CAT 2024 Slot 3 QA Question 8

Type-in-the-answer (no negative marking) · Arithmetic · Mean, Median & Mode · Try it, then check the answer and solution below.

CAT 2024 Slot 3QAArithmetic • Mean, Median & ModeModerate

The average of three distinct real numbers is 28. If the smallest number is increased by 7 and the largest number is reduced by 10, the order of the numbers remains unchanged, and the new arithmetic mean becomes 2 more than the middle number, while the difference between the largest and the smallest numbers becomes 64. Then, the largest number in the original set of three numbers is

TITA Answer:

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⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: 70

Answer and solution

Answer: 70

📌 Core Concept

The problem involves three distinct numbers

a , b , and

with

a < b < c$. Their average is given, and modifications to the smallest and largest numbers affect their mean and difference. We use algebraic equations to solve for the largest number.

🔢 Step-by-Step Solution

Given:

Average of

a , b , c

is 28:

\frac{a + b + c}{3} = 28 \implies a + b + c = 84

After modifying

a

and

c

New numbers:

a + 7 , b , c - 10

Order remains

a + 7 < b < c - 10

New mean is 2 more than

b

\frac{(a + 7) + b + (c - 10)}{3} = b + 2

Difference between new largest and smallest is 64:

(c - 10) - (a + 7) = 64

Simplify the mean equation:

\frac{a + b + c - 3}{3} = b + 2 Substitute a + b + c = 84: \frac{84 - 3}{3} = b + 2 \implies 27 = b + 2 \implies b = 25

Solve for

c - a

(c - 10) - (a + 7) = 64 \implies c - a - 17 = 64 \implies c - a = 81

Substitute

b = 25

and

c = a + 81

into

a + b + c = 84

a + 25 + (a + 81) = 84 \implies 2a + 106 = 84 \implies 2a = -22 \implies a = -11

c = a + 81 = -11 + 81 = 70

⚡ 30-Second Shortcut

Recognize that the new mean relates directly to the middle number.

Use the difference equation to find the relationship between

c

and

a

Substitute known values to solve for

a

and

c

🎯 Final Answer

The largest number in the original set is

\boxed{70}

Correct Answer: 70

CAT 2024 Slot 3 QA Question 8

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