{- 16}{7} \times (\frac{- 8}{9} + \frac{- 7}{6})$

$= \frac{- 16}{7} \times (\frac{- 16 + (- 21)}{18}) = \frac{- 16}{7} \times \frac{- 37}{18}$

$= \frac{- 16 \times - 37}{7 \times 18} = \frac{592}{126}$

$R . H . S . = (\frac{- 16}{7} \times \frac{- 8}{9}) + (\frac{- 16}{7} \times \frac{7}{6})$

$= \frac{- 16 \times (- 8)}{7 \times 9} + \frac{- 16 \times (- 7)}{7 \times 6} = \frac{128}{63} + \frac{112}{42}$

$= \frac{256 + 336}{126} = \frac{592}{126}$

$∴ L . H . S . = R . H . S .$

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Similar questions

Re-arrange suitably and find the sum in each of the following :

(i) $\frac{11}{12} + \frac{- 17}{3} + \frac{11}{2} + \frac{- 25}{2}$

(ii) $\frac{- 6}{7} + \frac{- 5}{6} + \frac{- 4}{9} + \frac{- 15}{7}$

(iii) $\frac{3}{5} + \frac{7}{3} + \frac{9}{5} + \frac{- 13}{15} + \frac{- 7}{3}$

(iv) $\frac{4}{13} + \frac{- 5}{8} + \frac{- 8}{13} + \frac{9}{13}$

(v) $\frac{2}{3} + \frac{- 4}{5} + \frac{1}{3} + \frac{2}{5}$

(vi) $\frac{1}{8} + \frac{5}{12} + \frac{2}{7} + \frac{7}{12} + \frac{9}{7} + \frac{- 5}{16}$

Prove that:
(i) $\frac{1}{3 + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + 1} = 1$
(ii) $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} = 2$

Name the property of multiplication illustrated by each of the following statements :

(i) $\frac{- 15}{8} \times \frac{- 12}{7} = \frac{- 12}{7} \times \frac{- 15}{8}$

(ii) $(\frac{- 2}{3} \times \frac{7}{9}) \times \frac{- 9}{5} = \frac{- 2}{3} \times (\frac{7}{9} \times \frac{- 9}{5})$

(iii) $\frac{- 3}{4} \times (\frac{- 3}{4} \times \frac{- 5}{6}) + (\frac{- 3}{4} \times \frac{7}{8})$

(iv) $\frac{- 16}{9} \times 1 = 1 \times \frac{- 16}{9} = \frac{- 16}{9}$

(v) $\frac{- 11}{15} \times \frac{15}{- 11} = \frac{15}{- 11} \times \frac{- 11}{15} = 1$

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

Fill in the blanks.

(i) $(\frac{- 3}{17}) + (\frac{- 12}{5}) = (\frac{- 12}{5}) + (. . . . .)$

(ii) $- 9 + \frac{- 21}{8} = (. . . . . .) + (- 9)$

(iii) $(\frac{- 8}{13} + \frac{3}{7}) + (\frac{- 13}{4}) = (. . . . . .) + [\frac{3}{7} + (\frac{- 13}{4})]$

(iv) $- 12 + (\frac{7}{12} + \frac{- 9}{11}) = (- 12 + \frac{7}{12}) + (. . . . .)$

(v) $\frac{19}{- 5} + (\frac{- 3}{11} + \frac{- 7}{8}) = {\frac{19}{- 5} + (. . . . .)} + \frac{- 7}{8}$

(vi) $\frac{- 16}{7} + . . . . . = . . . . . . + \frac{- 16}{7} = \frac{- 16}{7}$

Verify each of the following :

(i) $\frac{3}{7} \times \frac{- 5}{9} = \frac{- 5}{9} \times \frac{3}{7}$

(ii) $\frac{- 8}{7} \times \frac{13}{9} = \frac{13}{9} \times \frac{- 8}{7}$

(iii) $\frac{- 12}{5} \times \frac{7}{- 36} = \frac{7}{- 36} \times \frac{- 12}{5}$

(iv) $- 8 \times \frac{- 13}{12} = \frac{- 13}{12} \times (- 8)$

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