Average, Mixture and Alligation Question 1 - CAT 2024
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:
Step-by-Step Explanation
Let the three original distinct numbers in ascending order be x, y, and z (where x < y < z).
- Original Sum: Since the average of the three numbers is 28, their sum is:
x + y + z = 28 × 3 = 84
- Analyze the Changes:
The smallest number becomes x + 7 and the largest becomes z − 10. The middle number remains y.
New Sum = x + 7 + y + z − 10 = x + y + z − 3 = 84 − 3 = 81
New Arithmetic Mean = 81 / 3 = 27
- Find the Middle Number (y):
The new mean is 2 more than the middle number:
27 = y + 2 ⟹ y = 25
Substituting y = 25 back into the original sum:
x + 25 + z = 84 ⟹ x + z = 59 --- (Equation 1)
- Use the New Difference to Solve:
The difference between the modified largest and smallest numbers is 64:
z − 10 − x − 7 = 64 ⟹ z − x − 17 = 64 ⟹ z − x = 81 --- (Equation 2)
- Add Equation 1 and Equation 2 together:
x + z + z − x = 59 + 81
⟹ 2z = 140
⟹ z = 70
The largest number in the original set is 70.
Correct Answer: 70
💡 FundaMakers Tip:
When handling word problems involving modifications to a dataset, track the Net Change in Sum first. Adding 7 and subtracting 10 means a net change of -3 to the total sum. Knowing the new sum instantly gives you the new average, allowing you to isolate the middle variable y without ever setting up a tedious three-variable system.