The LCM of two numbers is 3 times their HCF. The sum of LCM and HCF is 120. If one of the numbers is 90, then the other number is

1. A

   4

2. B

   30

3. C

   90

4. D

   120

Appeared in: UPPSC 2022

Explanation

• The problem provides two conditions: LCM = 3 × HCF and LCM + HCF = 120.
• Substituting the first condition into the second gives 3 × HCF + HCF = 120, which simplifies to 4 × HCF = 120, so HCF = 30.
• Using the HCF, the LCM is calculated as LCM = 3 × 30 = 90.
• The core principle is that the product of two numbers equals the product of their HCF and LCM.
• Given one number is 90, the equation is 90 × (Other Number) = 30 × 90.
• Solving for the other number gives 30.

Why Other Options Were Wrong

• Option A: The number 4 is the result of dividing the sum (120) by the HCF (30), or the ratio between the combined parts (4H) and the sum. It is part of the intermediate calculation but is not the final answer.
• Option C: 90 is the value of the LCM and is also the first number given in the problem. It is a common mistake to confuse the LCM or the given number with the unknown second number.
• Option D: 120 is the sum of the HCF and LCM as stated in the problem. It is a given value used in the calculation, not the result.

Related Visual

Clinical Relevance

• Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Relationship between HCF, LCM, and the product of two numbers as background academic context rather than a clinical decision trigger.
• This question tests general intelligence and numerical ability, which are components of many nursing entrance and recruitment exams.
• Strong numeracy skills are essential for nurses in clinical practice for tasks such as medication dosage calculation, interpreting lab results, and managing fluid balances.
• What if the sum of HCF and LCM was 80? Following the same logic, 4H = 80, so H = 20 and L = 60. If one number was 60, the other would be 20.

How to Approach the Question

• First, carefully read the problem and identify all the given information and the relationships between the variables (LCM, HCF, and the numbers).
• Translate the word problem into algebraic equations. Let HCF be 'H' and LCM be 'L'. The problem states L = 3H and L + H = 120.
• Solve this system of two linear equations to find the numerical values of H and L.
• Recall the fundamental theorem relating HCF, LCM, and the two numbers: Number 1 × Number 2 = HCF × LCM.
• Substitute the known values (one of the numbers, HCF, and LCM) into this theorem.
• Solve the final equation to find the unknown number.

Concept Tested & Keywords

• Concept Tested: Relationship between HCF, LCM, and the product of two numbers.
• Stem keywords: LCM, HCF, sum of LCM and HCF, product of two numbers
• Lead-in keywords: the other number is

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Qu0QBs2cDXEw7XEtDLqldb

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