Alok lent out a certain sum. He lent 1/3 part of his sum at 7% SI, 1/4 part at 8% SI and remaining part at 10% SI. If 510 is his total interest, then find the money lent out.

Answer

To solve the problem step by step, we will first define the total amount of money Alok lent out as \( X \).

### Step 1: Define the parts of the total amount lent
Alok lent:
- \( \frac{1}{3} \) of \( X \) at 7% Simple Interest (SI)
- \( \frac{1}{4} \) of \( X \) at 8% SI
- The remaining amount at 10% SI

### Step 2: Calculate the amounts lent at each interest rate
1. Amount lent at 7%:
\[ \text{Amount at 7%} = \frac{X}{3} \]
\[ \text{Interest from this amount} = \frac{X}{3} \times \frac{7}{100} = \frac{7X}{300} \]

2. Amount lent at 8%:
\[ \text{Amount at 8%} = \frac{X}{4} \]
\[ \text{Interest from this amount} = \frac{X}{4} \times \frac{8}{100} = \frac{8X}{400} = \frac{2X}{100} = \frac{2X}{50} = \frac{4X}{200} \]

3. Remaining amount lent at 10%:
\[ \text{Remaining amount} = X - \left(\frac{X}{3} + \frac{X}{4}\right) \]
To combine \( \frac{X}{3} \) and \( \frac{X}{4} \), we find a common denominator, which is 12:
\[ \frac{X}{3} = \frac{4X}{12}, \quad \frac{X}{4} = \frac{3X}{12} \]
\[ \text{Total amount lent at 7% and 8%} = \frac{4X}{12} + \frac{3X}{12} = \frac{7X}{12} \]
\[ \text{Remaining amount} = X - \frac{7X}{12} = \frac{5X}{12} \]
\[ \text{Interest from this amount} = \frac{5X}{12} \times \frac{10}{100} = \frac{5X}{120} = \frac{X}{24} \]

### Step 3: Set up the equation for total interest
The total interest from all parts is given as 510:
\[ \frac{7X}{300} + \frac{2X}{50} + \frac{X}{24} = 510 \]

### Step 4: Find a common denominator and solve for \( X \)
The least common multiple (LCM) of 300, 50, and 24 is 600. We will convert each term to have a denominator of 600:
1. Convert \( \frac{7X}{300} \):
\[ \frac{7X}{300} = \frac{14X}{600} \]
2. Convert \( \frac{2X}{50} \):
\[ \frac{2X}{50} = \frac{24X}{600} \]
3. Convert \( \frac{X}{24} \):
\[ \frac{X}{24} = \frac{25X}{600} \]

Now, we can combine these:
\[ \frac{14X + 24X + 25X}{600} = 510 \]
\[ \frac{63X}{600} = 510 \]

### Step 5: Solve for \( X \)
Multiply both sides by 600:
\[ 63X = 510 \times 600 \]
\[ 63X = 306000 \]
\[ X = \frac{306000}{63} = 4857.14 \approx 4860 \]

### Conclusion
The total amount of money lent out by Alok is approximately \( 4860 \).