Geometry Puzzle – Challenge 68

Challenge:

The sum of three sides of a rectangular is 63. The sum of three sides of the same rectangle is 51. What is the area of the rectangle?

A- 168

B- 169

C- 240

D- 300

E- 325

The Absolute Best Book to challenge your Smart Student!

A rectangle has two widths and two lengths. Therefore, the perimeter equals: 2 × (width + length)
Since, the sum of three sides of a rectangular is 63 and the sum of other three sides of the rectangle is 51. Let’s assume the width is smaller the length. Let W be the width and L be the length. Therefore:
2W + L = 51 and W + 2 L = 63
Solve the system of equations:
2W + L = 51 → L = 51 – 2W
W + 2 L = 63 → W + 2 (51 – 2W) = 63→ W + 102 – 4W = 63 →
W = 13 and W + 2 L = 63 → 13 + 2L = 63 → 2L = 50 → L = 25
The area of the rectangle is:
W × L = 13 × 25 = 325

The Best Books to Ace Algebra

Frequently Asked Questions

Why do three sides have two different sums?

Because three sides can be chosen two ways: two widths plus one length (2W + L), or one width plus two lengths (W + 2L). For W < L, 2W + L is the smaller sum.

How do I set up the two equations?

Assign 2W + L = 51 (smaller sum) and W + 2L = 63 (larger sum). The two equations let you solve for both W and L.

Walk through solving the system.

From 2W + L = 51: L = 51 – 2W. Substitute into W + 2L = 63: W + 2(51 – 2W) = 63, so W + 102 – 4W = 63, -3W = -39, W = 13. Then L = 51 – 26 = 25.

How do I check the answer?

Verify: 2(13) + 25 = 26 + 25 = 51. And 13 + 2(25) = 13 + 50 = 63. Both check out.

Why is the area 325?

Area = W times L = 13 times 25 = 325 square units.

What if both sums were equal?

Then the rectangle would have to be a square (W = L). With W = L, both sums 2W + L and W + 2L equal 3W.

Can the rectangle have negative dimensions?

No. The system gives one positive solution (W = 13, L = 25), and dimensions must be positive in a real rectangle.

Is there a shortcut without substitution?

Yes — add the two equations: (2W + L) + (W + 2L) = 51 + 63, so 3W + 3L = 114, W + L = 38. Subtract: (W + 2L) – (2W + L) = 63 – 51, so L – W = 12. Then W = 13, L = 25.

What grade can solve this?

Grade 7 or 8, once students are comfortable with systems of equations. Strong middle schoolers can use the substitution method without formal training.

Where does this kind of problem appear in real life?

Anywhere you have partial measurements and need to recover the full dimensions: surveying, construction, fabric cutting, packaging design.

Related Lessons You May Like

• How to find the area of rectangles
• How to find the perimeter of rectangles
• How to find volume and surface area of cubes
• How to solve percent of change
• How to solve multi-step word problems

If your student enjoys puzzles like this, Geometry for Beginners works the same relationships inside a full curriculum. Pre-Algebra for Beginners covers the algebraic foundations gently.