Problem
Bigger addend, bigger sum
We have the inequality 367 + (box) less than 941. We must find the largest whole number that fits in the box so the total stays below 941.
OperationsBase-ten numbers
Your answer
Watch out It's tempting to stop at 574 since 367 + 574 = 941, but that's equal to the limit, not less than it — you must go one whole number lower.
How to solve
Strategy Work Backwards — The result of the addition is bounded, so we work backwards from 941: find what plus 367 gives exactly 941, then step down by one to satisfy the strict 'less than'. A quick guess-and-check confirms the boundary case.
1STEP 1
Find the boundary value
First find the box value that would make the sum exactly 941 by subtracting 367 from 941. This is the largest sum allowed if 'less than' were 'at most'.
941 - 367 = 574
Subtraction undoes addition, so it tells us exactly how big the box could be at the boundary.
3.NBT.A.2Work BackwardsSubtracting 367 from 941 gives exactly the box value that would make 367 + box equal 941.
Why?
If some box value added onto 367 reaches 941, then taking that same 367 back off 941 must leave precisely the box's value behind.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Working out 941 minus 367 lands on one definite number because each column is subtracted in its own place, unbundling a ten from the next place whenever a column is too small to take from.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
2STEP 2
Check the boundary
If the box were 574, the sum would be 367 + 574 = 941, but the statement needs the sum to be LESS than 941, not equal to it. So 574 is too big.
367 + 574 = 941 is not less than 941
Testing the boundary shows equality fails the strict inequality, so we must step down.
3.NBT.A.2Guess And Check3STEP 3
Step down by one
The next whole number below 574 is 573. Then 367 + 573 = 940, which is less than 941. So 573 is the greatest value that works.
367 + 573 = 940 less than 941
Since whole numbers increase by 1, the largest one keeping the sum under 941 is just one below the boundary.
3.OA.A.4Guess And CheckAnswer
573
573 is a three-digit number near 600, and 367 + 573 = 940, which sits just one below 941. Using 574 would hit 941 exactly, so 573 is indeed the largest that stays strictly under. The magnitude makes sense.
Takeaway
Find the exact match by subtracting, then drop down one because the sum must stay UNDER the target!
- Find the boundary value
- Check the boundary
- Step down by one
Where next?
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