A and B together can do a work in 16 days, which B and C together can do in 32 days. After A has been working at it for 15 days and B for 1…

Let, A, B and C can do the work in x, y and z days respectively.

A and B together can do a work in 16 days

∴ 1/x + 1/y = 1/16 ...... (1)

B and C together can do in 32 days

∴ 1/y + 1/z = 1/32 ...... (2)

After A has been working at it for 15 days and B for 13 days, C finished it in 10 days

∴ 15/x + 13/y + 10/z = 1 ...... (3)

(3) – 15 × (1)

$(\Rightarrow \frac{{15}}{x} + \frac{{13}}{y} + \frac{{10}}{z} - \frac{{15}}{x} - \frac{{15}}{y} - = 1 - \frac{{15}}{{16}}\;)$

$(\Rightarrow \frac{{10}}{z} - \frac{2}{y} = \frac{1}{{16}})$ ...... (4)

2 × (2) + (4)

$(\begin{array}{l} \Rightarrow \frac{2}{y} + \frac{2}{z} + \frac{{10}}{z} - \frac{2}{y} = \frac{2}{{32}} + \frac{1}{{16}}\;\\ \Rightarrow \frac{{12}}{z} = \frac{1}{8} \end{array})$

⇒ z = 96 days

Putting z in equation (2)

$(\begin{array}{l} \frac{1}{y} + \frac{1}{{96}} = \frac{1}{{32}}\;\\ \Rightarrow \frac{1}{y} = \frac{1}{{32}} - \frac{1}{{96}}\\ \Rightarrow \frac{1}{y} = \frac{2}{{96}} \end{array})$

⇒ y = 48 days

Putting value of y in (1)

$(\begin{array}{l} \frac{1}{x} + \frac{1}{{48}} = \frac{1}{{16}}\;\\ \Rightarrow \frac{1}{x} = \frac{1}{{16}} - \frac{1}{{48}}\\ \Rightarrow \frac{1}{x} = \frac{2}{{48}} \end{array})$

⇒ x = 24 days

Let, A, B and C can do the work in x, y and z days respectively.

A and B together can do a work in 16 days

∴ 1/x + 1/y = 1/16 ...... (1)

B and C together can do in 32 days

∴ 1/y + 1/z = 1/32 ...... (2)

After A has been working at it for 15 days and B for 13 days, C finished it in 10 days

∴ 15/x + 13/y + 10/z = 1 ...... (3)

(3) – 15 × (1)

$(\Rightarrow \frac{{15}}{x} + \frac{{13}}{y} + \frac{{10}}{z} - \frac{{15}}{x} - \frac{{15}}{y} - = 1 - \frac{{15}}{{16}}\;)$

$(\Rightarrow \frac{{10}}{z} - \frac{2}{y} = \frac{1}{{16}})$ ...... (4)

2 × (2) + (4)

$(\begin{array}{l} \Rightarrow \frac{2}{y} + \frac{2}{z} + \frac{{10}}{z} - \frac{2}{y} = \frac{2}{{32}} + \frac{1}{{16}}\;\\ \Rightarrow \frac{{12}}{z} = \frac{1}{8} \end{array})$

⇒ z = 96 days

Putting z in equation (2)

$(\begin{array}{l} \frac{1}{y} + \frac{1}{{96}} = \frac{1}{{32}}\;\\ \Rightarrow \frac{1}{y} = \frac{1}{{32}} - \frac{1}{{96}}\\ \Rightarrow \frac{1}{y} = \frac{2}{{96}} \end{array})$

⇒ y = 48 days

Putting value of y in (1)

$(\begin{array}{l} \frac{1}{x} + \frac{1}{{48}} = \frac{1}{{16}}\;\\ \Rightarrow \frac{1}{x} = \frac{1}{{16}} - \frac{1}{{48}}\\ \Rightarrow \frac{1}{x} = \frac{2}{{48}} \end{array})$

⇒ x = 24 days