Depth · one concept from first principles · step 2 of 4
Too few equations: let digit limits decide
Two two-digit numbers satisfy: a = b + 37, a is a multiple of 3, and reversing b's digits gives a - 19. Find a and b.
ExpressionsOperations
Your answer
How to solve
Strategy Convert to Algebra — Writing b by its digits turns the two conditions into one relation between the digits. That leaves several candidates, so list them using the two-digit limits and keep the one that makes a a multiple of 3.
1STEP 1
Write b by its digits
Let b have tens digit m and ones digit n, so b = 10m + n and its reversal is 10n + m.
b = 10m + n, a = 10m + n + 37
Place value turns a digit swap into algebra.
8.EE.C.8Convert To Algebra2STEP 2
Use the reversal condition
The reversal is a - 19, which is b + 37 - 19 = b + 18.
10n + m = 10m + n + 18, 9n - 9m = 18, n = m + 2
Swapping digits always changes a number by a multiple of 9.
8.EE.C.8Convert To AlgebraThe reversal condition forces the ones digit of to be more than its tens digit.
Why?
The reversal is and is , so the reversal equals , which is .
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
Why?
A tens digit counts tens, so is and its reversal is .
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
So , which turns into and then .
Why?
Taking off both sides keeps the balance and leaves .
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
Why?
is groups of , so dividing both sides by gives .
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
3STEP 3
List what the digit limits allow
Since a = b + 37 ≤ 99, we need b ≤ 62, and b ≥ 10. With n = m + 2, b = 10m + m + 2.
b = 13, 24, 35, 46, 57 → a = 50, 61, 72, 83, 94
The next one, b = 68, would make a = 105, which has three digits.
8.EE.C.8Make A Systematic List4STEP 4
Keep the multiple of 3
The digit sums of the candidates for a are 5, 7, 9, 11, 13. Only 72 has a digit sum divisible by 3.
72 = 3 × 24
One extra condition picks one candidate out of five.
4.OA.B.4Eliminate PossibilitiesAnswer
a = 72, b = 35
72 - 35 = 37; reversing 35 gives 53, and 72 - 19 = 53; 72 is a multiple of 3; both are two-digit numbers.
Takeaway
When equations run out, the hidden limits (digits are 0 to 9, numbers have two digits) finish the job.
- Write b by its digits
- Use the reversal condition
- List what the digit limits allow
- Keep the multiple of 3
Where next?
Another one like thissuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.
The same shape shows up later on the AMC, as Branching State Enumeration, Casework by Position / Vertex Type, Small-Domain Constrained Search. Why the shapes repeat →