$A$ and $B$ can do a piece of work in...

Question

$A$ and $B$ can do a piece of work in $10$ days, $B$ and $C$ in $15$days and $A$ and $C$ can do in $12$ days. $A$, $B$ and $C$ work together to finish the work. If they are paid $Rs. 15000$, how should the money be divided?

My Attempt: In $10$ days, $A$ and $B$ can do $1$ work.

In $1$ day, $A$ and $B$ can do $\dfrac {1}{10}$ work.

In $15$ days, $B$ and $C$ can do $1$ work.

In $1$ day, $B$ and $C$ can do $\dfrac {1}{15}$ work.

In $12$ days, $A$ and $C$ can do $1$ work.

In $1$ day, $A$ and $C$ can do $\dfrac {1}{12}$

Now,

In $1$ day, $2(A+B+C)$ can do $\dfrac {1}{10} + \dfrac {1}{15} +\dfrac {1}{12}$ work.

In $1$ day, $A+B+C$ can do $\dfrac {1}{8}$ work.

How should I complete further?

Answer 1

Well, as you have found $A$ and $B$ and $C$ together, can do $\frac{1}{8}$ work.

However, just $A$ and $B$ can only do $\frac{1}{10}$ work. From here, we can conclude $C$ can do $\frac{1}{8}-\frac{1}{10}=\frac{1}{40}$ of the work.

Similarly, you can calculate how much of the work $A$ and $B$ can do.

The fair way to distribute the momeny will be to distribute them according to how much they contributed to the work.

Answer 2

$C$ alone can do $\frac18-\frac1{10}=\frac1{40}$ of the work in one day.

$A$ alone can do $\frac18-\frac1{15}=\frac7{120}$ of the work.

$B$ alone can do $\frac 18-\frac1{12}=\frac1{24}$ of the work.

So $A$ has made $$\frac{\frac1{40}}{\frac18}=\frac15$$ of the total work. He earns $15000/5=3000$.

Answer 3

In 1 day, C can do 1/8 - 1/10 = 1/40 work.

In 1 day, B can do 1/8 - 1/12 = 1/24 work.

In 1 day, A can do 1/8 - 1/15 = 7/120 work.

So the money should be divided by the ratio A:B:C = (7/120):(1/24):(1/40)

=(7/120):(5/120):(3/120)=7:5:3

Since 15000/(7+5+3)=1000, A should have 7000, B should have 5000 and C should have 3000.

Answer 4

A's 1 day work = (A+B+C)'s 1 day work - (B+C)'s 1 day work

= $\frac 18 - \frac 1{15}$

= $\frac 7{120}$

B's 1 day work = (A+B+C)'s 1 day work - (A+C)'s 1 day work

= $\frac 18 - \frac 1{12}$

= $\frac 1{24}$

C's 1 day work = (A+B+C)'s 1 day work - (A+B)'s 1 day work

= $\frac 18 - \frac 1{10}$

= $\frac 1{40}$

Now ratio of A:B:C=7:5:3

Then dividend of A, B and C = 7000, 5000, 3000