Rs 2500 is lent partially on 8% per annum simple interest and 5% per annum simple interest if total interest is Rs 519 after 3 years then find the ratio in which amount was lent.

Options

A. `3 : 2`
B. `9 : 16`
C. `17 : 8`
D. `16 : 8 `

Answer

To solve the problem step by step, we will denote the amounts lent at 8% and 5% as \( P_1 \) and \( P_2 \) respectively.

### Step 1: Set up the equations
Let:
- \( P_1 \) = amount lent at 8%
- \( P_2 \) = amount lent at 5%

From the problem, we know:
\[ P_1 + P_2 = 2500 \]

### Step 2: Calculate the interest from each amount
The simple interest formula is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

The interest from \( P_1 \) (at 8% for 3 years) is:
\[ \text{SI}_1 = \frac{P_1 \times 8 \times 3}{100} = \frac{24P_1}{100} = 0.24P_1 \]

The interest from \( P_2 \) (at 5% for 3 years) is:
\[ \text{SI}_2 = \frac{P_2 \times 5 \times 3}{100} = \frac{15P_2}{100} = 0.15P_2 \]

### Step 3: Set up the total interest equation
According to the problem, the total interest earned after 3 years is Rs. 519. Therefore, we can write:
\[ 0.24P_1 + 0.15P_2 = 519 \]

### Step 4: Substitute \( P_2 \) in terms of \( P_1 \)
From the first equation \( P_1 + P_2 = 2500 \), we can express \( P_2 \) as:
\[ P_2 = 2500 - P_1 \]

Now, substitute \( P_2 \) in the total interest equation:
\[ 0.24P_1 + 0.15(2500 - P_1) = 519 \]

### Step 5: Simplify the equation
Expanding the equation gives:
\[ 0.24P_1 + 375 - 0.15P_1 = 519 \]

Combining like terms:
\[ (0.24 - 0.15)P_1 + 375 = 519 \]
\[ 0.09P_1 + 375 = 519 \]

### Step 6: Solve for \( P_1 \)
Subtract 375 from both sides:
\[ 0.09P_1 = 519 - 375 \]
\[ 0.09P_1 = 144 \]

Now, divide by 0.09:
\[ P_1 = \frac{144}{0.09} = 1600 \]

### Step 7: Find \( P_2 \)
Now substitute \( P_1 \) back to find \( P_2 \):
\[ P_2 = 2500 - P_1 = 2500 - 1600 = 900 \]

### Step 8: Find the ratio
Now we can find the ratio of the amounts lent:
\[ \text{Ratio} = \frac{P_1}{P_2} = \frac{1600}{900} = \frac{16}{9} \]

Thus, the ratio in which the amounts were lent is \( 16:9 \).