Averages , Ratio and Proportion Question 7 - CAT 2024
There are four numbers such that average of first two numbers is 1 more than the first number, average of first three numbers is 2 more than average of first two numbers, and average of first four numbers is 3 more than average of 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.
- Condition 1 (First two numbers):
(a + b)/2 = a + 1 ⟹ a + b = 2a + 2 ⟹ b = a + 2
The average of the first two numbers is (a + 1).
- Condition 2 (First three numbers):
(a + b + c)/3 = (a + 1) + 2 = a + 3 ⟹ a + b + c = 3a + 9
Substitute b = a + 2 into the expression:
a + (a + 2) + c = 3a + 9 ⟹ c = a + 7
The average of the first three numbers is (a + 3).
- Condition 3 (First four numbers):
(a + b + c + d)/4 = (a + 3) + 3 = a + 6 ⟹ a + b + c + d = 4a + 24
Substitute b = a + 2 and c = a + 7 into the expression :
a + (a + 2) + (a + 7) + d = 4a + 24 ⟹ d = a + 15
The smallest number is a and the largest number is d = a + 15.
Difference = d − a = (a + 15) − a = 15
Correct Answer: 15
💡 FundaMakers Tip:
Instead of treating each number as an independent variable, express every subsequent number relative to the first number a. Following the step-by-step average updates reveals a clear arithmetic growth pattern, giving you the final difference directly without needing the values of the individual numbers.