The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save ₹2000 per month, find their monthly incomes.
Step-by-step solution
To find: The two monthly incomes
Idea: Write the incomes as 9x and 7x and the expenditures as 4y and 3y. Income − expenditure = saving gives two equations. Make the y-coefficients equal and subtract to eliminate y.
- Incomes: 9x and 7x; expenditures: 4y and 3y. Saving = income − expenditure: 9x − 4y = 2000 … (1), 7x − 3y = 2000 … (2).1 mark
- Step 1: Multiply (1) by 3 and (2) by 4 so the y-terms match: 27x − 12y = 6000 … (3), 28x − 12y = 8000 … (4).1 mark
- Step 2: (4) − (3): x = 2000 (y is eliminated).½ mark
- Step 3: In (1): 9(2000) − 4y = 2000 ⇒ 4y = 16000 ⇒ y = 4000.½ mark
- Incomes: 9 × 2000 = ₹18,000 and 7 × 2000 = ₹14,000.1 mark
Check: Incomes 18000 : 14000 = 9 : 7 ✓. Expenditures 16000 : 12000 = 4 : 3 ✓. Savings 2000 each ✓.
Answer to write in the exam
Let incomes = 9x, 7x; expenditures = 4y, 3y
9x − 4y = 2000 … (1); 7x − 3y = 2000 … (2)
(1) × 3: 27x − 12y = 6000 … (3); (2) × 4: 28x − 12y = 8000 … (4)
(4) − (3): x = 2000
From (1): 18000 − 4y = 2000 ⇒ y = 4000
∴ Incomes = ₹18,000 and ₹14,000
Common mistakes that cost marks
- Giving x = 2000 as the answer. x is one “part”; the incomes are 9x and 7x.
- Using the same letter for both ratios (incomes 9x, 7x and expenditures 4x, 3x). The two ratios have different part sizes.
- Subtracting (4) − (3) and writing −x = 2000.
How this can come in the exam
Two sisters earn in the ratio 5 : 4 and spend in the ratio 3 : 2. Each saves ₹3000 a month.
(i) Form two equations. (ii) Find each sister’s income. (iii) Find each sister’s spending.
Show answer
(i) 5x − 3y = 3000, 4x − 2y = 3000 (1 mark). (ii) (i)×2 − (ii)×3: 10x − 12x = 6000 − 9000 ⇒ x = 1500; incomes ₹7500 and ₹6000 (2 marks). (iii) y = 1500: spending ₹4500 and ₹3000 (1 mark).Try one yourself
Incomes are in the ratio 3 : 2 and expenditures in the ratio 5 : 3. Each saves ₹1000. Find the incomes.
Show answer
3x − 5y = 1000, 2x − 3y = 1000 ⇒ x = 2000, y = 1000. Incomes ₹6000 and ₹4000.
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