Problem
Reduce to one unknown to solve
Two numbers A and B: B is 197 more than A, and together they add to 801. I need to find each number.
OperationsBase-ten numbers
Your answer
Watch out Don't just halve 801 for both numbers — first remove the 197 difference, or you'll get two equal, wrong values.
How to solve
Strategy Work Backwards — If I imagine taking the extra 197 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 197 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.
1STEP 1
Remove the extra 197 from the total
B carries 197 more than A. If I subtract that 197 from the total 801, what remains would be two equal copies of A.
801 - 197 = 604
Peeling off the known difference is a single Grade 3 subtraction within 1000.
3.NBT.A.2Work BackwardsTaking the extra 197 off the total 801 leaves two equal copies of A.
Why?
The total is the two numbers joined together, and since B is just A with 197 added on, the total is really A, another A, and that 197; slide the 197 aside and take it back off, and only the two equal copies of A are left.
Why?
The total 801 is nothing but A and B combined, with no part left out and none counted twice.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Because B is A with 197 stacked on top, the joined pile holds two A-sized amounts and one extra 197, and we can regroup so the two A's sit together and the 197 stands apart.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
Why?
Subtracting the 197 exactly cancels the 197 that was added on to build B, so what remains is the two equal A's.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Split the rest into two equal parts to get A
The 604 is two equal amounts of A, so halve it to find A.
604 ÷ 2 = 302
Sharing a number into 2 equal groups is basic Grade 3 division thinking.
3.OA.A.4Work Backwards3STEP 3
Add the 197 back to get B
B is 197 more than A, so add 197 to 302.
302 + 197 = 499
Putting the difference back is one more three-digit addition.
3.NBT.A.2Work BackwardsAnswer
A = 302, B = 499
Check both conditions: 302 + 499 = 801 (matches the total) and 499 - 302 = 197 (matches the difference). Both hold, so the pair is correct.
Takeaway
Take the gap off the total, share what's left in half, then add the gap back — pure Grade 3 thinking!
- Remove the extra 197 from the total
- Split the rest into two equal parts to get A
- Add the 197 back to get B
Where next?
Another one like thissuggested
Step back and solidifysuggested