In an examination, a boy was asked to multiply a given number by 7/19. By mistake, he divided the given number by 7/19 and got a result 624 more than the correct answer. The sum of digits of the given number is (a) 10 (b) 11 (c) 13 (d) 14

Let the given number be \(x\). According to the problem:

– The correct operation was to multiply \(x\) by \(\frac{7}{19}\): \(x \times \frac{7}{19}\).
– By mistake, the boy divided \(x\) by \(\frac{7}{19}\), which is equivalent to multiplying \(x\) by \(\frac{19}{7}\): \(x \times \frac{19}{7}\).
– The mistake led to a result that was 624 more than the correct answer.

Thus, we have:

\[
x \times \frac{19}{7} = x \times \frac{7}{19} + 624
\]

Rearrange the equation to isolate \(x\):

\[
x \times \frac{19}{7} – x \times \frac{7}{19} = 624
\]

Factoring \(x\) out:

\[
x \left(\frac{19}{7} – \frac{7}{19}\right) = 624
\]

To solve for \(x\), simplify the expression in the parentheses:

\[
x \left(\frac{19^2 – 7^2}{7 \times 19}\right) = 624
\]

Calculate the difference of squares:

\[
19^2 – 7^2 = (19 + 7)(19 – 7) = 26 \times 12 = 312
\]

Substitute back into the equation:

\[
x \left(\frac{312}{7 \times 19}\right) = 624
\]

Simplify the fraction:

\[
x \left(\frac{312}{133}\right) = 624
\]

Solving for \(x\):

\[
x = 624 \times \frac{133}{312} = 2 \times 133 = 266
\]

The sum of the digits of the given number \(266\) is \(2 + 6 + 6 = 14\).

Answer: (d) 14.