Rohan invests a total sum of money into two different funds, Fund A and Fund B. He places $4,000 in Fund A for 4 years at a certain annual rate of simple interest, and the remaining amount in Fund B for 5 years at an annual simple interest rate that is 3% higher than that of Fund A. The principal deposited in Fund A is 20% less than the principal deposited in Fund B. If the total simple interest earned from both funds together is $2,800, what is the annual rate of interest for Fund A?

The correct answer is 5%.

Step 1: Find the principal deposited in Fund B.

Let $P_A$ be the principal deposited in Fund A and $P_B$ be the principal deposited in Fund B.

We are given that:

$P_A=\$\mathrm{4,000}$

The principal deposited in Fund A is 20% less than that in Fund B, which means:

$P_A=P_B-0.20P_B=0.80P_B$

Substitute the value of $P_A$ to solve for $P_B$:

$4000=0.80P_B$

$P_B=\frac{4000}{0.80}=\$\mathrm{5,000}$

Step 2: Express the simple interest earned from each fund in terms of the rate of interest.

The formula for Simple Interest is:

$\text{Simple Interest (SI)}=\frac{P\times R\times T}{100}$

Let the annual interest rate for Fund A be $r\%$.

Then, the annual interest rate for Fund B is $(r+3)\%$.

For Fund A:

Principal ($P_A$) = $4,000, Time ($T_A$) = 4 years, Rate ($R_A$) = $r\%$

${\text{SI}}_A=\frac{4000\times r\times 4}{100}=160r$

For Fund B:

Principal ($P_B$) = $5,000, Time ($T_{\mathrm{B<!--\; population-->}}$) = 5 years, Rate ($R_B$) = $(r+3)\%$

${\text{SI}}_B=\frac{5000\times (r+3)\times 5}{100}=250(r+3)=250r+750$

Step 3: Set up the equation for total simple interest and solve for $r$.

The total simple interest earned from both funds together is $2,800:

${\text{SI}}_A+{\text{SI}}_B=2800$

Substitute the expressions for ${\text{SI}}_A$ and ${\text{SI}}_B$:

$160r+250r+750=2800$

Combine the like terms:

$410r+750=2800$

Subtract 750 from both sides:

$410r=2050$

Divide by 410:

$r=\frac{2050}{410}=5$

Therefore, the annual rate of interest for Fund A is 5%.