Two numbers are in the ratio 15 : 11 and their Highest Common Factor (HCF) is 13. Using this information, find the greater of the two numbers.

Introduction / Context:
This problem combines the concept of ratios with the idea of Highest Common Factor (HCF). Questions of this type appear often in aptitude exams where you are given a ratio and HCF, and you must deduce the actual numbers. Once the numbers are obtained, it is easy to answer additional questions such as finding the greater number, their sum, or their Least Common Multiple.

Given Data / Assumptions:

• The ratio of the two numbers is 15 : 11.
• The HCF of the two numbers is 13.
• We need to find the greater of the two numbers.
• Both numbers are positive integers.

Concept / Approach:
When two numbers are in the ratio m : n and their HCF is h, the actual numbers can be written as: first number = h * m second number = h * n. This works because the HCF represents the greatest common factor that has been factored out, and the ratio tells us how many multiples of that factor each number contains. After computing both numbers from this representation, we simply take the larger one as the answer.

Step-by-Step Solution:
Step 1: Use the ratio and HCF to express the numbers. Ratio = 15 : 11, HCF = 13. Number 1 = 15 * 13. Number 2 = 11 * 13. Step 2: Compute each number. Number 1 = 15 * 13 = 195. Number 2 = 11 * 13 = 143. Step 3: Identify the greater number. Between 195 and 143, the greater number is 195.

Verification / Alternative check:
We can verify our numbers satisfy both the HCF and ratio conditions. First, check the ratio: 195 : 143. Divide both by 13, the common factor. We get 15 : 11, which matches the given ratio. Next, check the HCF. The common factor is 13, and there is no larger factor that divides both 195 and 143. Therefore, the HCF is indeed 13. Since both conditions are satisfied, the values of the numbers are correct, and 195 is confirmed as the greater number.

Why Other Options Are Wrong:
125, 175, 143, and 169: None of these values fit correctly as the greater number in a pair that maintains both the given ratio 15 : 11 and HCF 13. For example, if 143 were considered the greater number, then dividing by 13 gives 11, so the smaller should be 15 * 13 = 195, which contradicts the assumption that 143 is the greater number. Similar reasoning rules out the other options when checked against the ratio and HCF conditions.

Common Pitfalls:
A frequent mistake is to treat the numbers in the ratio directly as the actual numbers rather than as multiples of the HCF. Some students also wrongly assume that the HCF is equal to one of the numbers given in the options. Forgetting to multiply by the HCF or mixing up which ratio term corresponds to the greater number can also lead to incorrect answers. Always express the numbers as h * m and h * n, compute them explicitly, and then compare to decide which is greater.

Final Answer:
The greater number is 195.