Question

The ratio of a two-digit natural number to a number formed by reversing its digits is $4:7.$ Which of the following is the sum of all the such numbers?

• A. $120$
• B. $330$
• C. $72$
• D. $108$

Answer 1

The correct option is A $120$

Let the two digit number be $10a+b$ and the number formed by reversing its digits be $10b+a.$
Then, $\frac{10a+b}{10b+a}=\frac{4}{7}$
$\Rightarrow 70a+7b=40b+4a$
$\Rightarrow 66a=33b$
$\Rightarrow b=2a$
When $a=1,b=2;$ number $=12$
When $a=2,b=4;$ number $=24$
When $a=3,b=6;$ number $=36$
When $a=4,b=8;$ number $=48$
When $a=5,b=10;$ that is not a single digit (reject it)

Sum of all such numbers $=12+24+36+48=120$