Two numbers are in the ratio 3 : 5 and their Highest Common Factor (HCF) is 20. Using this information, find the Least Common Multiple (LCM) of the two numbers.

Introduction / Context:
This question combines the idea of ratios with the concepts of HCF and LCM. Problems like this are common in aptitude tests because they check whether a student can move from a ratio to actual numbers and then apply number theory formulas. Once we express the two numbers in terms of their ratio and HCF, it becomes straightforward to calculate the Least Common Multiple, which is required here.

Given Data / Assumptions:

• The ratio of the two numbers is 3 : 5.
• HCF of the two numbers = 20.
• We need to find the LCM of the two numbers.
• Both numbers are positive integers.

Concept / Approach:
If two numbers are in the ratio m : n and their HCF is h, then the numbers can be expressed as: first number = 3 * 20 second number = 5 * 20 more generally, number1 = h * m and number2 = h * n. Once we know the explicit values of both numbers, we can find the LCM using either prime factorization or the relationship: LCM = (product of the numbers) / HCF. This is often the quickest approach for two numbers when HCF is known.

Step-by-Step Solution:
Step 1: Use the ratio and HCF to find the actual numbers. Ratio = 3 : 5, HCF = 20. Number 1 = 3 * 20 = 60. Number 2 = 5 * 20 = 100. Step 2: Use the relationship between product, HCF, and LCM. Number 1 * Number 2 = HCF * LCM. 60 * 100 = 20 * LCM. 6000 = 20 * LCM. LCM = 6000 / 20 = 300. Therefore, the LCM of the two numbers is 300.

Verification / Alternative check:
To verify, we can compute the LCM by prime factorization. The numbers are 60 and 100. Prime factorization gives: 60 = 2^2 * 3 * 5, 100 = 2^2 * 5^2. The LCM takes the highest powers of each prime: 2^2, 3, and 5^2. Thus, LCM = 2^2 * 3 * 5^2 = 4 * 3 * 25 = 300. This matches the value found using the relationship formula, confirming that 300 is correct.

Why Other Options Are Wrong:
30: This is far smaller than both numbers and cannot be their LCM since LCM must be at least as large as the largest number, which is 100.
60: While 60 is one of the numbers, it does not evenly divide 100 in a way that makes it the least common multiple of both 60 and 100.
10: This is actually a factor of both numbers, but it is a common factor rather than the least common multiple.
240: Although 240 is a common multiple of 60, it is not divisible by 100, so it cannot be the LCM of 60 and 100.

Common Pitfalls:
A common mistake is to treat the HCF as if it were one of the numbers directly or to misapply the ratio without multiplying by the HCF. Another error is forgetting the relationship LCM = (product) / HCF and instead attempting to list multiples manually, which is inefficient. Keeping the connection between ratio, HCF, and actual numbers clear makes these problems straightforward to solve.

Final Answer:
The LCM of the two numbers is 300.