There are two pumps to fill a tank with water. First pump can fill the empty tank in 8 hours, while the second in 10 hours. If both the pumps are opened at the same time and kept open for 4 hours, the part of tank that will be filled up is :

Options

A. `(9)/(10)`
B. `(1)/(10)`
C. `(2)/(5)`
D. `(1)/(5)`

Answer

To solve the problem step by step, we will first determine the rate at which each pump fills the tank, then calculate the total amount filled when both pumps are used together for a specified time.

### Step 1: Determine the filling rates of each pump
- The first pump can fill the tank in 8 hours.
- The second pump can fill the tank in 10 hours.

**Filling rate of the first pump:**
\[ \text{Rate of first pump} = \frac{1 \text{ tank}}{8 \text{ hours}} = \frac{1}{8} \text{ tanks/hour} \]

**Filling rate of the second pump:**
\[ \text{Rate of second pump} = \frac{1 \text{ tank}}{10 \text{ hours}} = \frac{1}{10} \text{ tanks/hour} \]

### Step 2: Calculate the combined filling rate of both pumps
To find the combined rate when both pumps are opened together, we add their individual rates:
\[ \text{Combined rate} = \frac{1}{8} + \frac{1}{10} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 8 and 10 is 40.
\[ \frac{1}{8} = \frac{5}{40}, \quad \frac{1}{10} = \frac{4}{40} \]
\[ \text{Combined rate} = \frac{5}{40} + \frac{4}{40} = \frac{9}{40} \text{ tanks/hour} \]

### Step 3: Calculate the total amount filled in 4 hours
Now, we will calculate how much of the tank is filled when both pumps are running for 4 hours:
\[ \text{Amount filled in 4 hours} = \text{Combined rate} \times \text{Time} \]
\[ \text{Amount filled in 4 hours} = \frac{9}{40} \times 4 = \frac{36}{40} \text{ tanks} \]

### Step 4: Determine the part of the tank filled
To find the part of the tank that is filled, we express the amount filled as a fraction of the total tank capacity:
\[ \text{Part of tank filled} = \frac{36}{40} = \frac{9}{10} \]

Thus, the part of the tank that will be filled up after 4 hours is \( \frac{9}{10} \).

### Final Answer:
The part of the tank that will be filled up is \( \frac{9}{10} \).

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