The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. Find both the numbers.
Step-by-step solution
To find: The number and the number with its digits interchanged
Idea: A two-digit number with tens digit t and units digit u is 10t + u; reversed it is 10u + t. Their sum is 11(t + u).
- Let the digits be x and x + 3. Number = 10x + (x + 3); reversed number = 10(x + 3) + x.1 mark
- Sum: 11x + 3 + 11x + 30 = 143 ⇒ 22x + 33 = 143.1 mark
- 22x = 110 ⇒ x = 5. The digits are 5 and 8, so the two numbers are 58 and 85.1 mark
Check: 58 + 85 = 143 ✓ and 8 − 5 = 3 ✓.
Answer to write in the exam
Let the digits be x and x + 3
Number = 10x + (x + 3) = 11x + 3; reversed = 10(x + 3) + x = 11x + 30
(11x + 3) + (11x + 30) = 143 ⇒ 22x = 110 ⇒ x = 5
Digits 5 and 8
∴ The numbers are 58 and 85
Common mistakes that cost marks
- Writing the number as x + (x + 3), forgetting that the tens digit is worth 10 times its face value.
- Giving only one number. The question asks for both the original and the reversed number.
- Writing the digits as x and 3x (“differ by 3” means subtract, not multiply).
How this can come in the exam
The sum of a two-digit number and the number formed by reversing its digits is always divisible by
- 9
- 10
- 11
- 2
Show answer
(C) 11
(10t + u) + (10u + t) = 11(t + u).
The sum of the digits of a two-digit number is 9. When the digits are reversed, the new number is 27 more than the original. Find the number.
Show answer
Let the tens digit be t; units 9 − t (1 mark). (10(9 − t) + t) − (10t + 9 − t) = 27 ⇒ 81 − 18t = 27 (1 mark) ⇒ t = 3; the number is 36 (1 mark).Try one yourself
The digits of a two-digit number differ by 1, and the number plus its reverse is 121. Find both numbers.
Show answer
11(t + u) = 121 ⇒ t + u = 11; digits 5 and 6: the numbers are 56 and 65.
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