Two numbers are in the ratio 4:5 and their LCM is 240. What is their sum?

Appeared in: AIIMS CRE, SNO-2024

Explanation

The numbers are in the ratio 4:5, so they can be represented as 4x and 5x, where x is a common multiplier.
The Least Common Multiple (LCM) of 4x and 5x is 20x, since 4 and 5 are co-prime.
Given that the LCM is 240, we can set up the equation 20x = 240.
Solving for x gives x = 240 / 20 = 12.
The sum of the numbers is 4x + 5x = 9x. Substituting x = 12, the sum is 9 * 12 = 108.

Option A: This would be the sum if the common multiplier 'x' was 10 (since 9 * 10 = 90). If x=10, the numbers would be 40 and 50, and their LCM would be 200, not 240.
Option C: The sum of the numbers (4x + 5x = 9x) must be a multiple of 9. Since 100 is not divisible by 9, it cannot be the correct answer.
Option D: The sum of the numbers must be a multiple of 9. Since 120 is not divisible by 9, it can be eliminated. This value might be chosen if one mistakenly confuses the LCM value (240) and tries to relate it to the sum without proper calculation.

Visual 1: Flowchart: Step 1: Represent numbers as 4x and 5x. -> Step 2: Find LCM of 4x and 5x (20x). -> Step 3: Equate LCM to given value (20x = 240) and solve for x. -> Step 4: Calculate the sum (9x).

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving problems involving Ratios and Least Common Multiple (LCM) as background academic context rather than a clinical decision trigger.
This question tests quantitative aptitude and logical reasoning, which are essential skills evaluated in many nursing entrance examinations.
These mathematical skills are foundational for clinical practice, particularly in areas like medication dosage calculation, fluid balance monitoring, and interpreting data from medical devices.
What if? If the question stated the HCF (Highest Common Factor) was 12, the numbers would be 412=48 and 512=60. The sum would still be 108. In this problem, the common multiplier 'x' is, in fact, the HCF.

How to Approach the Question

First, identify the key information provided: the ratio of the numbers (4:5) and their LCM (240).
Represent the unknown numbers using the given ratio and a common multiplier, 'x'. The numbers are 4x and 5x.
Calculate the LCM of these algebraic expressions. Since 4 and 5 have no common factors, the LCM is 4 * 5 * x = 20x.
Set this expression equal to the given LCM value: 20x = 240.
Solve the equation for 'x' to find the common multiplier: x = 12.
Finally, calculate what the question asks for – the sum. The sum is 4x + 5x = 9x. Substitute the value of x: 9 * 12 = 108.

Concept Tested & Keywords

Concept Tested: Solving problems involving Ratios and Least Common Multiple (LCM)
Stem keywords: ratio, LCM, sum, numbers
Lead-in keywords: What is
Negative lead-in flag: false

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