Question

2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

Class 10
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Answer 1

Given :

• 2 women and 5 men can together finish an embroidery work in 4 days
• 3 women and 6 men can finish the same work in 3 days.

To Find :

• Time taken by 1 woman alone to finish the work
• Time taken by 1 man alone to finish the work

Solution :

Let the time taken by 1 woman = x days

The time taken by 1 man = y days

Let Woman's one day work = $\sf \dfrac{1}{x}$

Man's one day work = $\sf \dfrac{1}{y}$

As by given, 2 women and 5 men can together finish the work in 4 days,

$\sf \dfrac{2}{x}+\dfrac{5}{y}=\dfrac{1}{4}$

As by given 3 women and 6 men can finish the work in 3 days,

$\sf \dfrac{3}{x}+\dfrac{6}{y}=\dfrac{1}{3}$

Let $\sf \dfrac{1}{x}=a$ and $\sf \dfrac{1}{y}=b$

$\implies \sf \dfrac{2}{x}+\dfrac{5}{y}=\dfrac{1}{4}$

$\implies \sf 2 \dfrac{1}{x}+5 \dfrac{1}{y}=\dfrac{1}{4}$

$\implies \sf 2a+5b=\dfrac{1}{4}$

$\implies \sf 4(2a+5b)=1$

$\implies \sf 8a+20b=1$ __________(1)

$\implies \sf \dfrac{3}{x}+\dfrac{6}{y}=\dfrac{1}{4}$

$\implies \sf 3 \dfrac{1}{x}+6 \dfrac{1}{y}=\dfrac{1}{3}$

$\implies \sf 3a+6b=\dfrac{1}{3}$

$\implies \sf 3(3a+6b)=1$

$\implies \sf 9a+18b=1$ __________(2)

Let's solve the linear equations by using Elimination method

$\implies \sf 8a+20b=1$ _____(1) × 9

$\implies \sf 9a+18b=1$ ______(2) × 8

$\: \:\sf {\cancel{72a}}+180b=9$

(-)$\sf {\cancel{72a}}+144b=8$

____________

$\:\:\:\sf 36b = 1$$\implies {\sf b = \dfrac{1}{36}}$

Substitute the value of b in place of eqn (1)

$\implies \sf 8a+20b=1$

$\implies \sf 8a+20(\dfrac{20}{36})=1$

$\implies \sf \dfrac{288a+20}{36}=1$

$\implies \sf 288a+20=36$

$\implies \sf 288a=36-20$

$\implies \sf 288a=16$

$\implies \sf a=\dfrac{{\cancel{16}}^1}{{\cancel{288}}_{18}}$ $\implies {\sf a = \dfrac{1}{18}}$

$\sf \rightarrow x = \dfrac{1}{a}$

$\sf \rightarrow x = \dfrac{1}{\dfrac{1}{18}}$

$\boxed{\sf x = 18 \: days }$

$\sf \rightarrow y = \dfrac{1}{b}$

$\sf \rightarrow x = \dfrac{1}{\dfrac{1}{36}}$

$\boxed{\sf x = 36 \: days }$

Required Answer :

Time taken by one woman alone to complete the work is $\underline{\sf 18 \: days }$

Time taken by one man alone to complete the work is $\underline{\sf 36\: days }$