Mohan 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 Mohan deposited more than ₹1000 in each saving scheme at the start. What is the value of M?
- 7500
- 6000
- To solve this, the other interest rate must also be given.
- 4500
- 12000
The Correct Option is B
Solution and Explanation
Step 1: Split the money
Mohan divides ₹M in the ratio \(1:2\). So, the smaller part = \(\dfrac{M}{3}\) and the larger part = \(\dfrac{2M}{3}\).
Step 2: Assign interest rates
Suppose the smaller part is deposited at rate \(r\). Then the larger part is deposited at rate \(\dfrac{r}{2}\). It is given that one of the rates is \(10\%\). So possibilities are:
- Case 1: \(r = 10\%\) and larger part at \(5\%\).
- Case 2: \(r = 20\%\) and larger part at \(10\%\).
Step 3: Write compound interest formula
For amount \(P\) at rate \(R\%\) compounded yearly for \(2\) years: \[ \text{CI} = P \left(1 + \frac{R}{100}\right)^2 - P \]
Step 4: Check Case 1
Smaller deposit = \(\dfrac{M}{3}\) at \(10\%\). \[ \text{CI}_1 = \frac{M}{3}\left[(1.1)^2 - 1\right] = \frac{M}{3}(0.21) = 0.07M \] Larger deposit = \(\dfrac{2M}{3}\) at \(5\%\). \[ \text{CI}_2 = \frac{2M}{3}\left[(1.05)^2 - 1\right] = \frac{2M}{3}(0.1025) \approx 0.06833M \] Total interest = \(0.07M + 0.06833M = 0.13833M\).
Given total interest = 830. \[ 0.13833M = 830 \quad \Rightarrow \quad M \approx 6000 \]
Step 5: Verify deposits
Smaller deposit = \(\dfrac{6000}{3} = 2000 > 1000\). Larger deposit = \(\dfrac{2 \times 6000}{3} = 4000 > 1000\). ✅ Hence valid.
Step 6: Case 2 check (for completeness)
If smaller deposit = \(\dfrac{M}{3}\) at \(20\%\) and larger = \(\dfrac{2M}{3}\) at \(10\%\), then total interest turns out much larger than 830, so this case is invalid.
Final Answer:
\[ \boxed{M = 6000} \]
Top Questions on Ratio and Proportion
Health insurance plays a vital role in ensuring financial protection and access to quality healthcare. In India, however, the extent and nature of health insurance coverage vary significantly between urban and rural areas. While urban populations often have better access to organized insurance schemes, employer-provided coverage, and awareness about health policies, rural populations face challenges such as limited outreach of insurance schemes, inadequate infrastructure, and lower awareness levels. This urban-rural divide in health insurance coverage highlights the broader issue of healthcare inequality, making it essential to analyze the factors contributing to this gap and explore strategies for more inclusive health protection. A state-level health survey was conducted.
The survey covered 1,80,000 adults across urban and rural areas. Urban residents formed 55% of the sample (that is, 99,000 people) while rural residents made up 45% (that is, 81,000 people). In each area, coverage was classified under four heads – Public schemes, Private insurance, Employer-provided coverage, and Uninsured. In urban areas, Public coverage accounted for 28% of the urban population, Private for 22%, Employer for 18%, and the remaining 32% were Uninsured. In rural areas, where formal coverage is generally lower, Public coverage stood at 35%, Private at 10%, Employer at 8%, while 47% were Uninsured.
For this survey, “Insured” includes everyone covered by Public + Private + Employer schemes, and “Uninsured” indicates those with no coverage at all. Officials noted that public schemes remain the backbone of rural coverage, while employer and private plans are relatively more prevalent in urban centres. (250 words)
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