The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?
Step-by-step solution
Idea: Write the number as 10x + y (x = tens digit, y = units digit) and its reverse as 10y + x. The sum condition gives x + y = 6; the difference condition gives x − y = 2 (if x is larger) or y − x = 2 (if y is larger).
- Let the tens digit be x and the units digit y, and first take x as the larger digit. The number is 10x + y and its reverse is 10y + x.½ mark
- Sum: (10x + y) + (10y + x) = 66 ⇒ 11(x + y) = 66 ⇒ x + y = 6 … (1).1 mark
- Digits differ by 2: x − y = 2 … (2).½ mark
- Eliminate y: (1) + (2) gives 2x = 8, x = 4; then y = 2. The number is 42. Check: 42 + 24 = 66 ✓ and 4 − 2 = 2 ✓.1 mark
- How many? Assuming the tens digit is the larger, 42 is the only answer. But nothing in the question forces that. If the units digit is larger, (2) becomes y − x = 2, giving x = 2, y = 4: the number 24, and 24 + 42 = 66 ✓. So, without the assumption, there are two such numbers: 42 and 24.1 mark
Check: 42 + 24 = 66 ✓ and the digits 4, 2 differ by 2 ✓. 24 + 42 = 66 ✓, digits 2, 4 differ by 2 ✓.
Answer to write in the exam
Let number = 10x + y; reversed = 10y + x
(10x + y) + (10y + x) = 66 ⇒ 11(x + y) = 66 ⇒ x + y = 6 … (1)
Digits differ by 2: x − y = 2 … (2)
(1) + (2): 2x = 8 ⇒ x = 4, y = 2 ⇒ number = 42
If y − x = 2: x = 2, y = 4 ⇒ number = 24
∴ The number is 42 (or 24); there are two such numbers
Common mistakes that cost marks
- Writing the number as x + y or xy instead of 10x + y.
- Forgetting to divide by 11: x + y = 66 is impossible for two digits.
- Missing the second number by assuming the tens digit must be the larger one.
How this can come in the exam
A two-digit number plus its reverse is 121, and the tens digit is 5 more than the units digit. The number is
- 83
- 38
- 92
- 74
Show answer
(A) 83
x + y = 11, x − y = 5 ⇒ x = 8, y = 3: 83 (83 + 38 = 121).
The sum of the digits of a two-digit number is 8, and the number is 18 more than its reverse. Find the number.
Show answer
x + y = 8 (½). (10x + y) − (10y + x) = 18 ⇒ x − y = 2 (1). x = 5, y = 3 (1). Number 53; check 53 − 35 = 18 ✓ (½).Try one yourself
The sum of a two-digit number and its reverse is 99, and the digits differ by 3. Find all such numbers.
Show answer
x + y = 9, digits differ by 3 ⇒ (6, 3) or (3, 6): 63 and 36.
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