3 small pumps and a large pump are filling a tank. Each of the three small pumps works at 2/3rd the rate of the large pump. If all 4 pumps work at the same time, they should fill the tank in what fraction of the time that it would have taken the large pump alone?

Options

A. `4//7`
B. `1//3`
C. `2//3`
D. `3//4`

Answer

To solve the problem, we need to determine how long it takes for all four pumps (three small pumps and one large pump) to fill the tank compared to the time it would take for the large pump to fill it alone.

### Step-by-step Solution:

1. **Define the rate of the large pump**:
Let the rate of the large pump be \( R \) (in tanks per hour).

**Hint**: Define the rate of the large pump as a variable to simplify calculations.

2. **Define the rate of the small pumps**:
Since each small pump works at \( \frac{2}{3} \) the rate of the large pump, the rate of each small pump is \( \frac{2}{3}R \).

**Hint**: Remember that the rate of each small pump is a fraction of the large pump's rate.

3. **Calculate the total rate of the three small pumps**:
The combined rate of the three small pumps is:
\[ 3 \times \frac{2}{3}R = 2R \]

**Hint**: When combining rates, simply multiply the rate of one pump by the number of pumps.

4. **Calculate the total rate when all four pumps are working together**:
The total rate when all four pumps (three small and one large) are working together is:
\[ 2R + R = 3R \]

**Hint**: Add the rates of the small pumps to the rate of the large pump to find the total rate.

5. **Determine the time taken by the large pump to fill the tank alone**:
The time taken by the large pump to fill the tank alone is:
\[ \text{Time}_{\text{large}} = \frac{1 \text{ tank}}{R} \]

**Hint**: Time is the reciprocal of the rate when filling one tank.

6. **Determine the time taken by all four pumps to fill the tank together**:
The time taken by all four pumps to fill the tank together is:
\[ \text{Time}_{\text{all}} = \frac{1 \text{ tank}}{3R} \]

**Hint**: Again, use the reciprocal of the combined rate to find the time.

7. **Find the fraction of time taken by all pumps compared to the large pump**:
To find the fraction of the time taken by all pumps compared to the large pump, we divide the time taken by all pumps by the time taken by the large pump:
\[ \text{Fraction} = \frac{\text{Time}_{\text{all}}}{\text{Time}_{\text{large}}} = \frac{\frac{1}{3R}}{\frac{1}{R}} = \frac{1}{3} \]

**Hint**: When comparing times, divide the time taken by one scenario by the time taken by another.

### Final Answer:
The four pumps together will fill the tank in \( \frac{1}{3} \) of the time that it would have taken the large pump alone.