Ravi 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 Rs. 510 is his annual total interest, then find the money lent out.

Answer

To solve the problem, we will break it down step by step.

### Step 1: Define the total sum lent out
Let the total sum lent out by Ravi be \( S \).

### Step 2: Calculate the amounts lent at different interest rates
- The amount lent at 7% SI is \( \frac{1}{3}S \).
- The amount lent at 8% SI is \( \frac{1}{4}S \).
- The remaining amount lent at 10% SI can be calculated as follows:
\[ \text{Remaining amount} = S - \left(\frac{1}{3}S + \frac{1}{4}S\right) \]

### Step 3: Find a common denominator for the fractions
To combine \( \frac{1}{3}S \) and \( \frac{1}{4}S \), we need a common denominator. The least common multiple of 3 and 4 is 12.
- Convert \( \frac{1}{3}S \) to twelfths:
\[ \frac{1}{3}S = \frac{4}{12}S \]
- Convert \( \frac{1}{4}S \) to twelfths:
\[ \frac{1}{4}S = \frac{3}{12}S \]

### Step 4: Calculate the remaining amount
Now we can calculate the remaining amount:
\[ \text{Remaining amount} = S - \left(\frac{4}{12}S + \frac{3}{12}S\right) = S - \frac{7}{12}S = \frac{5}{12}S \]

### Step 5: Calculate the interest from each part
Now, we can calculate the interest from each part:
1. Interest from \( \frac{1}{3}S \) at 7%:
\[ \text{Interest}_1 = \frac{1}{3}S \times \frac{7}{100} = \frac{7}{300}S \]
2. Interest from \( \frac{1}{4}S \) at 8%:
\[ \text{Interest}_2 = \frac{1}{4}S \times \frac{8}{100} = \frac{8}{400}S = \frac{1}{50}S \]
3. Interest from \( \frac{5}{12}S \) at 10%:
\[ \text{Interest}_3 = \frac{5}{12}S \times \frac{10}{100} = \frac{50}{1200}S = \frac{5}{120}S = \frac{1}{24}S \]

### Step 6: Combine the interests
Now, we can combine all the interests:
\[ \text{Total Interest} = \frac{7}{300}S + \frac{1}{50}S + \frac{1}{24}S \]

### Step 7: Find a common denominator for the total interest
The least common multiple of 300, 50, and 24 is 1200. We convert each term:
1. \( \frac{7}{300}S = \frac{28}{1200}S \)
2. \( \frac{1}{50}S = \frac{24}{1200}S \)
3. \( \frac{1}{24}S = \frac{50}{1200}S \)

Now, combine them:
\[ \text{Total Interest} = \left(\frac{28 + 24 + 50}{1200}\right)S = \frac{102}{1200}S \]

### Step 8: Set the total interest equal to Rs. 510
According to the problem, the total interest is Rs. 510:
\[ \frac{102}{1200}S = 510 \]

### Step 9: Solve for \( S \)
To find \( S \), we can rearrange the equation:
\[ S = 510 \times \frac{1200}{102} \]
Calculating this gives:
\[ S = 510 \times \frac{1200}{102} = 510 \times 11.7647 \approx 6000 \]

### Final Answer
The total sum lent out by Ravi is Rs. 6000.

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