Ratio of Incomes and Expenditures | Class 10 Maths | Linear Equations in Two Variables
This problem is based on the application of Linear Equations in Two Variables. We are given the ratios of the incomes and expenditures of two persons along with their monthly savings. Using this information, we form two linear equations and solve them to determine their monthly incomes.
Question
The ratio of the incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them saves ₹2000 per month, find their monthly incomes.
Given Data
- Ratio of incomes = 9 : 7
- Ratio of expenditures = 4 : 3
- Monthly savings of each person = ₹2000
Concept Used
The basic relation between income, expenditure and savings is:
Income − Expenditure = Savings
Since the ratios are given, we assume the quantities in terms of variables and then form linear equations.
Step 1: Assume the Variables
Let the common multiplier for the incomes be x.
Therefore,
- First person's income = 9x
- Second person's income = 7x
Let the common multiplier for the expenditures be y.
- First person's expenditure = 4y
- Second person's expenditure = 3y
Step 2: Form the Equations
Using the relation:
Income − Expenditure = Savings
For the first person:
9x − 4y = 2000 ...(1)
For the second person:
7x − 3y = 2000 ...(2)
Step 3: Eliminate One Variable
Multiply Equation (1) by 3:
27x − 12y = 6000
Multiply Equation (2) by 4:
28x − 12y = 8000
Subtract the first equation from the second:
x = 2000
Step 4: Find Monthly Incomes
First person's income:
9x = 9 × 2000 = ₹18,000
Second person's income:
7x = 7 × 2000 = ₹14,000
Verification
First person's expenditure:
4y = 9x − 2000
Second person's expenditure:
3y = 7x − 2000
Both equations satisfy the given saving of ₹2000, confirming that the obtained incomes are correct.
Final Answer
First person's monthly income = ₹18,000
Second person's monthly income = ₹14,000
Key Formula
Income − Expenditure = Savings
Important Points to Remember
- Whenever ratios are given, assume the quantities using a common variable.
- Convert the given information into linear equations.
- Use elimination or substitution method to solve the equations.
- Always verify the final answer using the original conditions.
- Savings are always calculated as Income minus Expenditure.
Common Mistakes Students Make
- Using the same variable for income and expenditure ratios.
- Writing expenditure minus income instead of income minus expenditure.
- Forgetting that both persons save the same amount (₹2000).
- Making calculation mistakes while eliminating variables.
- Not checking whether the final answers satisfy both equations.
Exam Tips
- Read the question carefully and identify what each ratio represents.
- Write variables clearly before forming equations.
- Keep all equations neatly numbered for easy elimination.
- Use the elimination method when coefficients can be made equal easily.
- These word problems are frequently asked in Class 10 Board Exams.