Average, Mixture and Alligation Question 4 - CAT 2024
There are four numbers such that the average of the first two numbers is 1 more than the first number, the average of the first three numbers is 2 more than the average of the first two numbers, and the average of the first four numbers is 3 more than the average of the first three numbers. Then, the difference between the largest and the smallest numbers is:
Step-by-Step Explanation
Let the four numbers be a, b, c, and d. We evaluate the cumulative averages step-by-step:
- Average of the first two numbers:
(a + b) / 2 = a + 1 ⟹ a + b = 2a + 2 ⟹ b = a + 2
- Average of the first three numbers:
The average of the first two numbers is a + 1. The average of the first three numbers is 2 more than that, which equals a + 1 + 2 = a + 3.
(a + b + c) / 3 = a + 3 ⟹ a + b + c = 3a + 9
- Substitute b = a + 2 into the expression:
a + a + 2 + c = 3a + 9 ⟹ 2a + 2 + c = 3a + 9 ⟹ c = a + 7
- Average of the first four numbers:
The average of the first three numbers is a + 3. The average of all four numbers is 3 more than that, which equals a + 3 + 3 = a + 6.
(a + b + c + d) / 4 = a + 6 ⟹ a + b + c + d = 4a + 24
- Substitute the expressions for b and c:
a + a + 2 + a + 7 + d = 4a + 24 ⟹ 3a + 9 + d = 4a + 24 ⟹ d = a + 15
The smallest number is a and the largest is d. Their absolute difference is:
d − a = a + 15 − a = 15
Correct Answer: 15
💡 FundaMakers Tip:
Notice how each subsequent variable is explicitly forced to expand relative to the base value a by a predictable mathematical gap. Write the values out as adjustments from the baseline a: b = a + 2, c = a + 7, d = a + 15. Recognizing this linear cumulative build allows you to find the final spread without computing intermediate averages.