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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.

Please help me to continue from here.

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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.

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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?

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  • $\begingroup$ how did you get the condition for equality in thr beginning of your solution? $\endgroup$ – pi-π Sep 30 '16 at 15:39
  • $\begingroup$ 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). $\endgroup$ – KonKan Sep 30 '16 at 15:41
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Look it is given that A is 25% more efficient than B. means if A took 4x days to do a work B will take 5x days to do the same work. Given 5x - 4x = 15 x = 15 number of days taken by A and B to complete the work alone = 4*15 and 5*15 so if work together , number of days required to complete the work = (4*15*5*15)/(4*15 + 15*5) = 20*15/9 60/3 days using d = AB/(A+B)

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