Scale Factor Effects - Pre Algebra

Two similar parallelograms have perimeters in the ratio 7:4. If the larger parallelogram has an area of 147 square centimeters, what is the area of the smaller parallelogram?
Answer: 48 square centimeters Perimeter ratio is 7:4 (large:small), so the area ratio is (7/4)² = 49/16. If the large area is 147, then small area = 147 × (16/49) = 147 × 16 ÷ 49 = 2352 ÷ 49 = 48 square centimeters. Choice A incorrectly uses 4/7 instead of (4/7)². Choice B assumes incorrect linear relationship. Choice D uses wrong calculations.
A rectangular garden is scaled up by a factor of 3.5. If the original garden had a perimeter of 24 feet and an area of 32 square feet, what is the ratio of the new area to the new perimeter? Give your answer in simplest form.
Answer: $\frac{14}{3}$ When a figure is scaled by factor 3.5: perimeter scales by 3.5, area scales by (3.5)² = 12.25. New perimeter = 24 × 3.5 = 84 feet. New area = 32 × 12.25 = 392 square feet. Ratio = 392/84. Choice B uses original values. Choice C shows incorrect scaling formula. Choice D incorrectly scales both by area factor.
A regular hexagon is enlarged by a scale factor of $\frac{5}{3}$ . If the ratio of the new area to the original area is expressed as $\frac{a}{b}$ in lowest terms, what is the value of $a + b$ ?
Answer: 34 Area scales by the square of the linear scale factor: (5/3)² = 25/9. Since gcd(25,9) = 1, this is already in lowest terms, so a = 25, b = 9, and a + b = 34. Choice B gives just the value of a. Choice C uses 5 + 9. Choice D uses 5 + 3.
The perimeter of a scaled polygon is 1.8 times the original perimeter. By what factor did the area change?
Answer: 3.24 If perimeter scales by factor 1.8, then area scales by (1.8)² = 3.24. Choice A uses the perimeter factor for area. Choice C incorrectly calculates (1.8/2)². Choice D uses 1.8 × 2 instead of 1.8².
A square is enlarged so that its new area is 2.25 times the original area. If the original perimeter was 20 cm, what is the new perimeter?
Answer: 30 cm When you see questions about scaling geometric figures, remember that area and perimeter scale differently. Area scales with the square of the scaling factor, while perimeter scales linearly with the scaling factor. Let's work through this step-by-step. First, find the scaling factor. Since the new area is 2.25 times the original area, and area scales as the square of the linear scaling factor, we have: $(\text{scaling factor})^2 = 2.25$ . Taking the square root: $\text{scaling factor} = \sqrt{2.25} = 1.5$ . Since the original perimeter was 20 cm, the new perimeter is: $20 \times 1.5 = 30 \text{ cm}$ . Now let's examine why the other answers are wrong. Choice A (25 cm) represents adding 5 cm to the original perimeter, which doesn't follow any scaling relationship. Choice B (45 cm) would result from multiplying the perimeter by 2.25 directly - this is the most common trap, as students often mistakenly apply the area scaling factor to the perimeter. Choice D (50 cm) would mean the perimeter was scaled by 2.5, which doesn't match our area relationship. The key insight is recognizing the relationship between area scaling and linear scaling. When area increases by a factor of $k^2$ , all linear dimensions (including perimeter) increase by a factor of $k$ . Always take the square root of the area scaling factor to find how linear measurements change.
A triangle with area 48 square inches is scaled to create a new triangle with area 12 square inches. If the original triangle had a perimeter of 32 inches, what is the difference between the original and new perimeters?
Answer: 16 inches When you see a question about scaling shapes, remember that area and perimeter scale differently. This is a key concept that often trips up students. To find how the triangle was scaled, you need to compare the areas. The original area is 48 square inches and the new area is 12 square inches. Since area scales by the square of the scale factor, you have: $\frac{\text{new area}}{\text{original area}} = \frac{12}{48} = \frac{1}{4}$ This means the scale factor squared equals $\frac{1}{4}$ , so the actual scale factor is $\frac{1}{2}$ . The new triangle is half the size of the original in each dimension. Since perimeter scales directly by the scale factor (not squared like area), the new perimeter is: $32 \times \frac{1}{2} = 16$ inches. The difference between the original and new perimeters is: $32 - 16 = 16$ inches. Looking at the wrong answers: Choice A (8 inches) incorrectly uses half the scale factor. Choice B (20 inches) might come from subtracting the scale factor times the original perimeter incorrectly. Choice C (24 inches) could result from using $\frac{3}{4}$ as the scale factor instead of $\frac{1}{2}$ . Remember this key relationship: when a shape is scaled by factor $k$ , the perimeter scales by $k$ but the area scales by $k^2$ . Always find the linear scale factor first by taking the square root of the area ratio, then apply it directly to the perimeter.
Two similar rectangles have areas in the ratio 16:25. If the smaller rectangle has a perimeter of 36 units, what is the perimeter of the larger rectangle?
Answer: 45 units Area ratio is 16:25, so linear scale factor is √(25/16) = 5/4 = 1.25. Larger perimeter = 36 × 1.25 = 45 units. Choice A incorrectly uses √(16/25) then adds. Choice C uses area ratio directly on perimeter. Choice D incorrectly assumes area ratio equals perimeter ratio.
A circle is scaled down by a factor of 0.4. If the original circle had an area of 100π square meters, what is the ratio of the new perimeter to the original perimeter?
Answer: 0.4 When a shape is scaled by a factor, different properties change in different ways. Linear measurements (like radius, diameter, and perimeter) change by the scale factor itself, while area changes by the square of the scale factor. Since the circle is scaled down by a factor of 0.4, all linear measurements become 0.4 times their original size. This includes the radius, diameter, circumference, and perimeter. Therefore, the ratio of the new perimeter to the original perimeter is simply 0.4. You can verify this makes sense: if the original circle had radius $r$ , its perimeter was $2\pi r$ . After scaling by 0.4, the new radius is $0.4r$ , so the new perimeter is $2\pi(0.4r) = 0.4(2\pi r)$ . The ratio is $\frac{0.4(2\pi r)}{2\pi r} = 0.4$ . Choice A (6.25) incorrectly uses $\frac{1}{0.4^2} = \frac{1}{0.16} = 6.25$ , which would be relevant if you were comparing original area to new area. Choice B (0.16) represents $0.4^2$ , the factor by which the area changes—this is the ratio of new area to original area, not perimeters. Choice C (2.5) equals $\frac{1}{0.4}$ , which reverses the ratio and gives you original perimeter to new perimeter instead. Remember: scaling affects linear measurements by the scale factor and areas by the scale factor squared. Don't confuse which property the question is asking about, and pay attention to whether the ratio requested is new-to-original or original-to-new.
Triangle ABC is scaled to create triangle DEF. If the scale factor is $\frac{2}{3}$ , and triangle DEF has a perimeter of 18 inches, what was the area scaling factor from triangle ABC to triangle DEF?
Answer: $\frac{4}{9}$ Area scaling factor is the square of the linear scale factor. Since the scale factor is 2/3, the area scaling factor is (2/3)² = 4/9. Choice A is the linear scale factor. Choice C is the reciprocal of the correct answer. Choice D is the reciprocal of the linear scale factor.

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