Question

A water tank has inlets of two types A and B. All inlets of type A when open, bring in water at the same rate. All inlets of type B, when open, bring in water at the same rate. The empty tank is completely filled in 30 minutes if 10 inlets of type A and 45 inlets of type B are open, and in 1 hour if 8 inlets of type A and 18 inlets of type B are open. In how many minutes will the empty tank get completely filled if 7 inlets of type A and 27 inlets of type B are open?

Answer 1

Correct Answer: 48

Step 1: Define Variables 

Let the rate of filling of each Type A pipe be \( a \) and each Type B pipe be \( b \).

Two scenarios are given:

• First case: 10 Type A pipes and 45 Type B pipes fill the tank in 30 minutes
• Second case: 8 Type A pipes and 18 Type B pipes fill the tank in 60 minutes

Step 2: Form Equations

Total work done in each case = 1 tank (work = 1 unit).

\[ 30(10a + 45b) = 1 \tag{1} \] \[ 60(8a + 18b) = 1 \tag{2} \]

Step 3: Eliminate Constants

From equations (1) and (2):

\[ 30(10a + 45b) = 60(8a + 18b) \] \[ 10a + 45b = 16a + 36b \] \[ 45b - 36b = 16a - 10a \Rightarrow 9b = 6a \Rightarrow \boxed{a = 1.5b} \]

Step 4: Compute Total Work

Using equation (1):

\[ \text{Total work} = 30(10a + 45b) = 30(15b + 45b) = 30 \times 60b = 1800b \]

Step 5: New Scenario – 7 Type A Pipes and 27 Type B Pipes

Rate of work = \( 7a + 27b \). Using \( a = 1.5b \):

\[ 7a + 27b = 7 \times 1.5b + 27b = 10.5b + 27b = 37.5b \]

Step 6: Time Required

Time = Total work / Rate of work

\[ \text{Time} = \frac{1800b}{37.5b} = \boxed{48 \text{ minutes}} \]