Central Tendency & DispersionPYQ July 21Question 2863 of 473
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There are n\displaystyle n numbers. When 50 is subtracted from each of these number the sum of the numbers so obtained is −10\displaystyle -10. When 46 is subtracted from each of the original n\displaystyle n numbers, then the sum of numbers so obtained is 70. What is the mean of the original n\displaystyle n numbers?

Options

A56.8\displaystyle 56.8
B25.7\displaystyle 25.7
C49.5\displaystyle 49.5
D53.8\displaystyle 53.8
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Correct Answer

✅ Option c — 49.5\displaystyle 49.5

All Options:

  • A56.8\displaystyle 56.8
  • B25.7\displaystyle 25.7
  • C49.5\displaystyle 49.5
  • D53.8\displaystyle 53.8

Detailed Solution & Explanation

**Step 1: Set up equations using the given conditions.** Let the n\displaystyle n original numbers have sum S\displaystyle S and mean xˉ\displaystyle \bar{x}, so S=nxˉ\displaystyle S = n\bar{x}. **Condition 1:** When 50 is subtracted from each number: ∑(xi−50)=−10  ⟹  S−50n=−10⋯(1)\sum(x_i - 50) = -10 \implies S - 50n = -10 \quad \cdots(1) **Condition 2:** When 46 is subtracted from each number: ∑(xi−46)=70  ⟹  S−46n=70⋯(2)\sum(x_i - 46) = 70 \implies S - 46n = 70 \quad \cdots(2) **Step 2: Subtract equation (1) from equation (2).** (S−46n)−(S−50n)=70−(−10)(S - 46n) - (S - 50n) = 70 - (-10) 4n=80  ⟹  n=204n = 80 \implies n = 20 **Step 3: Find S\displaystyle S using equation (1).** S−50(20)=−10  ⟹  S=1000−10=990S - 50(20) = -10 \implies S = 1000 - 10 = 990 **Step 4: Compute the mean.** xˉ=Sn=99020=49.5\bar{x} = \frac{S}{n} = \frac{990}{20} = 49.5 **Note:** The computed mean is 49.5, which matches Option C, not Option A (56.8). Let us recheck. **Recheck:** S=990\displaystyle S = 990, n=20\displaystyle n = 20, mean =49.5\displaystyle = 49.5. So the correct answer by computation is **49.5**. Hence, **Option C** is the correct answer.

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