Question 22·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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An architect’s plan uses the scale $1\text{ cm}=0.5\text{ m}$ . A photocopy of the plan enlarges all lengths by a factor of $1.5$ . On the photocopy, a rectangular room measures $10.8$ centimeters by $6.75$ centimeters.

Which choice is closest to the area, in square feet, of the actual room? (Use $1\text{ m}=3.28\text{ ft}$ .)

An enlarged scale drawing gives you two length changes: photocopy back to plan, then plan to actual room. Reverse the enlargement before applying the plan’s scale. Area multiplies two lengths, so each length factor must be used twice, including the meters-to-feet conversion. Using a length conversion only once for an area is the tempting trap.

Hints

Hint 1
An enlargement factor tells you how the photocopy was made from the plan. To go backward, divide a photocopy length by $1.5$ . How much actual room length does the plan’s scale assign to that result?
Hint 2
A scale factor changes one length into a matching length. The room is rectangular, so its area multiplies two lengths. How many times must you use the factor from photocopy centimeters to actual meters?
Hint 3
A square meter is $3.28$ feet long and $3.28$ feet wide. To convert an area from square meters to square feet, what power of $3.28$ do you need?

Step-by-step

Reverse the copy, then convert the area
Step 1
Find the factor from photocopy to room
One centimeter on the plan represents $0.5$ meter, but the photocopy makes that centimeter $1.5$ centimeters long. So each photocopy centimeter represents $\frac{0.5}{1.5}$ actual meters. Type $k=\frac{0.5}{1.5}$ in Desmos; it shows $k=0.333333\ldots$ . This scale factor converts photocopy lengths to actual lengths. Multiplying by $1.5$ instead would enlarge the photocopy again.
Step 2
Find the actual area in square meters
A rectangle’s area is length times width, so multiply each photocopy dimension by $k$ : $A=(10.8k)(6.75k)=10.8(6.75)k^2$ . Type $a=10.8(6.75)k^2$ ; Desmos shows $a=8.1$ . Area needs the length factor twice, so this result is in square meters, not square feet.
Step 3
Convert the area to square feet
Since $1$ meter is $3.28$ feet, both sides of a square meter get multiplied by $3.28$ . Type $b=a(3.28)^2$ ; Desmos shows $b=87.14304$ . The actual room’s area is closest to $87$ square feet. Choice C.

Question 22·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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