Question

A and B can do a work in 8 days ,B and C can do it in 12 days and A and C can do it in 9 days. How much time will A,B,C take to do work separately​

Answer 1

$\sf\small\underline\red{Let:-}$

$\sf{\implies (A+B)'s\:one\:day\:work=\frac{1}{8}}$

$\sf{\implies (B+C)'s\:one\:day\:work=\frac{1}{12}}$

$\sf{\implies (A+C)'s\:one\:day\:work=\frac{1}{9}}$

$\sf\small\underline\red{Given:-}$

$\sf{\implies Work\:done\:_{(A+B)}=8\:days}$

$\sf{\implies Work\:done\:_{(B+C)}=12\:days}$

$\sf{\implies Work\:done\:_{(A+C)}=9\:days}$

$\sf\small\underline\red{To\: Find:-}$

$\sf{\implies Work\:done\: separately\:_{(A+B+C)}=?}$

$\sf\small\underline\red{Solution:-}$

$\sf{\implies Calculate\: value\:of\:A\:,B\:,C\:by\:eq}$

$\tt{\implies A+B=\frac{1}{8}------(i)}$

$\tt{\implies B+C=\frac{1}{12}------(ii)}$

$\tt{\implies A+C=\frac{1}{9}------(iii)}$

• From eq (i) and (ii) subtracting:-]

$\tt{\implies A+B=\frac{1}{8}}$

$\tt{\implies B+C=\frac{1}{12}}$

• By solving we get here:-]

$\tt{\implies A-C=\dfrac{1}{8}-\dfrac{1}{12}}$

$\tt{\implies A-C=\dfrac{3-2}{8}}$

$\tt{\implies A-C=\dfrac{1}{24}----(iv)}$

• From eq (iii) and (iv) adding:-]

$\tt{\implies A+C=\frac{1}{9}}$

$\tt{\implies A-C=\frac{1}{24}}$

• By solving we get here:-]

$\tt{\implies 2A=\dfrac{8+3}{72}}$

$\tt{\implies 2A=\dfrac{11}{72}}$

$\tt{\implies A=\dfrac{11}{144}}$

$\tt{\implies A=13\dfrac{1}{11}\:days}$

• Putting the value of A in eq (i):-]

$\tt{\implies A+B=\frac{1}{8}}$

$\tt{\implies \frac{11}{144}+B=\frac{1}{8}}$

$\tt{\implies B=\frac{1}{8}-\frac{11}{144}}$

$\tt{\implies B=\frac{18-11}{144}}$

$\tt{\implies B=\frac{7}{144}}$

$\tt{\implies B=20\frac{4}{7}\:days}$

• Putting the value of B in eq (ii):-]

$\tt{\implies \frac{7}{144}+C=\frac{1}{12}}$

$\tt{\implies C=\frac{1}{12}-\frac{7}{144}}$

$\tt{\implies C=\frac{12-7}{144}}$

$\tt{\implies C=\frac{5}{144}}$

$\tt{\implies C=28\frac{4}{5}\:days}$

$\sf\large{Hence,}$

$\bf{\implies A\:_{(can\: word\: separately)}=13\dfrac{1}{11}\:days}$

$\bf{\implies B\:_{(can\: word\: separately)}=20\dfrac{4}{7}\:days}$

$\bf{\implies C\:_{(can\: word\: separately)}=28\dfrac{4}{5}\:days}$

Answer 2

Given :-

A and B can do a work in 8 days ,B and C can do it in 12 days and A and C can do it in 9 days

To Find :-

Time will A,B,C take to do work separately​

Solution :-

When A + B do together

$\sf A+B = \dfrac{1}{8}$

When B + C do together

$\sf B+C=\dfrac{1}{12}$

When A and C do together

$\sf A+C=\dfrac{1}{9}$

Let us separate A + C

$\sf A-C = (A+B)-(B+C)$

$\sf A-C = \dfrac{1}{8} - \dfrac{1}{12}$

By taking LCM of 8 and 12 as 24

$\sf A-C = \dfrac{3-2}{24}$

$\sf A-C = \dfrac{1}{24}$

According to the question

$\sf A+\cancel{C} = \dfrac{1}{9}$

$\sf A-\cancel{C} = \dfrac{1}{24}$

$\sf A+A = \dfrac{1}{9} + \dfrac{1}{24}$

$\sf 2A = \dfrac{8 + 3}{72}$

$\sf 2A = \dfrac{11}{72}$

$\sf 72(2A) = 11$

$\sf 144A = 11$

$\sf A = \dfrac{11}{144}$

Now

For B

$\sf \dfrac{11}{144} + B =\dfrac{1}{8}$

$\sf B = \dfrac{1}{8} - \dfrac{11}{144}$

$\sf B = \dfrac{18-11}{144}$

$\sf B = \dfrac{7}{144}$

For C

$\sf A+C = \dfrac{1}{9}$

$\sf \dfrac{11}{144} + C = \dfrac{1}{9}$

$\sf C = \dfrac{1}{9} - \dfrac{11}{144}$

$\sf C = \dfrac{16-11}{144}$

$\sf C = \dfrac{5}{144}$

Days taken

A = 144/11

B = 144/7

C = 144/5