Difficulty: Medium
Correct Answer: 84
Explanation:
Introduction:
This question checks your understanding of how H.C.F., L.C.M., and the ratio of two numbers are related. It is a typical aptitude problem where you must reconstruct the original numbers from their H.C.F., L.C.M., and ratio.
Given Data / Assumptions:
Concept / Approach:
If two numbers are in the ratio m : n and have H.C.F. h, then the numbers can be written as: h * m and h * n Their L.C.M. is then: h * m * n (when m and n are co-prime) We use the ratio and H.C.F. to get the numbers, and check consistency with the given L.C.M.
Step-by-Step Solution:
Step 1: Let the numbers be 21 * 1 and 21 * 4, since the ratio is 1 : 4. Step 2: So the numbers are 21 and 84. Step 3: Check the H.C.F. of 21 and 84. H.C.F.(21, 84) = 21 (correct). Step 4: Compute their L.C.M. using the product relation. Product = 21 * 84 = 1764. Using HCF * LCM = product: 21 * LCM = 1764 ⇒ LCM = 1764 / 21 = 84. Step 5: The calculated L.C.M. matches the one given. Step 6: Therefore, the larger number is 84.
Verification / Alternative check:
Besides the H.C.F. and L.C.M. check, the ratio condition is satisfied: 21 : 84 = 1 : 4 (dividing both by 21). Thus, the numbers 21 and 84 correctly meet all given constraints and 84 is indeed the larger number.
Why Other Options Are Wrong:
48 and 12: Neither pair with any other number in the ratio 1 : 4 gives H.C.F. 21 and L.C.M. 84. 108: With the smaller number in 1 : 4 ratio, L.C.M. and H.C.F. would not match the given values. 21: This is the smaller number, not the larger one.
Common Pitfalls:
A common mistake is to ignore the ratio and try to guess numbers randomly from the options. Others may misapply the relation between H.C.F., L.C.M., and the product of numbers. Correctly interpreting the ratio and using the H.C.F. as a common factor leads quickly to the solution.
Final Answer:
The larger of the two numbers is 84.
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