Question

The arithmetic mean of 20 numbers is 16. If each number is multiplied by 3, then the new mean is

• A. 60
• B. 48
• C. 23
• D. 19

Hint:

Multiplying all observations by a constant multiplies the mean by the same constant.

Answer 1

The Correct Option is B

The given problem involves finding the new arithmetic mean of numbers after they have been modified by a specific operation. Let's solve it step-by-step:

1. The arithmetic mean of a set of numbers is given by the formula: \(\overline{x} = \frac{\text{Sum of the numbers}}{\text{Number of numbers}}\).
2. We are told that the arithmetic mean of 20 numbers is 16. This implies: \(\overline{x} = 16 = \frac{\text{Sum}}{20}\).
3. To find the total sum of these 20 numbers, we multiply both sides of the equation by 20: \(\text{Sum} = 16 \times 20\).
   • \(16 \times 20 = 320\)
4. Each number is then multiplied by 3:
   • The new series of numbers then can be described as each original number being multiplied by 3.
5. The sum of the new series is: \(\text{New Sum} = 3 \times \text{Original Sum} = 3 \times 320\).
6. By calculating \(3 \times 320\), we get:
   • \(3 \times 320 = 960\)
7. The number of terms remains unchanged, at 20, hence the new mean is: \(\text{New Mean} = \frac{\text{New Sum}}{20} = \frac{960}{20}\).
8. Calculate \(\frac{960}{20}\):
   • \(\frac{960}{20} = 48\)

Hence, the correct answer is 48.