Question

A two digit number is such that the unit digit mu tiplied by 3 is equal to three more than the sum of the digits. When the digits are reversed the resulting number is 18 less than the original number. Find the units digit of the original number.

• A. 8
• B. 5
• C. 4
• D. 2
• E. 1

Answer 1

The Correct Option is B

The correct answer is option (B):

5

Let's break down this problem step-by-step to find the units digit of the original two-digit number.

First, let's represent the digits. Let 'x' be the tens digit and 'y' be the units digit of the original number. The original number can then be represented as 10x + y. The reversed number would be 10y + x.

Now, let's translate the problem's clues into equations:

Clue 1: "The unit digit multiplied by 3 is equal to three more than the sum of the digits."
This translates to: 3y = x + y + 3

Clue 2: "When the digits are reversed, the resulting number is 18 less than the original number."
This translates to: 10y + x = 10x + y - 18

Now we have a system of two equations:

Equation 1: 3y = x + y + 3
Equation 2: 10y + x = 10x + y - 18

Let's simplify these equations:

From Equation 1:
3y = x + y + 3
2y = x + 3
x = 2y - 3 (Equation 3)

From Equation 2:
10y + x = 10x + y - 18
9y - 9x = -18
y - x = -2 (Equation 4)

Now, we can substitute the value of x from Equation 3 into Equation 4:

y - (2y - 3) = -2
y - 2y + 3 = -2
-y = -5
y = 5

We've found that y = 5. Since 'y' represents the units digit of the original number, the units digit is 5.

Now, let's check our answer by finding the tens digit, x:

x = 2y - 3
x = 2(5) - 3
x = 10 - 3
x = 7

The original number is 75. The reversed number is 57.

Let's verify the original clues:

Clue 1: 3 * 5 = 7 + 5 + 3 → 15 = 15. Correct!
Clue 2: 57 = 75 - 18 → 57 = 57. Correct!

Therefore, the units digit of the original number is 5.