Geometry Puzzle – Challenge 69

Challenge:

A length of a rectangle is increased by \(20\%\). What percent of its width should be decreased to keep the area of the rectangle the same?

A- \(15\%\)

B- \(16\frac{2}{3}\%\)

C- \(20\%\)

D- \(25\%\)

E- \(30\%\)

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The area of a rectangle is the product of its width and length. Let x be the area of the rectangle.
Area of a rectangle = Width × Length = x
Length → 1.2Length → Area of a rectangle = Width × 1.2Length = x
Let W1 be the width of the first rectangle and W2 be the width of the second rectangle.
W1 × 1.2L = W2 × L → W1 × 1.2 = W2 → W1 \(= \frac{1}{1.2}\)W2 →
W1 \(= \frac{5}{6 }\)W2
To keep the area of the rectangle the same \(\frac{1}{6 }\) or \(16\frac{2}{3}\) percent of the width should be decreased.

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Frequently Asked Questions

How do you calculate percentage change in area?

Subtract the old area from the new area, divide by the old area, and multiply by 100. For a rectangle 10 by 10 (area 100) whose length becomes 12, the new area is 120, so the change is (120 − 100) ÷ 100 × 100 = 20%. In this puzzle the width shrinks to 5/6 of its size, so the area change is 0%.

What is the formula for percentage change in area of a rectangle?

If the length changes by a and the width by b (as decimals), the area changes by (1 + a)(1 + b) − 1. In this challenge, a = 0.2 and the width must change by b = −1/6, so 1.2 × 5/6 − 1 = 0, no change in area. A 20% increase and 20% decrease would give 1.2 × 0.8 − 1 = −0.04, a 4% loss.

What is the percentage change in area of a square?

Because both sides of a square change together, square the multiplier. If each side increases by 10%, the area changes by 1.1² = 1.21, a 21% increase. If each side increases by 20%, the area grows by 1.2² = 1.44, or 44%. That is very different from this rectangle puzzle, where only the length grows by 20%.

What is the percentage change in area of a circle?

A circle’s area is πr², so the area changes by the square of the radius multiplier. If the radius increases by 20%, the area becomes 1.2² = 1.44 times as large, a 44% increase. If the radius drops by 50%, the area becomes 0.25 of the original, a 75% decrease. π cancels, so you never need to compute it.

How do you calculate the change in area?

Find the new area and subtract the original area. A rectangle 10 cm by 6 cm has area 60 cm². If its length increases by 20% to 12 cm and the width stays 6 cm, the new area is 72 cm², a change of 12 cm². To bring it back to 60 cm², the width must drop to 5 cm, which is the 1/6 (16 2/3%) decrease in this puzzle.

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.