Problem Breakdown

Sapna invests a total sum of ₹21,900. This amount is divided into two parts, each invested at a different simple interest rate: one at 14% per annum and the other at 10% per annum. The key condition is that after 2 years, the interest earned from both parts is equal. The objective is to determine the specific amount invested at the 14% rate.

Investment Calculation

Let $P_1$ represent the principal amount invested at 14% per annum, and $P_2$ represent the principal amount invested at 10% per annum. The total investment is given as ₹21,900.

This leads to the equation: $P_1 + P_2 = 21,900$

The time period ($T$) for both investments is 2 years.

The formula for calculating Simple Interest (SI) is: $SI = \frac{P \times R \times T}{100}$ where $P$ is the principal, $R$ is the rate of interest per annum, and $T$ is the time period in years.

Interest earned from the first part ($SI_1$) at 14% is: $SI_1 = \frac{P_1 \times 14 \times 2}{100}$

Interest earned from the second part ($SI_2$) at 10% is: $SI_2 = \frac{P_2 \times 10 \times 2}{100}$

The problem states that the interests earned are equal: $SI_1 = SI_2$

Substituting the expressions for $SI_1$ and $SI_2$:

$\frac{P_1 \times 14 \times 2}{100} = \frac{P_2 \times 10 \times 2}{100}$

Simplify the equation by cancelling out common terms (100 and 2):

$14 P_1 = 10 P_2$

Further simplification by dividing both sides by 2 yields:

$7 P_1 = 5 P_2$

This equation establishes the ratio between the two principal amounts:

$\frac{P_1}{P_2} = \frac{5}{7}$ So, the ratio $P_1 : P_2$ is $5 : 7$.

The sum of the ratio parts is $5 + 7 = 12$. This means the total investment of ₹21,900 is divided into 12 equal parts.

To find the sum invested at 14% ($P_1$), we take 5 parts out of the total 12 parts:

$P_1 = \frac{5}{12} \times 21,900$

Performing the calculation:

$P_1 = \frac{5 \times 21,900}{12} = \frac{109,500}{12}$

$P_1 = 9,125$

Thus, the sum invested at 14% per annum is ₹9,125.