Question

Assertion: The mean of 15 numbers is 25. If 6 is subtracted from each number, the mean of new numbers is 19.
Reason: Mode = 3(median) − 2(mean).
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true and Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Answer 1

(b) Both Assertion (A) and Reason (R) are true and Reason (R) is not a correct explanation of Assertion (A).

Assertion (A):
Mean of 15 numbers = 25

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{numbers}\mathrm{be}{x}_1,x_2,...x_{15}. \\ \mathrm{Given}: \\ \frac{x_1+x_2+...+x_{15}}{15}=25 \end{gathered}$$
$\Rightarrow x_1+x_2+...+x_{15}=375$
When 6 is subtracted from each number, we get:
$$\begin{gathered} \frac{\left(x_1-6\right)+\left(x_2-6\right)+.....+\left(x_{15}-6\right)}{15} \\ =\frac{\left(x_1+x_2+....+x_{15}\right)-6\times 15}{15} \\ =\frac{375-90}{15} \\ =\frac{285}{15} \\ =19 \end{gathered}$$

This means that Assertion (A) is true.

Reason (R):
Mode = 3(Median) $-$ 2(Mean)
This is the empirical formula.
Hence, Reason (R) is true.

But Reason (R) is not the correct explanation of Assertion (A).