Problem
Write it as an equation with a single unknown
Two numbers have a known sum and a known difference. We need the larger one divided by the smaller one.
ExpressionsNumber system
Your answer
How to solve
Strategy Convert to Algebra — Adding the sum and the difference wipes out the smaller number entirely, leaving twice the larger one. That turns two unknowns into one in a single move.
1STEP 1
Write the two facts as equations
Call the larger number ■ and the smaller ▲. Then the sum and the difference are two sentences about them.
■ + ▲ = 65.5, ■ - ▲ = 55.5
Two facts, two unknowns.
6.EE.B.6Convert To Algebra2STEP 2
Add the two equations
The smaller number is added in one and taken away in the other, so adding them cancels it and leaves twice the larger.
65.5 + 55.5 = 121
Twice the larger number is 121.
6.NS.B.3Identify SubproblemsAdding the sum and the difference leaves twice the larger number.
Why?
The smaller number is added in the sum and taken away in the difference, so putting the two together cancels it out.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The larger number is added in both, so it appears twice in the total.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Find each number
Halve to get the larger, then take it off the sum to get the smaller.
121 ÷ 2 = 60.5, 65.5 - 60.5 = 5
The numbers are 60.5 and 5.
6.EE.B.7Work Backwards4STEP 4
Divide the larger by the smaller
Now the question can be answered directly.
60.5 ÷ 5 = 12.1
The quotient is 12.1.
6.NS.B.3Identify SubproblemsAnswer
12.1
Check both facts: 60.5 + 5 = 65.5 and 60.5 - 5 = 55.5, exactly as given.
Takeaway
Adding a sum and a difference cancels one of the two numbers straight away.
- Write the two facts as equations
- Add the two equations
- Find each number
- Divide the larger by the smaller