1. I drew two pictures of a school. If the larger picture is twice as wide and 3 times as high as the smaller picture, by what factor has the area increased?

The length and width scale factors are 2 and 3, so the area changed by a factor of 2x3=6

2. I have two pictures of a star. The smaller star has an area of 5 cm². If the larger star is 3 times as wide and 4 times as high as the smaller star, what is its area?

the larger star has area 3x4x5=60 cm².

3. I took the same 5 cm² picture of a star and made it 1 1/2 times as wide and twice as high. What is the area of this new star?

5 x 1.5 x 2 = 15 cm².

4. I took a picture of a hot dog and stretched it in several different ways. The area of the original hot dog was 3 cm², and the area of all of the other hot dogs was 12 cm².

3x area change factor = 12, so the area changes by a factor of 4

a. The first time, I used a vertical scale factor of 2; what was my horizontal scale factor?

2 (because 2x2=4)

b. The second time, I used a horizontal scale factor of 8, what was the vertical scale factor?

I need a factor n for which 8n = 4, so n=1/2

c. The last time, I used a horizontal scale factor of 6, what was the vertical scale factor?

I need a factor n for which 6 n = 4, so n=4/5=2/3

5. I have two pictures of whales. The larger picture was made from the smaller picture, by enlarging by a factor of 2 vertically, and 1.5 horizontally. If the area of the larger whale is 45 cm², what is the area of the smaller whale?

smaller x 2 x 1.5 = 45, so smaller = 45 / (2 x1.5)= 45 / 3 = 15 cm².

OR, you could reason, the length scale factors are 2 and 1.5, so the area scale factor is 2x1.5 = 3. So then, the area of the smaller = 45 / 3 = 15 cm².

6. I have two pictures of a tree. the smaller picture was made from the larger picture by compressing by a scale factor of 1/2 vertically, and 2/3 horizonally. If the smaller picture has an area of 6 cm², what is the area of the larger tree?

The area scale factor is 1/2 x 2/3 = 1/3. Larger x 1/3 = Smaller= 6, so Larger = 3 x 6 = 18 cm².

7. I have two similar (proportional) pictures of a moon and star. The larger picture was enlarged by 200% on a standard copy machine from the smaller one:

A 200% enlargement means a length scale factor of 2, and an area scale factor of 2x2=4

a. If the area of the smaller star is 5 cm², what is the area of the larger star? 5 x 4 = 20 cm².

b. If the area of the larger moon is 48 cm², what is the area of the smaller moon? 48 / 4 = 12 cm².

8. I have a small picture of the famous painting the Mona Lisa. My picture is 1/5 as wide as the original painting. If my picture is 24 in², what is the area of the oringinal painting?

Mine is 1/5 as wide as the original, and the original is 5x as wide as mine. The area scale factor would be 1/25 or 25, thus the area of the original painting is 24 x 25 = 600 in². OR 24 / (1/25) = 600 in².

9. I have two similar/proportional pictures of a cake. If the smaller cake has area 8 cm², and the area of the large cake is 72cm², what is the (length) scale factor that compares the large one to the small one.

The area scale factor comparing large/small is 72/8=9, so the length scale factor is 3 because 3x3=9.

10. A newspaper used pictures of houses in a graph showing how house prices have gone up. To show that house prices went up 150%, they showed a house that had been increased by a length scale factor of 1.5.

a. By what factor did the area increase?

The area increased by a factor of 1.5x1.5=2.25

b. Do you think this graphic is misleading or not?

Lots of people think it's misleading, because the area being 2.25 times as large makes us think the ratio of the two houses is bigger than the newspapers data really shows. Also, this picture of the house is a 3-D sort of picture, so it actually makes it look like it is 1.5x1.5x1.5=3.375 bigger (that's the factor the volume would increase by)

11. I have a large plate and a small plate (of the same shape). The diameter of the large plate is 1 1/2 times the diameter of the small plate. The plates are of the same thickness and made of the same material, so that the weight is proportional to the area of the plates. If the small plate weighs 4oz, how much does the large plate weigh?

The area scale factor is 1.5 x 1.5 = 2.25, and that applies to the weight (because weight is proportional to area in this case), so the weight of the large plate should be 4x2.25=9 oz.

12. John painted a santa that was 10 inches high, and it used 2 oz of paint. Then he was asked to paint another (similar/proportional) santa that is 30 inches high. How much paint will he use for the larger santa?

The length scale factor is 30/10=3, so the area scale factor is 3x3=9

The amount of paint he uses will depend on the area of the painting, so he will use about 2x9=18 oz of paint

13. Jan built a 1/50 scale model of the new library before it was built.

a. If the length along one side of the scale model is 3 feet, what is the length of the corresponding side of the library?

3x50=150 feet

b. If Jan used 2 square feet of glass for the windows in the model, how many square feet of glass will the library have?

Glass is measured in square feet, which is an area, so we use the area scale factor which is 50x50=2500.

The amount of glass in the new library will be about 2x2500=5000 ft².