Incomes of Mahesh and Kamal are in the ratio 1 : 2 and their expenses are in the ratio 1 : 3. Which one of the following is correct?
It is not possible to determine who saves more.
Understanding the Ratio Problem: Income, Expense, and Savings
The question provides us with the ratios of incomes and expenses for two individuals, Mahesh and Kamal. We are asked to compare their savings based on this information.
Let's represent the incomes and expenses using variables based on the given ratios:
- Income of Mahesh : Income of Kamal = \(1 : 2\). Let their incomes be \(I_M\) and \(I_K\). We can write this as \(\frac{I_M}{I_K} = \frac{1}{2}\). This means \(I_K = 2 \times I_M\). Let's assume \(I_M = x\). Then \(I_K = 2x\) for some positive value \(x\).
- Expense of Mahesh : Expense of Kamal = \(1 : 3\). Let their expenses be \(E_M\) and \(E_K\). We can write this as \(\frac{E_M}{E_K} = \frac{1}{3}\). This means \(E_K = 3 \times E_M\). Let's assume \(E_M = y\). Then \(E_K = 3y\) for some positive value \(y\).
Savings for anyone is calculated as Income minus Expense. So, for Mahesh and Kamal, their savings are:
- Savings of Mahesh (\(S_M\)) = \(I_M - E_M = x - y\)
- Savings of Kamal (\(S_K\)) = \(I_K - E_K = 2x - 3y\)
We need to determine which of \(S_M\) or \(S_K\) is greater, or if they are equal, or if it's not possible to determine.
Comparing Savings: Mahesh vs Kamal
We need to compare \(x - y\) with \(2x - 3y\). The relative values of \(S_M\) and \(S_K\) depend on the relationship between \(x\) and \(y\). The problem gives us the ratios of incomes and expenses separately, but it does not provide any information about how the income level relates to the expense level for either person, or how the income scale (\(x\)) relates to the expense scale (\(y\)).
Let's look at different possible scenarios based on different values of \(x\) and \(y\):
| Scenario | Assume \(x\) | Assume \(y\) | \(I_M = x\) | \(I_K = 2x\) | \(E_M = y\) | \(E_K = 3y\) | \(S_M = x-y\) | \(S_K = 2x-3y\) | Comparison (\(S_M\) vs \(S_K\)) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 100 | 20 | 100 | 200 | 20 | 60 | \(100-20=80\) | \(200-60=140\) | \(S_K > S_M\) |
| 2 | 100 | 40 | 100 | 200 | 40 | 120 | \(100-40=60\) | \(200-120=80\) | \(S_K > S_M\) |
| 3 | 100 | 50 | 100 | 200 | 50 | 150 | \(100-50=50\) | \(200-150=50\) | \(S_K = S_M\) |
| 4 | 100 | 60 | 100 | 200 | 60 | 180 | \(100-60=40\) | \(200-180=20\) | \(S_M > S_K\) |
| 5 | 100 | 30 | 100 | 200 | 30 | 90 | \(100-30=70\) | \(200-90=110\) | \(S_K > S_M\) |
As demonstrated by the examples, the comparison between Mahesh's savings (\(S_M\)) and Kamal's savings (\(S_K\)) changes depending on the chosen values for \(x\) and \(y\). We can have situations where Kamal saves more, where they save equally, or where Mahesh saves more.
Since the problem only gives the income ratio and the expense ratio independently, and doesn't provide a link between the actual income and expense amounts (or the scale factors \(x\) and \(y\)), we cannot definitively determine who saves more.
Conclusion
Based on the analysis, the relative magnitude of Mahesh's savings (\(x-y\)) and Kamal's savings (\(2x-3y\)) depends on the specific values of \(x\) (income scale) and \(y\) (expense scale). Without knowing the relationship between \(x\) and \(y\), it is impossible to determine whether Mahesh saves more, Kamal saves more, or if they save equally.
Revision Table: Key Concepts
| Concept | Explanation | Application in this Problem |
|---|---|---|
| Ratio | A comparison of two quantities by division. Expressed as \(a:b\) or \(a/b\). | Used to relate incomes (\(1:2\)) and expenses (\(1:3\)). |
| Income | Money received. | \(I_M = x\), \(I_K = 2x\). |
| Expense | Money spent. | \(E_M = y\), \(E_K = 3y\). |
| Savings | Income minus Expense. | \(S_M = I_M - E_M\), \(S_K = I_K - E_K\). |
| Undetermined Value | When a variable or outcome cannot be uniquely determined from the given information. | The relationship between \(x\) and \(y\) is undetermined, leading to the inability to compare savings. |
Additional Information: Ratio Problems and Variables
When dealing with ratio problems involving different quantities (like income and expense here), it's crucial to remember that the ratios might be based on different base amounts. In this case, the income ratio \(1:2\) uses one scale factor (let's say \(x\)), meaning the actual incomes are \(1x\) and \(2x\). The expense ratio \(1:3\) uses potentially a different scale factor (let's say \(y\)), meaning the actual expenses are \(1y\) and \(3y\).
For us to be able to compare savings (\(x-y\) and \(2x-3y\)), we would need more information linking \(x\) and \(y\). For example, if the problem stated that Mahesh saves a certain percentage of his income, or that the total income equals the total expense, or a direct relationship between \(x\) and \(y\) (like \(x = 5y\)), then we might be able to determine the comparison.
Since no such link is provided, the scales of income and expense can vary independently, leading to different outcomes for the savings comparison.
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