Three numbers are in the ratio 4:5:6 and their average is 25. What is the largest of these three numbers?

Introduction / Context:
This question combines the concept of ratios with the concept of average. Three numbers are given in a fixed ratio, and their average is known. From this, we can find the actual numbers. We are asked specifically for the largest number among the three. This type of problem is very common in aptitude tests for reinforcing proportional reasoning.

Given Data / Assumptions:
- Three numbers are in the ratio 4:5:6.
- Their average is 25.
- All numbers are real and consistent with the ratio.
- We need to find the largest number among them.

Concept / Approach:
When numbers are in a ratio, we can represent them using a common multiplier k. Thus, the three numbers can be written as 4k, 5k, and 6k. The average of three numbers is their sum divided by 3. Using the given average, we form an equation involving k and solve for it. Once we know k, we can compute each number and identify the largest one.

Step-by-Step Solution:
Step 1: Let the three numbers be 4k, 5k, and 6k.Step 2: Their sum is 4k + 5k + 6k = 15k.Step 3: The average of three numbers is total sum divided by 3, so average = 15k / 3.Step 4: Simplify 15k / 3 to get 5k.Step 5: The average is given as 25, thus 5k = 25.Step 6: Solve for k: k = 25 / 5 = 5.Step 7: Substitute k = 5 into the expressions for the numbers to get 4k = 4 * 5 = 20, 5k = 5 * 5 = 25, and 6k = 6 * 5 = 30.Step 8: The largest of these numbers is 30.

Verification / Alternative check:
We can verify by checking both the ratio and the average. The three numbers 20, 25, and 30 are indeed in the ratio 4:5:6 because dividing each by 5 yields 4, 5, and 6. Their sum is 20 + 25 + 30 = 75, and the average is 75 / 3 = 25, matching the given average. This confirms that the numbers are correctly found and that 30 is the largest number.

Why Other Options Are Wrong:
Values such as 40, 50, 60, or 24 do not fit as the largest number in a set that has ratio 4:5:6 with average 25. For example, if 40 were the largest number corresponding to 6k, then k would be 40 / 6, and the resulting numbers would not have an average of 25. Similar contradictions occur with the other options when checked carefully against the given average and ratio conditions.

Common Pitfalls:
Some learners mistakenly divide the average by each ratio part separately rather than expressing all numbers in terms of k. Others may incorrectly average the ratio values directly. The correct approach is to treat 4k, 5k, and 6k as the real numbers and use the given average to find k. This systematic method avoids confusion and errors.

Final Answer:
The largest of the three numbers is 30.