# Solving Area Problems Using Scale Drawings

Examples, videos, and solutions to help Grade 7 students solve area problems related to scale drawings and percent.

### Lesson 15 Student Outcomes

• Students solve area problems related to scale drawings and percent by using the fact that an area, A', of scale drawing is k2 times the corresponding area, A, in the original drawing, where k is the scale factor.

Lesson 15 Classwork
Opening Exercise
For each diagram, Drawing 2 is a scale drawing of Drawing 1. Complete the accompanying charts.
For each drawing: identify the side lengths, determine the area, and compute the scale factor.
Convert each scale factor into a fraction and percent, examine the results, and write a conclusion relating scale factors to area.

Key Points: Overall Conclusion
If the scale factor is represented by k, then the area of the scale drawing is k2 times the corresponding area of the original drawing.

Example 1
What percent of the area of the large square is the area of the small square?

Example 2
What percent of the area of the large disk lies outside the smaller disk?

Example 3
If the area of the shaded region in the larger figure is approximately 21.5 square inches, write an equation that relates the areas using scale factor and explain what each quantity represents. Determine the area of the shaded region in the smaller scale drawing.

Example 4
Use Figure 1 below and the enlarged scale drawing to justify why the area of the scale drawing is k2 times the area of the original figure.
Explain why the expressions (kl)(kw) and k2lw are equivalent. How do the expressions reveal different information about this situation?

Exercise 1

1. The Lake Smith basketball team had a team picture taken of the players, the coaches, and the trophies from the season. The picture was 4 inches by 6 inches. The team decides to have the picture enlarged to a poster and then enlarged again to a banner measuring 48 inches by 72 inches.
a. Sketch drawings to illustrate the original picture and enlargements.
b. If the scale factor from the picture to the poster is 500%, determine the dimensions of the poster.
c. What scale factor is used to create the banner from the picture?
d. What percent of the area of the picture is the area of the poster? Justify your answer using the scale factor AND by finding the actual areas.
e. Write an equation involving the scale factor that relates the area of the poster to the area of the picture.
f. Assume you started with the banner and wanted to reduce it to the size of the poster. What would the scale factor as a percent be?
g. What scale factor would be used to reduce the poster to the size of the picture?

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