Abhishek had 50,000 of savings which he decided to invest in two deposits. The first deposit yielded 10% interest compounded annually while the second deposit yielded 20% interest compounded annually. The difference in interests earned from the two deposits after 2 years is 15500. What is the value of the smaller of the two deposits?

Let the value of the first deposit be \( x \). Thus, the second deposit will be \( 50,000 - x \), as the total savings amount to 50,000.

### Step 1: Interest from the first deposit
The first deposit earns 10% interest compounded annually. The formula for compound interest is:

\[
A = P \left( 1 + \frac{r}{100} \right)^t
\]

For the first deposit:
- \( P = x \) (the principal of the first deposit)
- \( r = 10\% \) (interest rate)
- \( t = 2 \) years (time period)

The amount after 2 years is:

\[
A_1 = x \left( 1 + \frac{10}{100} \right)^2 = x \times (1.1)^2 = x \times 1.21
\]

So, the interest from the first deposit is:

\[
\text{Interest from first deposit} = A_1 - x = 1.21x - x = 0.21x
\]

### Step 2: Interest from the second deposit
The second deposit earns 20% interest compounded annually. Applying the same formula for compound interest:

For the second deposit:
- \( P = 50,000 - x \)
- \( r = 20\% \)
- \( t = 2 \) years

The amount after 2 years is:

\[
A_2 = (50,000 - x) \left( 1 + \frac{20}{100} \right)^2 = (50,000 - x) \times (1.2)^2 = (50,000 - x) \times 1.44
\]

The interest from the second deposit is:

\[
\text{Interest from second deposit} = A_2 - (50,000 - x) = 1.44(50,000 - x) - (50,000 - x) = 0.44(50,000 - x)
\]

### Step 3: Difference in interests
The problem states that the difference in interests from the two deposits after 2 years is 15,500:

\[
0.44(50,000 - x) - 0.21x = 15,500
\]

Expanding both sides:

\[
0.44(50,000) - 0.44x - 0.21x = 15,500
\]
\[
22,000 - 0.65x = 15,500
\]

### Step 4: Solve for \( x \)

Now solve for \( x \):

\[
22,000 - 15,500 = 0.65x
\]
\[
6,500 = 0.65x
\]
\[
x = \frac{6,500}{0.65} = 10,000
\]

### Final Answer:
The value of the smaller of the two deposits is \( \boxed{10,000} \).

Would you like further details or clarifications?

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Here are 5 related questions to explore:
1. How would the solution change if the interest was compounded monthly instead of annually?
2. What would be the total interest earned if both deposits were invested for 3 years instead of 2?
3. How is the formula for compound interest derived?
4. How would the deposits change if the interest rates were swapped between the two accounts?
5. What happens if the interest rates are not compounded annually but continuously?

**Tip:** Always make sure to check if interest rates are compounded annually or at other intervals (monthly, quarterly, etc.), as this can significantly impact the final amount.

Learn how to calculate the value of the smaller deposit when 50,000 is invested in two deposits with different compound interest rates. This problem involves applying compound interest formulas and solving a linear equation, suitable for students in Grades 10-12.

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