Two numbers are in the ratio 2 : 3 and their sum is 75. What is the larger of the two numbers?

Introduction / Context:
This is a straightforward question on numbers in a given ratio with their sum known. The two numbers share a fixed proportional relationship, and the total of the two is given. These two pieces of information are enough to determine each number exactly. Such questions train the ability to convert ratio based descriptions into actual values using the concept of total parts.

Given Data / Assumptions:
- Ratio of the numbers is 2 : 3.
- Sum of the two numbers is 75.
- We need to find the larger of the two numbers.

Concept / Approach:
When two numbers are in the ratio m : n, we can write them as 2k and 3k for some positive value k. Their sum then becomes 2k + 3k which is 5k. If the sum is known, we solve 5k equal to that sum to get k, and then each number is found by substituting k back into 2k and 3k. The larger number corresponds to the bigger coefficient in the original ratio, which here is 3k.

Step-by-Step Solution:
Step 1: Let the two numbers be 2k and 3k, following the ratio 2 : 3. Step 2: Their sum is 2k + 3k = 5k. Step 3: Given that 2k + 3k = 75, so 5k = 75. Step 4: Solve for k to get k = 75 / 5 = 15. Step 5: The first number is 2k = 2 * 15 = 30. Step 6: The second number is 3k = 3 * 15 = 45. Step 7: The larger of the two numbers is 45.

Verification / Alternative check:
Check that the numbers satisfy the original conditions. The ratio 30 : 45 simplifies to 2 : 3 when both terms are divided by 15. Their sum is 30 + 45 = 75, which matches the given total. Since both conditions are satisfied, the calculation is correct. Therefore 45 is confirmed as the larger number.

Why Other Options Are Wrong:
30 is the smaller number, not the larger, although it is part of the correct pair. 50 and 48 would make the sum or the ratio inconsistent with the given data. For instance, pairing 25 and 50 would give a sum of 75 but a ratio of 1 : 2, not 2 : 3. Thus only 45 fits as the larger number in the correct ratio based pair whose sum is 75.

Common Pitfalls:
Some students mistakenly divide the sum directly by one of the ratio terms or miscalculate the total parts. For example, they might divide 75 by 3 only, ignoring the fact that 2 parts and 3 parts together make 5 parts. Another pitfall is to swap which term of the ratio represents which number, but in this question we only need the larger number, which is always linked to the bigger ratio term 3k.

Final Answer:
The larger of the two numbers is 45.