Question

There are 50 numbers. Each number is subtracted from 53 and the mean of the numbers so obtained is found to be -3.5. The mean of the given numbers is

(a) 46.5
(b) 49.5
(c) 53.5
(d) 56.5

Answer 1

Let the 50 numbers be x₁, x₂, x₃,..., x₅₀.

When each number is subtracted from 53, the resulting numbers are 53 − x₁, 53 − x₂, 53 − x₃,..., 53 − x₅₀.

It is given that, the mean of resulting numbers 53 − x₁, 53 − x₂, 53 − x₃,..., 53 − x₅₀ is −3.5.

We know

Mean = $\frac{\mathrm{Sum}\mathrm{of}\mathrm{observations}}{\mathrm{Number}\mathrm{of}\mathrm{observations}}$

$\therefore -3.5=\frac{\left(53-x_1\right)+\left(53-x_2\right)+\left(53-x_3\right)+...+\left(53-x_{50}\right)}{50}$

$\Rightarrow \frac{\underset{50\mathrm{times}}{\underbrace{53+53+53+...+53}}-\left(x_1+x_2+x_3+...+x_{50}\right)}{50}=-3.5$

$\Rightarrow \frac{53\times 50}{50}-\frac{x_1+x_2+x_3+...+x_{50}}{50}=-3.5$

$\Rightarrow \frac{x_1+x_2+x_3+...+x_{50}}{50}=53+3.5=56.5$ .....(1)

Now,

Mean of 50 given numbers = $\frac{x_1+x_2+x_3+...+x_{50}}{50}$

∴ Mean of 50 given numbers = 56.5 [From (1)]

Thus, the mean of the given numbers is 56.5.

Hence, the correct answer is option (d).