Problem
Name both unknowns and write two equations
The same length is cut from two strings, and what is left is in a known ratio. We need the length cut.
Expressions
Your answer
How to solve
Strategy Convert to Algebra — Cutting the same amount off both leaves their difference unchanged at 20, and the new ratio says that difference is two parts. Reading the difference gets there without algebra, and the equation confirms it.
1STEP 1
Name the cut and write what is left
Call the cut length ■; then the remainders are 30 - ■ and 50 - ■.
50 - 30 = 20
Their difference stays 20 cm whatever is cut.
6.EE.B.6Convert To AlgebraCutting the same length from both strings leaves their difference unchanged.
Why?
Taking the same amount from each shortens both by that amount, so whatever one had over the other it still has.
🧱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?
Subtracting the same number from both sides of a comparison leaves the comparison saying the same thing.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Read the ratio as parts
A ratio of 1 : 3 means the remainders are one part and three parts, so their difference is two parts.
3 - 1 = 2
Two parts make 20 cm.
7.EE.B.4Guess And Check3STEP 3
Find the remainders
One part is half of the difference, and the shorter remainder is one part.
20 ÷ 2 = 10
String A has 10 cm left.
7.EE.B.4Work Backwards4STEP 4
Work back to the cut
Subtract what is left from what string A started as.
30 - 10 = 20
20 cm was cut from each.
7.EE.B.4Work BackwardsAnswer
20 cm
Check the remainders: 30 - 20 = 10 and 50 - 20 = 30, and 10 : 30 is 1 : 3.
Takeaway
Cutting the same amount from two lengths leaves their difference untouched.
- Name the cut and write what is left
- Read the ratio as parts
- Find the remainders
- Work back to the cut