KA-0123Eliminate impossible choices by bounding
4 points, difficulty 2 of 3Level 5–6about 105sA rectangle has an area of 36 and sides that are whole numbers. Which of these could NOT be its perimeter?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
List every factor pair of 36.
2The strategy
Whole-number sides with area 36 means the sides are a factor pair of 36. List all the pairs, work out each perimeter, and see which offered value never appears.
3The full solution
The pairs are 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6, giving perimeters 74, 40, 30, 26 and 24. There is no 28.
Solution
The reliable way
Whole-number sides multiplying to 36 must be a factor pair. The pairs are 1 x 36, 2 x 18, 3 x 12, 4 x 9 and 6 x 6, giving perimeters of 74, 40, 30, 26 and 24. Every offered value appears except 28, so 28 is impossible. The transferable idea: bounding and listing the possibilities rules choices out far faster than trying to construct the awkward one.
The elegant way
A perimeter of 28 needs sides adding to 14 and multiplying to 36, and no pair of whole numbers does both.
Why this is on the test: Ruling choices out by what is possible is a different skill from computing an answer, and it is the one a could-not question demands.