An amount of Rs. 12200 is partly invested in scheme A at 10% p.a. on compound interest for two years and in scheme B at the same rate on simple interest for four years. If the interest received from both the schemes is equal, then find the amount invested in scheme A?

The correct answer is Rs. 8000.

Let's break down the problem step-by-step to understand how to solve it.

Step 1: Understand the Given Information

Total amount available for investment = Rs. 12,200
Let the amount invested in Scheme A be $P_{\mathrm{A}}$.
Then, the amount invested in Scheme B is $P_{\mathrm{B}}=12200-P_{\mathrm{A}}$.
Rate of interest for both schemes ($R$) = 10% per annum (p.a.)

Step 2: Calculate Interest from Scheme A (Compound Interest)

Scheme A offers compound interest for a period of $n=2$ years.
The formula for the total amount $A$ under compound interest is:

$A=P_{\mathrm{A}}{\left(1+\frac{R}{100}\right)}^n$

Substituting the given values into the formula:

$A=P_{\mathrm{A}}{\left(1+\frac{10}{100}\right)}^2=P_{\mathrm{A}}{\left(1.1\right)}^2=1.21P_{\mathrm{A}}$

The Compound Interest ($CI$) earned from Scheme A is:

$CI=A-P_{\mathrm{A}}=1.21P_{\mathrm{A}}-P_{\mathrm{A}}=0.21P_{\mathrm{A}}$

Step 3: Calculate Interest from Scheme B (Simple Interest)

Scheme B offers simple interest for a period of $T=4$ years.
The Simple Interest ($SI$) formula is:

$SI=\frac{P_{\mathrm{B}}\times R\times T}{100}$

Substituting the given values:

$SI=\frac{P_{\mathrm{B}}\times 10\times 4}{100}=\frac{40P_{\mathrm{B}}}{100}=0.40P_{\mathrm{B}}$

Step 4: Equate the Interests and Find the Ratio

According to the problem statement, the interest earned from both schemes is equal:

$CI=SI$

$0.21P_{\mathrm{A}}=0.40P_{\mathrm{B}}$

Multiplying both sides by 100 to eliminate decimals gives:

$21P_{\mathrm{A}}=40P_{\mathrm{B}}$

Therefore, the ratio of investments is:

$\frac{P_{\mathrm{A}}}{P_{\mathrm{B}}}=\frac{40}{21}$

Step 5: Calculate the Amount Invested in Scheme A

The total ratio parts equal:

$40+21=61$

Now, find the portion of the total amount (Rs. 12,200) allocated to Scheme A:

$P_{\mathrm{A}}=\frac{40}{61}\times 12200$

Since $12200÷61=200$:

$P_{\mathrm{A}}=40\times 200=8000$

Thus, the amount invested in scheme A is Rs. 8000.