The annual incomes of two persons are in the ratio 9 ∶ 7 and their expenses are in the ratio 4 ∶ 3. If each of them saves Rs. 2000 per year, what is the difference in their annual incomes?
Rs. 4000
Solving Annual Income and Expense Ratio Problems
This problem involves two persons whose annual incomes and expenses are given in specific ratios, and they both save the same amount each year. We need to find the difference between their annual incomes.
Setting up the Equations based on Ratios
Let the ratio of annual incomes be \(9:7\). We can represent their annual incomes as \(9x\) and \(7x\), where \(x\) is a common multiplier.
Let the ratio of their annual expenses be \(4:3\). We can represent their annual expenses as \(4y\) and \(3y\), where \(y\) is another common multiplier.
The relationship between income, expense, and saving is: Income \(-\) Expense \(=\) Saving.
We are told that each person saves Rs. 2000 per year. Using the income and expense expressions, we can form two linear equations:
- For the first person: \(\text{Income}_1 - \text{Expense}_1 = \text{Saving}_1\)
- This gives us: \(9x - 4y = 2000\) (Equation 1)
- For the second person: \(\text{Income}_2 - \text{Expense}_2 = \text{Saving}_2\)
- This gives us: \(7x - 3y = 2000\) (Equation 2)
Solving the System of Linear Equations
We have a system of two linear equations with two variables (\(x\) and \(y\)):
\(9x - 4y = 2000 \quad (1)\)
\(7x - 3y = 2000 \quad (2)\)
We can solve this system using the elimination method. To eliminate \(y\), we can multiply Equation (1) by 3 and Equation (2) by 4:
- Multiply Equation (1) by 3: \(3 \times (9x - 4y) = 3 \times 2000\)
- This results in: \(27x - 12y = 6000\) (Equation 3)
- Multiply Equation (2) by 4: \(4 \times (7x - 3y) = 4 \times 2000\)
- This results in: \(28x - 12y = 8000\) (Equation 4)
Now, subtract Equation (3) from Equation (4):
\((28x - 12y) - (27x - 12y) = 8000 - 6000\)
\(28x - 12y - 27x + 12y = 2000\)
\(x = 2000\)
We have found the value of \(x\), which is the multiplier for the annual incomes.
Calculating the Annual Incomes
The annual incomes of the two persons are \(9x\) and \(7x\).
- Annual Income of the first person = \(9x = 9 \times 2000 = 18000\)
- Annual Income of the second person = \(7x = 7 \times 2000 = 14000\)
So, the annual incomes are Rs. 18000 and Rs. 14000.
Calculating the Difference in Annual Incomes
The question asks for the difference in their annual incomes.
\(\text{Difference} = \text{Income}_1 - \text{Income}_2 = 18000 - 14000 = 4000\)
The difference in their annual incomes is Rs. 4000.
Summary of Calculations
| Item | Person 1 | Person 2 | Ratio |
|---|---|---|---|
| Annual Income | \(9x = 18000\) | \(7x = 14000\) | \(9:7\) |
| Annual Expense | \(4y = 16000\) | \(3y = 12000\) | \(4:3\) |
| Annual Saving (Income - Expense) | \(18000 - 16000 = 2000\) | \(14000 - 12000 = 2000\) | Each saves 2000 |
| Difference in Annual Incomes | \(18000 - 14000 = 4000\) | ||
The difference in their annual incomes is Rs. 4000.
Revision Table: Income Expense Ratios
| Concept | Description | Formula/Representation |
|---|---|---|
| Ratio | Comparison of two quantities by division. | \(a:b\) or \(a/b\) |
| Income Ratio | Ratio of incomes of different entities. | \(\text{Income}_1 : \text{Income}_2\) |
| Expense Ratio | Ratio of expenses of different entities. | \(\text{Expense}_1 : \text{Expense}_2\) |
| Saving | Part of income not spent. | Saving = Income - Expense |
| Representing Ratios | Using a common variable multiplier for ratio components. | If ratio is \(a:b\), quantities are \(ak, bk\). |
| Setting up Equations | Using the relationship between Income, Expense, and Saving to form equations. | Income - Expense = Saving |
Additional Information: Ratio and Proportion Basics
Understanding ratios and proportions is fundamental to solving problems like this. A ratio is simply a way to compare two or more quantities. For example, an income ratio of \(9:7\) means that for every Rs. 9 earned by the first person, the second person earns Rs. 7.
When quantities are in a ratio \(a:b\), they can be represented as \(ax\) and \(bx\) for some common non-zero factor \(x\). Similarly, if another pair of quantities are in ratio \(c:d\), they can be represented as \(cy\) and \(dy\) using a potentially different common factor \(y\). Using these representations helps convert ratio problems into algebraic equations that can then be solved.
In this specific problem, the key was to recognize that while the incomes are related by one ratio and expenses by another, the *difference* (savings) relates the income of a person to their own expense. This allowed us to set up two equations, one for each person's saving, linking the income variable (\(x\)) and the expense variable (\(y\)). Solving this system provided the value of \(x\), which was necessary to find the actual incomes and their difference.
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