Problem
Find two lengths from sum and difference
Two bars, A and B, together measure 90 cm. A is 10 cm longer than B. Find each bar's length in centimeters.
Measurement & data
Your answer
Watch out Don't just halve the total (90 ÷ 2 = 45) — that ignores the 10 cm difference, so the two bars wouldn't differ by 10 cm.
How to solve
Strategy Draw a Diagram — The picture of the two stacked bars lets us see that if we remove the 10 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.
1STEP 1
Remove the difference from the sum
Cut off the 10 cm extra piece so both bars match — the total drops from 90 to 80 cm.
90 - 10 = 80
Taking the extra piece away leaves two equal bars, so the leftover 80 cm is just B counted twice.
2.MD.B.5Draw A DiagramCutting the 10 cm extra piece off the 90 cm total drops it to 80 cm, and that 80 cm is the shorter bar counted twice.
Why?
The 90 cm total is the two bars laid end to end, and the longer bar is the shorter bar plus the 10 cm piece, so the total is really two shorter-bar lengths and one 10 cm piece.
Why?
The two bars are joined end to end with no gap and no overlap, so their two lengths add up to the whole 90 cm.
🧱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?
The longer bar runs alongside the shorter bar and then keeps going for the 10 cm piece, so the longer bar is the shorter-bar length and the piece joined together.
🧱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?
The 10 cm piece is the one extra part added on top of the two shorter-bar lengths, so subtracting it undoes that adding and leaves exactly the two shorter-bar lengths, which is 80 cm.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Split into two equal bars to get B
The 80 cm is the shorter bar counted twice, so halving it gives bar B = 40 cm.
80 ÷ 2 = 40
Sharing 80 equally into two groups is a second-grade halving fact.
2.MD.B.5Identify Subproblems3STEP 3
Add the difference back to get A
Bar A is the shorter bar B plus the 10 cm piece added back — so A = 50 cm.
40 + 10 = 50
A is just B with its extra piece restored.
2.MD.B.5Identify SubproblemsAnswer
The longer bar A is 50 cm and the shorter bar B is 40 cm.
Check both facts: 50 + 40 = 90 cm (the sum) and 50 - 40 = 10 cm (the difference). Both match, and A (50) is longer than B (40), so the answer is sensible.
Takeaway
Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!
- Remove the difference from the sum
- Split into two equal bars to get B
- Add the difference back to get A
Where next?
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested