Problem
Eliminate to find two decimals from sum and difference
Two decimal numbers add up to 12.46 and differ by 5.68. Find both numbers.
Base-ten numbers
Your answer
Watch out It's tempting to just split the sum in half, as if the two numbers were equal, but they actually differ by 5.68.
How to solve
Strategy Draw a Diagram — Picture the two numbers as two bars. The sum is both bars together; the difference is the extra piece on the longer bar. Adding sum and difference gives two copies of the larger number, which is the classic sum-and-difference (bar model) method.
1STEP 1
Add sum and difference to get twice the larger
Sum plus difference cancels the shorter bar: 12.46 + 5.68 = 18.14, two larger numbers.
12.46 + 5.68 = 18.14 = 2 × (larger)
Sum plus difference doubles the bigger number - a bar-model staple.
5.NBT.B.7Draw A DiagramAdding the sum and the difference of the two numbers gives twice the larger number.
Why?
Sum plus difference regroups into the larger plus another larger, because the difference fills the smaller up to the size of the larger.
Why?
The difference is exactly what the smaller is missing to reach the larger, since the difference was made by taking the smaller away from the larger.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The sum is the larger and the smaller together, so the extra difference is free to group with the smaller instead of standing apart.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
Why?
The larger placed beside an equal larger is two copies of the larger, which means two times the larger.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Find the larger number
Half of 18.14 is the larger number: 18.14 / 2 = 9.07.
18.14 ÷ 2 = 9.07
Two equal larger bars share 18.14 evenly, so split it in half.
5.NBT.B.7Draw A Diagram3STEP 3
Find the smaller number
Subtract the larger from the total to get the smaller: 12.46 - 9.07 = 3.39.
12.46 - 9.07 = 3.39
Once one number is known, the other is whatever is left of the sum.
5.NBT.B.7Organize Information In More WaysAnswer
9.07 and 3.39
Check both clues: 9.07 + 3.39 = 12.46 (sum correct) and 9.07 - 3.39 = 5.68 (difference correct). Both match the givens.
Takeaway
Sum plus difference is two of the bigger number - cut it in half and the rest falls out!
- Add sum and difference to get twice the larger
- Find the larger number
- Find the smaller number
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