Question

It can take 12 hours to fill a swimming pool using two pipes. If the pipe of
larger diameter is used for four hours and the pipe of smaller diameter for
9 hours, only half of the pool can be filled. How long would it take for each
pipe to fill the pool separately?​

Answer 1

$\huge\sf\pink{Answer}$

☞ Smaller pipe takes 30 hours

☞ Larger pipe takes 20 hours

$\rule{110}1$

$\huge\sf\blue{Given}$

✭ It takes 12 hours to fill a swimming pool using 2 pipes

✭ The pipe of larger diameter is used for 4 hours and the pipe of smaller diameter is used for 9 hours, only half the pool gets filled

$\rule{110}1$

$\huge\sf\gray{To \:Find}$

◈ Time taken by each pipe to fill the pool seperately?

$\rule{110}1$

$\huge\sf\purple{Steps}$

Let the time taken by the pipe with larger diameter as a hours, and the let the time taken by the smaller one be b hours

Two pipes fill the full pipe in 12 hours,so then in one hour it will be filling $\sf\dfrac{1}{12} \ part$

Case 1

➳ $\sf\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{12} \quad -eq(1)$

➳ $\sf \dfrac{b+a}{ab} = \dfrac{1}{12}$

➳ $\sf ab=12(b+a)$

➳ $\sf ab=12b+12a\quad-eq(2)$

Case 2

➝ $\sf\dfrac{4}{a} + \dfrac{9}{b} = \dfrac{1}{2}$

➝ $\sf\dfrac{4b+9a}{ab} = \dfrac{1}{2}$

➝ $\sf ab = 2(4b+9a)$

➝ $\sf ab = 8b+18\quad-eq(3)$

$\bullet\:\underline{\textsf{As Per the Question}}$

➢ $\sf 12b+12a=8b+18a$

➢ $\sf 12b-8b=18a-12a$

➢ $\sf 4b=6a$

➢ $\sf b=\dfrac{6a}{4}$

➢ $\sf\green{b=\dfrac{3a}{2}}\quad-eq(4)$

Substituting the value of b in eq(1)

»» $\sf\dfrac{1}{a} + \dfrac{1}{\dfrac{3a}{2}}=\dfrac{1}{12}$

»» $\sf\dfrac{1}{a} + \dfrac{2}{3a} = \dfrac{1}{12}$

»» $\sf\dfrac{3+2}{3a} = \dfrac{1}{12}$

»» $\sf \dfrac{5}{3a} = \dfrac{1}{12}$

»» $\sf 3a=5\times 12$

»» $\sf a=\dfrac{60}{3}$

»» $\sf\orange{a=20 \ hours}$

Substituting the value of a in eq(4)

➠ $\sf b = \dfrac{3\times 20}{2}$

➠ $\sf b = \dfrac{60}{2}$

➠ $\sf\orange{b=30 \ hours}$

$\rule{170}3$