Averages , Ratio and Proportion Question 2 - CAT 2023
There are three persons A, B and C in a room. If a person D joins the room, the average weight of the persons in the room reduces by x kg. Instead of D, if person E joins the room, the average weight of the persons in the room increases by 2x kg. If the weight of E is 12 kg more than that of D, then the value of x is:
Step-by-Step Explanation
Let the initial average weight of A, B, and C (3 people) be Aₐᵥg.
- Apply the deviation shortcut for replacements:
When a new person joins a group, their value equals the old average plus the total change they bring to the group. The new group size is 4.
- When D joins: The average reduces by x kg.
D = Aₐᵥg + 4(−x) ⟹ D = Aₐᵥg − 4x
- When E joins instead: The average increases by 2x kg.
E = Aₐᵥg + 4(+2x) ⟹ E = Aₐᵥg + 8x
- Solve using the given weight difference:
We are given that E's weight is 12 kg more than D's weight:
E − D = 12
Substitute the expressions for E and D:
(Aₐᵥg + 8x) − (Aₐᵥg − 4x) = 12
12x = 12
⟹ x = 1
Correct Option: C
💡 FundaMakers Tip:
Use the Average Deviation Formula:
New Value = Old Average + (Nₙₑw × ΔAverage).
This completely eliminates the need to track the sum of A, B, and C. Since Nₙₑw = 4 in both cases, the gap between E and D is simply:
4 × (Difference in average changes)
= 4 × (2x − (−x))
= 12x
Setting 12x = 12 solves the question in under 10 seconds!