Question

please help me to solve this both sums​

Answer 1

Step-by-step explanation:

Question:-

A and B can do a piece of work in 20 days, B and C can do it in 15 days, C and A can do it in 12 days.

(i) How long will they take to finish the work together.

(ii) How long would each take to do the same work.

Solution:-

$\rm A \: and\: B \: can\: do \: the \: work \: in \: 20 \: days$

$\rm (A + B)'s\: 1 \: day \: work = \dfrac{1}{20}$

$\rm B \: and\: C \: can\: do \: the \: work \: in \: 15 \: days$

$\rm (B + C)'s\: 1 \: day \: work = \dfrac{1}{15}$

$\rm C \: and\: A \: can\: do \: the \: work \: in \: 12 \: days$

$\rm (C + A)'s\: 1 \: day \: work = \dfrac{1}{12}$

Time taken by A, B and C if they work together.

$\rm \dashrightarrow (A+B) + (B + C) + (C + A) = \dfrac{1}{20} + \dfrac{1}{15} + \dfrac{1}{12}$

$\rm \dashrightarrow A+B+ B + C+ C + A = \dfrac{3 + 4 + 5}{60}$

$\rm \dashrightarrow 2A + 2B + 2C= \dfrac{12}{60}$

$\rm \dashrightarrow 2(A + B + C) = \dfrac{1}{15}$

$\rm \dashrightarrow A+B+ C= \dfrac{1}{10}$

$\rm (A + B + C)'s \: 1 \: work \: = \dfrac{1}{10}$

$\rm \therefore A, \: B \: and \: C \: together \: can \: finish \: the \: work \: in \: 10\:days$

Now,

Time taken by A alone to finish the work:-

$\rm \dashrightarrow A's \: 1 \: day \: work = (A + B + C)'s \: 1 \: day \: work - (B + C)'s\: 1 \: day \: work$

$\rm = \dfrac{1}{10} - \dfrac{1}{15}$

$\rm = \dfrac{3 - 2}{30} = \dfrac{1}{30}$

$\rm \dashrightarrow A's \: 1 \: day \: work = \dfrac{1}{30}$

$\rm So, \: A \: alone\: can \: do \: the \: work \: in \: 30 \: days$

Time taken by B alone to finish the work:-

$\rm \dashrightarrow B's \: 1 \: day \: work = (A + B + C)'s \: 1 \: day \: work - (A + C)'s\: 1 \: day \: work$

$\rm = \dfrac{1}{10} - \dfrac{1}{12}$

$\rm = \dfrac{6 - 5}{60} = \dfrac{1}{60}$

$\rm \dashrightarrow B's \: 1 \: day \: work = \dfrac{1}{60}$

$\rm So, \: B \: alone\: can \: do \: the \: work \: in \: 60 \: days$

Time taken by C alone to finish the work:-

$\rm \dashrightarrow C's \: 1 \: day \: work = (A + B + C)'s \: 1 \: day \: work - (A + B)'s\: 1 \: day \: work$

$\rm = \dfrac{1}{10} - \dfrac{1}{20}$

$\rm = \dfrac{2 - 1}{20} = \dfrac{1}{20}$

$\rm \dashrightarrow C's \: 1 \: day \: work = \dfrac{1}{20}$

$\rm So, \: C \: alone\: can \: do \: the \: work \: in \: 20 \: days$