Rs P is invested in scheme A (compound interest `"@"10% `p.a. compounded annually) for 2 years and Rs Q is invested in scheme B (compound interest `"@"20%` p.a. compounded annually) for 2 years. The interests earned from schemes A and B were ₹420 and ₹1760 respectively. What is the ratio of P to Q?

Options

A. `1:4`
B. None of those given as options
C. Cannot be determined
D. `1:2`

Answer

To solve the problem step by step, we will use the formula for compound interest and the information provided in the question.

### Step 1: Understanding the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \(A\) = Amount after time \(t\)
- \(P\) = Principal amount (initial investment)
- \(r\) = Rate of interest per annum
- \(t\) = Time in years

### Step 2: Calculate the Amount for Scheme A
For Scheme A:
- Principal = \(P\)
- Rate = \(10\%\)
- Time = \(2\) years
- Interest earned = ₹420

Using the formula:
\[ A_A = P + 420 = P \left(1 + \frac{10}{100}\right)^2 \]
This simplifies to:
\[ A_A = P \left(1 + 0.1\right)^2 = P \left(1.1\right)^2 = P \times 1.21 \]
Thus, we have:
\[ P + 420 = 1.21P \]
Rearranging gives:
\[ 420 = 1.21P - P \]
\[ 420 = 0.21P \]
Now, solving for \(P\):
\[ P = \frac{420}{0.21} = 2000 \]

### Step 3: Calculate the Amount for Scheme B
For Scheme B:
- Principal = \(Q\)
- Rate = \(20\%\)
- Time = \(2\) years
- Interest earned = ₹1760

Using the formula:
\[ A_B = Q + 1760 = Q \left(1 + \frac{20}{100}\right)^2 \]
This simplifies to:
\[ A_B = Q \left(1 + 0.2\right)^2 = Q \left(1.2\right)^2 = Q \times 1.44 \]
Thus, we have:
\[ Q + 1760 = 1.44Q \]
Rearranging gives:
\[ 1760 = 1.44Q - Q \]
\[ 1760 = 0.44Q \]
Now, solving for \(Q\):
\[ Q = \frac{1760}{0.44} = 4000 \]

### Step 4: Finding the Ratio of P to Q
Now we have:
- \(P = 2000\)
- \(Q = 4000\)

The ratio of \(P\) to \(Q\) is:
\[ \text{Ratio} = \frac{P}{Q} = \frac{2000}{4000} = \frac{1}{2} \]

### Final Answer
The ratio of \(P\) to \(Q\) is \(1:2\).

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