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1. If the digit in the unit's place of a two-digit number is halved and the digit in the ten's place is doubled, the number thus obtained is equal to the number obtained by interchanging the digits. Which of the following is definitely true ?

A. Sum of the digits is a two-digit number

B. Digit in the unit's place is half of the digit in the ten's place

C. Digit in the unit's place and the ten's place are equal

D. Digit in the unit's place is twice the digit in the ten's place

Solution:

Let the ten's digit be x and the unit's digit be y
Then, number = 10x + y
New number :

\begin{aligned} = 10 \times 2 x + \frac{y}{2} \\ = 20 x + \frac{y}{2} \end{aligned}

\begin{aligned} ∴ 20 x + \frac{y}{2} = 10 y + x \\ \Leftrightarrow 40 x + y = 20 y + 2 x \\ \Leftrightarrow 38 x = 19 y \\ \Leftrightarrow y = 2 x \end{aligned}

So, the unit's digit is twice the ten's digit.

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