Geometry Puzzle – Challenge 72
A rectangle's length is three times its width. What is the ratio of its width to its perimeter? Let width = X. Then length = 3X, and perimeter = 2X + 2(3X) = 8X. So width-to-perimeter = X : 8X = 1 : 8.
Key takeaways:
- Perimeter of rectangle = 2(length + width).
- If length = 3X and width = X, perimeter = 2(3X + X) = 8X.
- Ratio width:perimeter = X : 8X = 1 : 8.
- Always simplify ratios by canceling common factors.
- Set up algebra with a variable first, then substitute.
- Ratio answers are unitless — only the relationship matters.
Sharing fun math puzzles like the following one with your kids is a great way to get them thinking mathematically.
Challenge:
The length of a rectangle is three times its width. What is the ratio of its width to its perimeter?
A- 1 : 4
B- 1 : 6
C- 1 : 8
D- 2 : 9
E- 3 : 10
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The correct answer is C.
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Let x be the width of the rectangle. So, the length of the rectangle is 3x.
Perimeter of a rectangle = 2 × Width + 2 × Length =
2x + 2 (3x) = 8x
The ratio of its width to its perimeter is 1 to 8.The Best Books to Ace Algebra
Frequently Asked Questions
How do I set up the rectangle perimeter formula?
P = 2L + 2W or P = 2(L + W), where L is length and W is width. The factor of 2 reflects the two pairs of equal sides.
Why use a variable instead of a specific number?
Because the answer is a ratio (unitless), it does not depend on the actual size. Using X keeps the algebra general, and the X cancels out of the final ratio.
Walk through the substitution.
Let width = X. Then length = 3X. Perimeter = 2(3X) + 2(X) = 6X + 2X = 8X. Ratio width:perimeter = X : 8X, which simplifies to 1 : 8.
Why does the X cancel out?
Both sides of the ratio share X as a factor. Dividing both sides by X removes it: (1 X)/(8 X) = 1/8.
What if the length were 4 times the width?
Length = 4X, perimeter = 2(4X) + 2(X) = 10X. Ratio width:perimeter = 1 : 10.
What is the ratio of length to perimeter for the original?
Length = 3X, perimeter = 8X, so length:perimeter = 3 : 8.
Can I check with a specific number?
Yes. Take W = 5, then L = 15. Perimeter = 2(15 + 5) = 40. Ratio width:perimeter = 5 : 40 = 1 : 8. Matches.
Why is this a useful kind of problem?
It builds the habit of using variables to express relationships and then simplifying. That habit shows up everywhere from algebra to physics to financial modeling.
What is the perimeter formula for non-rectangular shapes?
Perimeter is always the sum of all side lengths. For a triangle: P = a + b + c. For an n-sided polygon: sum of all n sides.
What grade level is this puzzle for?
Late elementary through middle school, once students understand perimeter formulas and basic variable expressions. It is also a nice warm-up for Algebra I.
Related Lessons You May Like
- How to find the area of rectangles
- How to find the perimeter of rectangles
- How to solve angles and angle measure
- How to solve percent of change
- How to find volume and surface area of cubes
If your student enjoys puzzles like this, Geometry for Beginners works the same kinds of relationships inside a full curriculum. Pre-Algebra for Beginners covers the algebraic foundations.
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