Question 42·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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On an original map, 1 inch in either direction represents an actual distance of 21 miles. A reproduction of the map is made so that every horizontal length is 20% shorter than on the original map, but the total printed area is 12% greater than on the original map. The horizontal and vertical dimensions are each scaled by a constant factor. On the reproduction, which choice gives the actual vertical distance, in miles, represented by 1 inch?

A map resized by different amounts horizontally and vertically has two scale factors, one for each direction. Multiply them to match the printed area change, then find the vertical factor. To get miles per inch on the new map, reverse that factor: a longer printed length means fewer miles per inch. Squaring the horizontal factor would wrongly assume both directions change alike.

Hints

Hint 1
A scale factor compares a new length with the matching original length. A length that becomes 20% shorter keeps 80% of its original size. What multiplier describes that horizontal change?
Hint 2
An area uses two lengths, one horizontal and one vertical. Multiply the two separate scale factors to get the printed area factor. What must the vertical factor be to make their product match the area change?
Hint 3
A unit rate gives a distance for one map inch. If the same actual distance takes up more vertical inches on the reproduction, does each new inch represent more miles or fewer?

Step-by-step

Find the vertical scale factor
Step 1
Turn both percent changes into factors
A scale factor is a new measurement divided by the matching original measurement. The horizontal length is 20% shorter, so subtract that loss from the whole length:
$1-0.20=0.80.$
The printed area is 12% greater, so add that gain to the whole area:
$1+0.12=1.12.$
Step 2
Find how much vertical lengths grow
An area multiplies a horizontal length by a vertical length, so multiply the horizontal factor by the vertical factor, not the horizontal factor by itself twice. Let $v$ be the vertical factor. The area change gives $0.80v=1.12$ . Type $0.8x_1\sim1.12$ in Desmos, using $x_1$ for $v$ ; $\sim$ tells Desmos to find the value that fits. Under PARAMETERS, it shows $x_1=1.4$ . So vertical lengths on the reproduction are $1.4$ times as long.
Step 3
Convert the vertical factor to miles per inch
The original $1$ vertical map inch represents $21$ miles. On the reproduction, those same $21$ miles take up $1.4$ vertical inches, so divide to find the unit rate in miles per new inch. Type $\frac{21}{x_1}$ below the regression; Desmos prints $15$ . One vertical inch on the reproduction represents $15$ actual miles. Choice B.

Question 42·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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