The incomes of α and β are in the ratio 9 : 4 and their expenditures are in the ratio 7 : 3. If each saves ₹4,500, what is the expenditure of β?

Appeared in: UP NHM CHO 7 Sept 2022(shift-1st)

Explanation

The problem is solved by translating the word problem into a system of two linear equations with two variables.
Let incomes be 9x and 4x, and expenditures be 7y and 3y. The two equations are 9x - 7y = 4500 and 4x - 3y = 4500.
Solving this system yields y = 22,500.
The expenditure of β is 3y, which calculates to 3 * 22,500 = 67,500.

Option A: This value is incorrect. It does not result from the correct algebraic solution of the system of equations derived from the problem's conditions.
Option B: This value is incorrect. It may arise from a calculation error, such as incorrectly solving the system of equations or misinterpreting the final variable to calculate.
Option D: This value is incorrect. It does not satisfy the conditions. If β's expenditure was ₹75,600, then y would be 25,200, which would lead to inconsistent savings amounts for α and β.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving linear equations derived from ratio and proportion problems as background academic context rather than a clinical decision trigger.
This question tests quantitative aptitude, a foundational skill required for many competitive examinations, including those for nursing positions.
While not a clinical skill, the logical reasoning and problem-solving abilities demonstrated by solving such problems are transferable to analyzing complex situations in a clinical setting.
What if the savings were different for each person? If α saved ₹4,500 and β saved ₹5,000, you would have two distinct equations (9x - 7y = 4500 and 4x - 3y = 5000) and would need to solve them simultaneously using substitution or elimination without equating them directly.

How to Approach the Question

First, identify the key information: the income ratio (9:4), the expenditure ratio (7:3), and the savings amount (₹4,500 for each).
Assign variables to represent the unknown quantities. Let the incomes be 9x and 4x, and the expenditures be 7y and 3y.
Use the fundamental relationship: Income - Expenditure = Savings. Formulate an equation for each person.
You will now have a system of two linear equations. Solve these equations simultaneously to find the value of the variables.
Carefully read the question again to determine what value you need to calculate. In this case, it's the expenditure of β (3y).
Substitute the value of 'y' you found back into the expression for β's expenditure to get the final answer.

Concept Tested & Keywords

Concept Tested: Solving linear equations derived from ratio and proportion problems.
Stem keywords: incomes, expenditures, ratio, saves
Lead-in keywords: what is

QJWPC5V2e7A63YmSVuHMJw

Practise the full UP NHM CHO 7 Sept 2022(shift-1st)

Attempt every question from this paper in a timed mock, then review the full solution for each one.