CAT 2024 Slot 1 QA Q11 Ratio, Mixtures & Averages

Question 11 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.

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

TITA: type in the answer. There are no options and no negative marking.

Answer: 15 Ratio, Mixtures & Averages TITA

Solution

Let the numbers be \(a, b, c, d\).
\(\frac{a+b}{2} = a + 1 \Rightarrow b = a + 2\).
\(\frac{a+b+c}{3} = (a + 1) + 2 = a + 3 \Rightarrow c = 3a + 9 - (2a + 2) = a + 7\).
\(\frac{a+b+c+d}{4} = (a + 3) + 3 = a + 6 \Rightarrow d = 4a + 24 - (3a + 9) = a + 15\).
Largest \(-\) smallest \(= (a + 15) - a = 15\).

Shortcut. Take \(a = 0\): the numbers are 0, 2, 7, 15.