Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

Name both unknowns and write two equations

String A is 30 cm30\ \text{cm} long and string B is 50 cm50\ \text{cm} long. The same length is cut from each, after which the lengths remaining are in the ratio 1:31 : 3. How many centimeters were cut from string A?

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
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
3STEP 3

Find the remainders

One part is half of the difference, and the shorter remainder is one part.

20 ÷ 2 = 10
4STEP 4

Work back to the cut

Subtract what is left from what string A started as.

30 - 10 = 20
Answer
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
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