If P and Q are in the ratio of 7:8 and the difference of their squares is 60, then what is the value of P?

Appeared in: UPPSC 2022

Explanation

The problem requires translating a word problem into an algebraic equation.
The ratio 7:8 is represented by letting P = 7x and Q = 8x, where 'x' is a common multiplier.
The condition 'difference of their squares is 60' is written as Q² - P² = 60, which becomes (8x)² - (7x)² = 60.
Solving this equation (64x² - 49x² = 60) yields 15x² = 60, so x² = 4, and x = 2.
Finally, the value of P is found by substituting x back into its expression: P = 7 * 2 = 14.

Option A: If P = 36, then the common multiplier x would be 36/7. This would make Q = 8 * (36/7) = 288/7. The difference of their squares, (288/7)² - 36², is approximately 385, which is not 60.
Option B: If P = 45, then the common multiplier x would be 45/7. This would make Q = 8 * (45/7) = 360/7. The difference of their squares, (360/7)² - 45², is approximately 619, which is not 60.
Option C: If P = 24, then the common multiplier x would be 24/7. This would make Q = 8 * (24/7) = 192/7. The difference of their squares, (192/7)² - 24², is exactly 180, which is not 60.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving algebraic problems involving ratios and the difference of squares as background academic context rather than a clinical decision trigger.
This question assesses quantitative aptitude, a key component of the general intelligence section in many nursing and other competitive entrance examinations.
Strong problem-solving and mathematical reasoning skills are foundational for clinical tasks such as accurate drug dosage calculation, interpreting lab results, and understanding statistical data in research.
What if the question asked for the sum of the numbers? After finding P=14 and Q=16, the sum would be 14 + 16 = 30.

How to Approach the Question

First, identify the given information: a ratio (P:Q = 7:8) and a mathematical relationship (Q² - P² = 60).
Represent the numbers in the ratio using a common variable 'x'. Let P = 7x and Q = 8x.
Substitute these expressions into the given equation: (8x)² - (7x)² = 60.
Simplify and solve the algebraic equation for 'x'. You can expand directly: 64x² - 49x² = 60.
Once 'x' is found, substitute its value back into the expression for P (P = 7x) to find the final answer.
As a final check, calculate Q (Q = 8x) and verify that the difference of the squares of your calculated P and Q is indeed 60.

Concept Tested & Keywords

Concept Tested: Solving algebraic problems involving ratios and the difference of squares.
Stem keywords: P and Q, ratio, 7:8, difference of their squares, 60
Lead-in keywords: what is the value

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