Two individuals, P and Q, each invested Rs Y at 40% per annum compound interest for two years. However, P's investment is compounded semi-annually, while Q's investment is compounded annually. After 2 years, P received Rs 568 more than Q. Find the value of Y.

The correct answer is 5000.

Let us solve the problem step-by-step by calculating the compound interest earned by both P and Q.

Step 1: Understand the given data
Principal invested by both P and Q = Rs $Y$
Annual rate of interest ($R$) = $40\%$
Time period ($t$) = $2$ years

Step 2: Calculate the amount received by Q (Compounded Annually)
When interest is compounded annually:
Rate per annum ($r$) = $40\%$
Number of periods ($n$) = $2$ years

$A_Q=Y{\left(1+\frac{40}{100}\right)}^2$

Simplifying the fraction inside the bracket:
$1+\frac{40}{100}=1+\frac{2}{5}=\frac{7}{5}=1.4$

So, the amount $A_Q$ is:
$A_Q=Y\times {\left(1.4\right)}^2=1.96Y$

Step 3: Calculate the amount received by P (Compounded Semi-Annually)
When interest is compounded semi-annually (half-yearly):
Half-yearly rate ($r_{\text{half}}$) = $\frac{40}{2}\%=20\%$
Number of half-yearly periods ($n$) = $2\times 2=4$

$A_P=Y{\left(1+\frac{20}{100}\right)}^4$

Simplifying the fraction inside the bracket:
$1+\frac{20}{100}=1+\frac{1}{5}=\frac{6}{5}=1.2$

So, the amount $A_P$ is:
$A_P=Y\times {\left(1.2\right)}^4=2.0736Y$

Step 4: Use the given difference in amounts to find Y
It is given that P received Rs 568 more than Q:
$A_P-A_Q=568$

Substitute the expressions for $A_P$ and $A_Q$:
$2.0736Y-1.96Y=568$

$0.1136Y=568$

$Y=\frac{568}{0.1136}$

$Y=\frac{5680000}{1136}=5000$

Thus, the value of Y is 5000.