Question

2 men and 5 women can do a piece of work in 4 days, while 1 man and 1 woman
can finish it in 12 days. How long it would take for Iman to do the work ?​

Answer 1

$\red{\underline{\underline{Answer:}}}$

$\sf{A \ man \ takes \ 18 \ days \ to \ do}$

$\sf{the \ work.}$

$\sf\orange{Given:}$

$\sf{\implies{2 \ men \ and \ 5 \ women \ can}}$

$\sf{do \ a \ piece \ of \ work \ in \ 4 \ days.}$

$\sf{\implies{1 \ man \ and \ 1 \ woman \ can \ finish}}$

$\sf{it \ in \ 12 \ days.}$

$\sf\pink{To \ find:}$

$\sf{How \ long \ will \ it \ take \ to \ one \ man}$

$\sf{to \ do \ the \ work.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{Let \ the \ work \ done \ by \ a \ man \ be \ x}$

$\sf{and \ work \ done \ by \ a \ woman \ be \ y.}$

$\sf{\therefore{Work \ done \ by \ man \ in \ a \ day=\frac{1}{x}}}$

$\sf{\therefore{Work \ done \ by \ woman \ in \ a \ day=\frac{1}{y}}}$

$\sf{According \ to \ first \ condition.}$

$\sf{\frac{2}{x}+\frac{5}{y}={1}{4}...(1)}$

$\sf{According \ to \ second \ condition.}$

$\sf{\frac{1}{x}+\frac{1}{y}=\frac{1}{12}}$

$\sf{Replace \ \frac{1}{x} \ by \ a \ and \ \frac{1}{y}}$

$\sf{by \ b \ in \ equations \ (1) \ and \ (2), \ we \ get}$

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

$\sf{\therefore{4(2a+5b)=1}}$

$\sf{\therefore{8a+20b=1...(3)}}$

$\sf{a+b=\frac{1}{12}}$

$\sf{\therefore{12(a+b)=1}}$

$\sf{\therefore{12a+12b=1...(4)}}$

$\sf{Multiply \ equation \ (3) \ by \ 3, \ we \ get}$

$\sf{24a+60b=3...(5)}$

$\sf{Multiply \ equation \ (4) \ by \ 5, \ we \ get}$

$\sf{60a+60b=5...(6)}$

$\sf{Subtract \ equation \ (5) \ from \ equation \ (6)}$

$\sf{60a+60b=5}$

$\sf{-}$

$\sf{24a+60b=3}$

_____________________

$\sf{36a=2}$

$\sf{\therefore{a=\frac{2}{36}}}$

$\sf{\therefore{a=\frac{1}{18}}}$

$\sf{But, \ a=\frac{1}{x}}$

$\sf{\therefore{\frac{1}{x}=\frac{1}{18}}}$

$\sf{\therefore{x=18}}$

$\sf\purple{\tt{\therefore{A \ man \ takes \ 18 \ days \ to \ do}}}$

$\sf\purple{\tt{the \ work.}}$

Answer 2

$\huge\underline\mathfrak\red{Question:}$

$\rm{2 \ men \ and \ 5 \ women \ can \ do \ a \ work \ in \ 4 \ days }$

$\rm{and \ 1 \ man \ and \ 1 \ women \ can \ do \ the \ same \ job \ in \ 12 }$

$\rm{days. \ How \ long \ would \ it \ take \ for \ 1 \ man \ to \ finish \ the \ work?}$

__________________________________________

$\huge\underline\mathfrak\blue{Answer:}$

$\rm\orange{time \ taken \ by \ 1 \ man \ to \ compete }$

$\rm\orange{the \ the \ work : \ 18 \ days }$

__________________________________________

$\sf\large\underline\pink{Given:}$

$\rm{Two \ men \ and \ five \ women \ do \ work \ in \ 4 \ days}$

$\rm{1 \ man \ and \ 1 \ woman \ </p><p>can}$

$\rm{finish \ it \ in \ 12 \ days}$

__________________________________________

$\sf\large\underline\pink{To \ find}$

$\rm{How \ long \ it \ would \ take \ for }$

$\rm{1 \ man \ to \ do \ the \ work}$

__________________________________________

$\huge\underline\mathfrak\red{Solution:}$

$\rm{let \ the \ work \ done \ by \ a \ man \is \ m:}$

$\rm{and, \ the \ work \ done \ by \ a \ woman \is \ n}$

$\rm{let \ the \ work \ done \ by \ a \ man }$

$\rm{in \ a \ day : \frac{1}{m} }$

$\rm{and, \ the \ work \ done \ by \ a \ woman \is \ n:}$

$\rm{in \ a \ day : \frac{1}{n} }$

$\rm{By \ the \ first \ condition :}$

$\frac{2}{m} + \frac{5}{n} = \frac{1}{4}$

.......................A)

$\rm{By \ the \ second \ condition :}$

$\frac{1}{m} + \frac{1}{n} = \frac{1}{12}$

..........................B)

$\rm{let \ \frac{1}{m} \ be \ p }$

$\rm{let \ \frac{1}{n} \ be q }$

$\rm{so, \ we \ have ,}$

$2p + 5q= \frac{1}{4}$

$8p + 20q = 1$

................................ a)

$\rm{also,}$

$p + q = \frac{1}{12}$

$12p + 12q = 1$

................................b)

$\rm{Multiply \ equation \ a) \ by 3}$

$\rm{we \ get }$

$24p + 60q = 3$

...........................i)

$\rm{Multiply \ equation \ a) \ by 5}$

$\rm{we \ get }$

$60p + 60q = 5$

...........................ii)

$\rm{now, \ perform \ {eq}^{n} ii) - {eq}^{n}i) }$

$\rm{we \ get }$

$36p = 2$

$p = \frac{2}{36}$

$p = \frac{1}{18}$

$\rm{but , \ as \ we \ know ,}$

$p = \frac{1}{m} = work \: complete \: by \: a \: man \\ \: in \: a \: day$

$\rm{hence ,}$

$m = 18$

$\rm\orange{time \ taken \ by \ 1 \ man \ to \ compete }$

$\rm\orange{the \ the \ work : \ 18 \ days }$

__________________________________________

$\rm\green{hope \ it \ helps }$ :))