Measures of Central Tendency and DispersionPYQ Sept 25Question 4176 of 473
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A set contains seven numbers whose average is 12. The average of four greater numbers is 18 and the average of four smaller numbers is 8. Which of the followings is the value of one of the numbers?

Options

A12
B16
C18
D20
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Correct Answer

✅ Option d — 20

All Options:

  • A12
  • B16
  • C18
  • D20

Detailed Solution & Explanation

Let the seven numbers sorted in ascending order be represented by x1≤x2≤x3≤x4≤x5≤x6≤x7\displaystyle x_1 \le x_2 \le x_3 \le x_4 \le x_5 \le x_6 \le x_7.
1. **Sum of all 7 numbers**:
Since their average is 12\displaystyle 12:
∑i=17xi=7×12=84\sum_{i=1}^7 x_i = 7 \times 12 = 84
x1+x2+x3+x4+x5+x6+x7=84— (Equation 1)x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 84 \quad \text{--- (Equation 1)}
2. **Sum of the 4 smaller numbers**:
The four smaller numbers are x1,x2,x3,\displaystyle x_1, x_2, x_3, and x4\displaystyle x_4. Since their average is 8\displaystyle 8:
x1+x2+x3+x4=4×8=32— (Equation 2)x_1 + x_2 + x_3 + x_4 = 4 \times 8 = 32 \quad \text{--- (Equation 2)}
3. **Sum of the 4 greater numbers**:
The four greater numbers are x4,x5,x6,\displaystyle x_4, x_5, x_6, and x7\displaystyle x_7. Since their average is 18\displaystyle 18:
x4+x5+x6+x7=4×18=72— (Equation 3)x_4 + x_5 + x_6 + x_7 = 4 \times 18 = 72 \quad \text{--- (Equation 3)}
4. **Adding Equation 2 and Equation 3**:
(x1+x2+x3+x4)+(x4+x5+x6+x7)=32+72(x_1 + x_2 + x_3 + x_4) + (x_4 + x_5 + x_6 + x_7) = 32 + 72
x1+x2+x3+2x4+x5+x6+x7=104x_1 + x_2 + x_3 + 2x_4 + x_5 + x_6 + x_7 = 104
We can rewrite this expression as:
(x1+x2+x3+x4+x5+x6+x7)+x4=104(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7) + x_4 = 104
Substitute the value of the total sum from Equation 1 (84\displaystyle 84):
84+x4=10484 + x_4 = 104
x4=104−84=20x_4 = 104 - 84 = 20
Thus, the 4th number (the middle number) is 20\displaystyle 20. Since 20\displaystyle 20 is one of the numbers in the set, it matches the choices.
Hence, **Option D** is the correct answer.

About This Chapter: Measures of Central Tendency and Dispersion

Paper

Paper 3: Quantitative Aptitude

Weightage

12-15 Marks

Key Topics

Mean, Median, Mode, Range, Mean Deviation, Standard Deviation

The core foundation of Statistics. This chapter covers Mean (Arithmetic, Geometric, Harmonic), Median, Mode, and their properties. It also explores measures of spread like Range, Mean Deviation, Standard Deviation, and Quartile Deviation.

View Official ICAI Syllabus

Exam Strategy Tip

Do not just memorize formulas; ICAI loves asking about the mathematical properties (e.g., 'sum of deviations from the AM is always zero'). You can usually eliminate 2 options just by knowing the properties.

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