# A is $25\%$ more efficient than B…

A is $25\%$ more efficient than B in work. If A takes $15$ days less than B to complete a work, in how many days can they finish the work if they work together?

My Attempt:

B can do $1$ work in $x$ days.

So B can do $\frac {1}{x}$ work per day.

Since A is $25\%$ more efficient than B, A can do $1.25$ work in $x$ days.

So A can do $\frac {1.25}{x}$ work per day.

Let the work be represented by a unit distance $1$ (it doesn't matter anyway) and the speeds of A and B as $\frac54v$ and $v$ per day (since A is 25% more efficient than B). We have $$\frac1{\frac54v}+15=\frac1v$$ $$\frac1{\frac54}+15v=1$$ $$v=\frac{1-\frac45}{15}=\frac1{75}$$ When A and B work together their combined speed is $\frac94v=\frac3{100}$; the time taken is then $\frac{100}3=33.333\dots$ days.
Notice that (continuing your thought): $$\textrm{Power}_A=\textrm{Work per day (for A)}=\frac{1.25}{x}=\frac{1}{x-15}\Leftrightarrow\frac{5}{4x}=\frac{1}{x-15}\Leftrightarrow x=75$$ Can you take it from here?
• I edited to make it more clear: both fractions represent A's power or in other words: "the work per day" produced by A. (Since B needs $x$ days to finish $1$ work, A will do the same in $x-15$ days). – KonKan Sep 30 '16 at 15:41