Ram lent 3 persons A, B and C at the rate of 15%,12% and 10% respectively. If he gets Rs 960 at the end of year then find the amount lent to A?

Options

A. Rs 1680
B. Rs 1550
C. Rs 1920
D. Rs 2020

Answer

To solve the problem step by step, we need to find the amount lent to person A. We know that Ram lent money to three persons A, B, and C at different interest rates and received a total of Rs 960 as interest at the end of the year.

### Step 1: Understand the Simple Interest Formula
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) = Simple Interest
- \(P\) = Principal amount (the amount lent)
- \(R\) = Rate of interest (in percentage)
- \(T\) = Time (in years)

### Step 2: Set Up the Equation for Total Interest
Let:
- \(A\) = Amount lent to person A
- \(B\) = Amount lent to person B
- \(C\) = Amount lent to person C

The interest earned from each person can be expressed as:
- Interest from A = \(\frac{A \times 15 \times 1}{100} = 0.15A\)
- Interest from B = \(\frac{B \times 12 \times 1}{100} = 0.12B\)
- Interest from C = \(\frac{C \times 10 \times 1}{100} = 0.10C\)

The total interest earned from all three persons is given as Rs 960:
\[ 0.15A + 0.12B + 0.10C = 960 \]

### Step 3: Express the Amounts Lent
We do not have the exact amounts lent to B and C, so we need to express them in terms of A. However, we can assume that the total amount lent (let's denote it as \(P\)) is the sum of amounts lent to A, B, and C:
\[ P = A + B + C \]

### Step 4: Solve for A
Since we need to find the amount lent to A, we can express B and C in terms of A. For simplicity, let's assume that Ram lent equal amounts to B and C. Thus, we can express:
\[ B = C = x \]
Then we can write:
\[ P = A + x + x = A + 2x \]

Substituting this into the interest equation:
\[ 0.15A + 0.12x + 0.10x = 960 \]
\[ 0.15A + 0.22x = 960 \]

### Step 5: Solve for x in terms of A
From the equation \(P = A + 2x\), we can express \(x\) as:
\[ x = \frac{P - A}{2} \]

Now substitute \(x\) back into the interest equation:
\[ 0.15A + 0.22\left(\frac{P - A}{2}\right) = 960 \]

### Step 6: Rearrange and Solve
This equation can be rearranged and solved for \(A\) once we have a value for \(P\). However, without additional information about the total amount lent or the specific amounts lent to B and C, we cannot isolate \(A\) directly.

### Conclusion
To find the exact amount lent to A, we need more information about either the total amount lent or the amounts lent to B and C.