Question

a can do a certain work in the same period in which b and c together can do it. if a and b together could do it in 10 days and c alone in 50 days, then in how many days could b alone do it?

Explanation:

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<reasoning>

Set up the rate equations

\[

$$\begin{aligned} &R_A = R_B + R_C \\ &R_A + R_B = \frac{1}{10} \\ &R_C = \frac{1}{50} \end{aligned}$$

\]

Solve for the individual rates

\[

$$\begin{aligned} &(R_B + R_C) + R_B = \frac{1}{10} \\ &2R_B + \frac{1}{50} = \frac{1}{10} \\ &2R_B = \frac{5}{50} - \frac{1}{50} = \frac{4}{50} \\ &R_B = \frac{2}{50} = \frac{1}{25} \end{aligned}$$

\]

Calculate the time for B alone

\[
T_B = \frac{1}{R_B} = 25
\]
</reasoning>

<answer>
25 days
</answer>

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Answer:

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<reasoning>

Set up the rate equations

\[

$$\begin{aligned} &R_A = R_B + R_C \\ &R_A + R_B = \frac{1}{10} \\ &R_C = \frac{1}{50} \end{aligned}$$

\]

Solve for the individual rates

\[

$$\begin{aligned} &(R_B + R_C) + R_B = \frac{1}{10} \\ &2R_B + \frac{1}{50} = \frac{1}{10} \\ &2R_B = \frac{5}{50} - \frac{1}{50} = \frac{4}{50} \\ &R_B = \frac{2}{50} = \frac{1}{25} \end{aligned}$$

\]

Calculate the time for B alone

\[
T_B = \frac{1}{R_B} = 25
\]
</reasoning>

<answer>
25 days
</answer>

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