A sum of Rs 44,200 is divided between John and Smith, 12 years and 14 years old respectively, in such a way that if their portions be invested at 10 percent per annum compound interest, they will receive equal amounts on reaching 16 years of age.
What will each receive, when 16 years old ?

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript and use the compound interest formula.

### Step 1: Determine the time periods for John and Smith
- John is currently 12 years old and will be 16 in 4 years (16 - 12 = 4).
- Smith is currently 14 years old and will be 16 in 2 years (16 - 14 = 2).

**Hint**: Calculate the difference between their current ages and the age they will be when they receive the amount.

### Step 2: Set up the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( A \) = Amount after time \( t \)
- \( P \) = Principal amount (initial investment)
- \( r \) = Rate of interest (10% in this case)
- \( t \) = Time period in years

### Step 3: Write the equations for both John and Smith
Let \( P_J \) be the principal amount John receives and \( P_S \) be the principal amount Smith receives. We know:
\[ P_J + P_S = 44200 \]

The amounts they will receive when they are 16 years old can be expressed as:
- For John:
\[ A_J = P_J \left(1 + \frac{10}{100}\right)^4 = P_J \left(\frac{11}{10}\right)^4 \]
- For Smith:
\[ A_S = P_S \left(1 + \frac{10}{100}\right)^2 = P_S \left(\frac{11}{10}\right)^2 \]

### Step 4: Set the amounts equal to each other
Since they will receive equal amounts when they turn 16:
\[ P_J \left(\frac{11}{10}\right)^4 = P_S \left(\frac{11}{10}\right)^2 \]

### Step 5: Substitute \( P_S \)
From the first equation, we can express \( P_S \):
\[ P_S = 44200 - P_J \]
Substituting this into the equality gives:
\[ P_J \left(\frac{11}{10}\right)^4 = (44200 - P_J) \left(\frac{11}{10}\right)^2 \]

### Step 6: Simplify the equation
Dividing both sides by \( \left(\frac{11}{10}\right)^2 \) (which is valid since it is not zero):
\[ P_J \left(\frac{11}{10}\right)^2 = 44200 - P_J \]
This simplifies to:
\[ P_J \cdot \frac{121}{100} = 44200 - P_J \]

### Step 7: Combine like terms
Rearranging gives:
\[ P_J \cdot \frac{121}{100} + P_J = 44200 \]
Factoring out \( P_J \):
\[ P_J \left(\frac{121}{100} + 1\right) = 44200 \]
\[ P_J \left(\frac{221}{100}\right) = 44200 \]

### Step 8: Solve for \( P_J \)
Multiplying both sides by \( \frac{100}{221} \):
\[ P_J = 44200 \cdot \frac{100}{221} \]
Calculating this gives:
\[ P_J = 20000 \]

### Step 9: Find \( P_S \)
Now, substitute \( P_J \) back to find \( P_S \):
\[ P_S = 44200 - 20000 = 24200 \]

### Step 10: Calculate the amounts they will receive when they are 16
- For John:
\[ A_J = 20000 \left(1 + \frac{10}{100}\right)^4 = 20000 \left(\frac{11}{10}\right)^4 \]
Calculating this gives:
\[ A_J = 20000 \cdot \frac{14641}{10000} = 29282 \]

- For Smith:
\[ A_S = 24200 \left(1 + \frac{10}{100}\right)^2 = 24200 \left(\frac{11}{10}\right)^2 \]
Calculating this gives:
\[ A_S = 24200 \cdot \frac{121}{100} = 29282 \]

### Final Answer
Both John and Smith will receive Rs 29,282 when they turn 16 years old.