Two numbers are in the ratio 15 : 11. If their highest common factor (HCF) is 13, what is the greater of the two numbers?

Introduction:
This question checks whether you understand how the highest common factor (HCF) relates to numbers given in a specific ratio. This is a common pattern in ratio and HCF questions asked in competitive examinations.

Given Data / Assumptions:

• The ratio of two numbers is 15 : 11.
• Their HCF is 13.
• We must find the greater number.

Concept / Approach:
If two numbers are in the ratio m : n and their HCF is h, then the actual numbers can be written as:
First number = m * hSecond number = n * hThis works because the HCF is the common factor that multiplies the ratio terms to give the actual numbers.

Step-by-Step Solution:
Step 1: Identify ratio terms.m = 15, n = 11, and HCF = 13Step 2: Compute the actual numbers.First number = 15 * 13 = 195Second number = 11 * 13 = 143Step 3: Identify the greater number.Between 195 and 143, the greater number is 195

Verification / Alternative check:
Check that the HCF of 195 and 143 is indeed 13. Factorization: 195 = 3 * 5 * 13, 143 = 11 * 13. The common factor is 13, so HCF = 13. Now, the ratio 195 : 143 simplifies by dividing both sides by 13, giving 15 : 11, which matches the given ratio. Thus, our computation is fully consistent.

Why Other Options Are Wrong:
125, 175, 143, 169: None of these, when paired with an appropriate second number using HCF = 13, will retain the exact ratio 15 : 11. For example, 143 is actually the smaller number, not the greater one. The other values do not follow from multiplying 15 or 11 by 13.

Common Pitfalls:
A common mistake is to forget that HCF must be a factor of both numbers, or to treat the ratio 15 : 11 as actual numbers rather than scaled versions. Always multiply the ratio terms by the HCF to recover the real numbers.

Final Answer:
The greater number is 195.