A, B. & C invested their respective savings in a scheme, which offered CI at 20% p.a. for two years and received total interest of Rs. 1694. If A & C invested double of their respective saving in another scheme, which offered Sl at 10% p.a. for two years and received total interest of Rs. 1100, then find difference between saving of A & C together & saving of B?

Options

A. A)1250 Rs.
B. B)1650 Rs.
C. C)1150 Rs.
D. D)1050 Rs.

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the Interest Calculation for A, B, and C
A, B, and C invested their savings in a scheme that offered Compound Interest (CI) at 20% per annum for 2 years, and the total interest received was Rs. 1694.

The formula for Compound Interest is:
\[ A = P(1 + r/n)^{nt} \]
Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = principal amount (the initial amount of money).
- \( r \) = annual interest rate (decimal).
- \( n \) = number of times that interest is compounded per year.
- \( t \) = the number of years the money is invested for.

Since we are calculating total interest, we can also use the formula for the total interest earned over 2 years:
\[ \text{Total Interest} = P \times \left( (1 + r)^t - 1 \right) \]

### Step 2: Calculate the Effective Rate of Interest
For 20% per annum over 2 years:
\[ \text{Effective Rate} = 20\% \times 2 + (20\%)^2 = 40\% + 4\% = 44\% \]
Thus, the total interest can be expressed as:
\[ 0.44 \times (P_A + P_B + P_C) = 1694 \]

### Step 3: Set Up the Equation
Let \( P_A \), \( P_B \), and \( P_C \) be the principal amounts invested by A, B, and C respectively. Therefore, we have:
\[ 0.44 \times (P_A + P_B + P_C) = 1694 \]
Dividing both sides by 0.44:
\[ P_A + P_B + P_C = \frac{1694}{0.44} = 3850 \]

### Step 4: Analyze the Second Investment
A and C invested double their savings in another scheme that offered Simple Interest (SI) at 10% per annum for 2 years, receiving total interest of Rs. 1100.

The formula for Simple Interest is:
\[ SI = \frac{P \times r \times t}{100} \]
For A and C:
\[ SI = \frac{2(P_A + P_C) \times 10 \times 2}{100} = 1100 \]
This simplifies to:
\[ \frac{40(P_A + P_C)}{100} = 1100 \]
Thus:
\[ P_A + P_C = \frac{1100 \times 100}{40} = 2750 \]

### Step 5: Calculate B's Principal
Now we can find B's principal:
\[ P_B = (P_A + P_B + P_C) - (P_A + P_C) = 3850 - 2750 = 1100 \]

### Step 6: Find the Difference Between A & C Together and B
Now, we need to find the difference between the savings of A and C together and the savings of B:
\[ P_A + P_C - P_B = 2750 - 1100 = 1650 \]

### Final Answer
The difference between the savings of A and C together and the savings of B is Rs. 1650.

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