Three pipes A, B and C when opened together can fill a tank in $\frac{5}{2}$ hours. Pipes B and C together take 3 hours to fill the tank while pipe A and C together take 4 hours to fill the tank. How long will the pipes A and B together take to fill the tank completely ?

$4\frac{8}{13}$

The correct answer is option (4) → $4\frac{8}{13}$

Let pipe A can fill the tank in x hour → work = $\frac{1}{x}$

Let pipe B can fill the tank in y hour → work = $\frac{1}{Y}$

Let pipe C can fill the tank in z hour → work = $\frac{1}{z}$

and,

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{2}{5}$  [Given]   ...(1)

$\frac{1}{y}+\frac{1}{z}=\frac{1}{3}$   ...(2)

$\frac{1}{x}+\frac{1}{z}=\frac{1}{4}$   ...(3)

Eq. (1) - Eq. (2),

$\frac{1}{x}=\frac{2}{5}-\frac{1}{3}=\frac{6-5}{15}=\frac{1}{15}$

Eq. (1) - Eq. (3),

$\frac{1}{y}=\frac{2}{5}-\frac{1}{4}=\frac{8-5}{20}=\frac{3}{20}$

$∴\frac{1}{x}+\frac{1}{y}=\frac{1}{15}+\frac{3}{20}=\frac{4+9}{60}=\frac{13}{60}$

∴ A and B together can fill tank in $\frac{60}{13}=4\frac{8}{13}hr$