Question

Qoban has some money (M) that he divides in the ratio of 1:2. He then deposits the smaller amount in a savings scheme that offers a certain rate of interest, and the larger amount in another savings scheme that offers half of that rate of interest. Both interests compound yearly. At the end of two years, the total interest earned from the two savings schemes is £ 830. It is known that one of the interest rates is 10% and that Qoban deposited more than £1000 in each saving scheme at the start. Find the value of M?

• A. 12000
• B. 8000
• C. 6000
• D. 9000
• E. 5700

Hint:

For compound interest problems with two parts, set up separate interest calculations and sum them to form an equation. Consider both possibilities for which part gets the higher rate.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Let the amounts be $x$ and $2x$. Total money $M = 3x$.
Rates are $r$ and $r/2$. We need to find which one is 10%.
Step 2: Key Formula or Approach:
Interest for 2 years (compounded annually) $= P((1 + R)^2 - 1)$.
Step 3: Detailed Explanation:
Case 1: Rate for smaller amount $x$ is 10%. Then rate for $2x$ is 5%.
$Interest = x(1.1^2 - 1) + 2x(1.05^2 - 1) = 0.21x + 2x(0.1025) = 0.21x + 0.205x = 0.415x$.
$0.415x = 830 \implies x = \frac{830}{0.415} = 2000$.
Money deposited $= 2000$ and $4000$ (both $> 1000$).
$M = 3x = 6000$.
Case 2: Rate for larger amount $2x$ is 10%. Then rate for $x$ is 20%.
$Interest = x(1.2^2 - 1) + 2x(1.1^2 - 1) = 0.44x + 0.42x = 0.86x$.
$0.86x = 830 \implies x = \frac{830}{0.86} \approx 965$.
Since $x<1000$, this case is rejected.
Step 4: Final Answer:
The value of M is 6000.