Question

Two pipes A and B can fill a water tank in 20 minutes and 24 minutes, respectively, and the third pipe C can empty at the rate of 3 gallons per minute. If A, B and C opened together, fill the tank in 15 minutes the capacity ( in gallons) of the tank is:
A. 180
B. 150
C. 120
D. 60

Answer 1

Hint: Assume the capacity of the tank to be x gallon. Using a unitary method to calculate rate of A and rate of B as (A+B+C) together can fill the tank in 15 minutes, rate of (A+B+C) will be $\dfrac{x}{15}$ gallons per minute. Put the values of the rate of A, B and C to get an equation in x and then solve the obtained equation to get the value of x.

Complete step-by-step answer:
Let us assume the capacity of the tank to be x gallons.
According to question, A fills the tank in 20 minutes i.e. A fills x gallons in 20 minutes. So, rate of A to fill $=\left( \dfrac{x}{20} \right)$ gallons per minute………………….(1)
According to the question, B fills the tank in 24 minutes. i.e. B fills x gallons in 24 minutes So, the rate of B to fill $=\left( \dfrac{x}{24} \right)$ gallons per minute……………. (2)
And, according to the question, when A, B and C can open together the tank gets filled in 15 minutes.
i.e. (A+B+C) fills x gallons in 15 minutes. So, the rate of (A+B+C) to fill =$\dfrac{x}{15}$ gallons per minute.
$\Rightarrow \left( \text{Rate of A} \right)+\left( \text{Rate of B} \right)+\left( \text{Rate of C} \right)=\dfrac{x}{15}$ .
Using eq (1) and (2), we will get,
$\Rightarrow \dfrac{x}{20}+\dfrac{x}{24}+\left( \text{Rate of C} \right)=\dfrac{x}{15}$ .
According to the question, C can empty at a rate of 3 gallons per minute. So,
$\Rightarrow \dfrac{x}{20}+\dfrac{x}{24}-3=\dfrac{x}{15}$ .
Taking terms containing ‘x’ to LHS and constant term to RHS, we will get,
$\Rightarrow \dfrac{x}{20}+\dfrac{x}{24}-\dfrac{x}{15}=3$
Taking LCM in LHS, we will get,
$\dfrac{6x+5x-8x}{120}=3$ .
$\Rightarrow \dfrac{3x}{100}=3$
On dividing both sides by 3, we will get,
$\Rightarrow \dfrac{x}{120}=1$ .
Multiplying both sides by 120, we will get,
$\Rightarrow x=120$ .
Hence the required capacity of the tank is 120 gallons and option (c) is the correct answer.

Note: In the equation $\left( \text{Rate of A} \right)+\left( \text{Rate of B} \right)+\left( \text{Rate of C} \right)=\dfrac{x}{15}$ , we have put rate of $c=-3$. We have used negative signs because ‘C’ is making the tank empty while A and B are filling the tank.