Number System 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 distinct real numbers in ascending order be x,y, and z (so x < y < z).
- Original state:
Average = (x + y + z)/(3) = 28 ⇒ x + y + z = 84
- Modified state:
The new numbers are (x + 7),y, and (z - 10).
New Sum = (x + 7) + y + (z - 10) = (x + y + z) - 3 = 84 - 3 = 81
New Mean = (81)/(3) = 27
The new mean is 2 more than the middle number (y):
27 = y + 2 ⇒ y = 25
Substitute y = 25 back into the original sum:
x + 25 + z = 84 ⇒ x + z = 59 --- (Equation 1)
- Use the final difference constraint:
The new difference between the largest and smallest numbers is 64:
(z - 10) - (x + 7) = 64 ⇒ z - x - 17 = 64 ⇒ z - x = 81 --- (Equation 2)
- Solve for z:
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
đź’ˇ FundaMakers Tip:
Use the Net Shift Method to find the new mean instantly. The net adjustment to the sum is + 7 - 10 = - 3. Spreading this change across 3 elements reduces the old mean by exactly 1 unit (28 - 1 = 27). Bypassing the full sum equations lets you calculate y = 25 immediately.
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