Write the number in place value and the puzzle turns into two ordinary equations.
Find the number
The digits of a two-digit number add to 11. Reversing them makes the number 27 larger. Find the number.
Reversing changes the value by exactly 9(u − t), so 9(u − t) = 27 gives u − t = 3. With t + u = 11 that makes t = 4 and u = 7, so the number is 47. D is the reversed number — the one that is 27 larger.
Two letters side by side mean multiply. That is not how numbers are written.
tens digit
t
×10
units digit
u
×1
the number
10t + u
Everything in this topic follows from that one line. The digits are the unknowns; the number itself is an expression built from them.
original = 10t + u
tens first
reversed = 10u + t
the digits change places
reversed − original = 9u − 9t
subtract
= 9(u − t)
always a multiple of 9
The gap between a two-digit number and its reverse is always nine times the difference of the digits. So “reversing increases it by 27” instantly means the digits differ by 3 — and a question claiming a gap of 20 has no solution at all.
9u − 9t = 27
expand E2 carefully
u − t = 3
divide by 9 — much simpler now
t + u = 11 and u − t = 3
add the two
2u = 14, u = 7, t = 4
the digits
the number is 47
10(4) + 7
u = 7 and t = 4 are working, not the answer. Questions here ask for “the number”, and sometimes for the reversed one — 74 will be sitting in the choices.
| Number | Written as |
|---|---|
| two-digit | 10t + u |
| three-digit | 100h + 10t + u |
| its reverse | 100u + 10t + h |
| the difference | 99(h − u) — the middle digit cancels |
Which is why reversing a three-digit number always changes it by a multiple of 99, and why the tens digit never affects that gap.
Digits are whole numbers from 0 to 9, and the leading digit cannot be 0. If your solution gives t = 12 or a fraction, the setup is wrong — no rounding will rescue it.
| The sentence | The equation |
|---|---|
| the digits add to 11 | t + u = 11 |
| the tens digit is twice the units | t = 2u |
| the number is 4 times the digit sum | 10t + u = 4(t + u) |
| the reverse is 27 more | 10u + t = 10t + u + 27 |
| the number exceeds the reverse by 18 | 10t + u = 10u + t + 18 |
Notice the last two point opposite ways. “Exceeds” tells you which side is bigger, and getting it backwards makes the digits come out negative.
Three digits: the digits add to 12, the middle digit is the sum of the other two, and reversing increases the number by 198.
h + t + u = 12 and t = h + u → 2t = 12, t = 6
substitute the middle digit
99(u − h) = 198 → u − h = 2
the 99 shortcut; the reverse is bigger
u + h = 6 and u − h = 2 → u = 4, h = 2
add the pair
the number is 264
462 − 264 = 198 ✓
What is different: three unknowns need three facts, and the middle digit drops out of the reversal entirely.
Only one fact: reversing a two-digit number increases it by 54. Find every such number.
9(u − t) = 54 → u = t + 6
one equation, two unknowns
t ≥ 1 and u ≤ 9 → t = 1, 2 or 3
the digit limits cut the list
17, 28, 39
71 − 17 = 82 − 28 = 93 − 39 = 54 ✓
What is different: there are three answers. t = 0 would give 06, which is not a two-digit number.
One equation, one answer anyway: a two-digit number is 8 times the sum of its digits.
10t + u = 8(t + u) → 2t = 7u
collect the digits
7u must be even and 2t ≤ 18 → u = 2, t = 7
u = 4 would need t = 14
the number is 72
72 = 8 × 9 ✓
What is different: the second “equation” is the rule that a digit runs from 0 to 9. Test the small values instead of looking for another fact.
A two-digit number plus its reverse is 132, and its digits differ by 2. Find the two numbers.
(10t + u) + (10u + t) = 11t + 11u = 11(t + u)
the sum is always 11 × the digit sum
11(t + u) = 132 → t + u = 12
divide by 11
digits 7 and 5 → 75 and 57
sum 12, difference 2; 75 + 57 = 132 ✓
Divided by the sum of its digits, a two-digit number gives quotient 5 and remainder 2. Its units digit is 1 more than its tens digit. Find it.
number = quotient × divisor + remainder
turn the division into an equation
10t + u = 5(t + u) + 2 → 5t = 4u + 2
collect the digits
u = t + 1: 5t = 4t + 6 → t = 6, u = 7
substitute the second fact
the number is 67
67 = 5 × 13 + 2 ✓
Number − reverse is 9 × (difference of digits); number + reverse is 11 × (sum of digits). Mixing them up gives digits that do not fit. And in a division item, the remainder must be smaller than the digit sum — a quick check on any candidate.
Digit items give two facts — a digit sum or difference, and a reversal gap, a reversal sum, or a multiple of the digit sum — and ask for the number. Use the shortcuts 9(u − t), 11(t + u) and 99(h − u) to get one fact in a line, then check the choices: the reversed number is always there as a trap.
Find the number
The digits of a two-digit number add to 9, and reversing them decreases the number by 45. Find it.
The change is 9(u − t) = −45, so t − u = 5. Together with t + u = 9 that gives t = 7 and u = 2 — the number is 72. A is the reversal. Check the direction of the change before picking.
