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Proportional
Scaling

Master the art of proportional reasoning to solve scaling problems: maps, models, recipes, and similar figures on the SSAT.

SSAT Middle Level

What Is Scaling?

Scaling means enlarging or reducing every measurement of an object by the same factor. When you zoom in on a photo, every part gets bigger by the same amount: the shape stays the same, only the size changes.

3 cm 2 cm
\(\times 3\)
 9 cm 6 cm

Key Vocabulary

Scale Factor
The constant multiplier that relates corresponding lengths: \( k = \frac{\text{new length}}{\text{original length}} \)
Proportion
An equation stating two ratios are equal: \( \frac{a}{b} = \frac{c}{d} \)
Similar Figures
Shapes with the same angles whose sides are in proportion: all ratios of corresponding sides are equal.

Why Does Scaling Matter?

Proportional scaling shows up everywhere in real life and on standardized tests. Understanding it unlocks a wide range of problem types.

Maps
1 inch = 50 miles means every inch on paper represents a real distance.
Recipes
Doubling a recipe means multiplying every ingredient by the same factor.
Blueprints
Architects scale buildings down so they fit on paper, keeping all proportions.

The Proportion Rule

If two quantities are proportional, their ratios form an equation you can cross-multiply to solve:

a b = c d
\( a \times d = b \times c \)
 cross-multiply

Finding the Scale Factor

The scale factor \( k \) tells you how many times larger (or smaller) the new figure is compared to the original:

4 cm 5 cm 5 cm
\( k = 2 \)
 8 cm 10 cm 10 cm

Scaling vs Adding

A common confusion: proportional scaling is multiplication, not addition. Adding the same number to every side does not preserve the shape.

Proportional (Multiply) Non-proportional (Add)
Original sides 3, 4, 5 3, 4, 5
Operation \(\times 2\) \(+ 2\)
New sides 6, 8, 10 5, 6, 7
Ratios equal? ✓ Yes: all \(\frac{2}{1}\) ✕ No: \(\frac{5}{3} \neq \frac{6}{4}\)
Shape preserved? ✓ Similar ✕ Distorted

How to Solve Scaling Problems

1
Identify corresponding parts. Match each measurement in the original to its counterpart in the scaled version.
2
Set up a proportion. Write \( \frac{\text{original}_1}{\text{new}_1} = \frac{\text{original}_2}{\text{new}_2} \).
3
Cross-multiply. Multiply the diagonals: \( a \times d = b \times c \).
4
Solve for the unknown. Divide both sides to isolate the variable.
5
Check your answer. Verify that all ratios are equal and the answer is reasonable.

Example 1: Map Scale

On a map, 2 inches represent 50 miles. If two cities are 7 inches apart on the map, what is the actual distance?

\( 2 \text{ in} = 50 \text{ mi} \)
\( 7 \text{ in} = ? \text{ mi} \)

Example 2: Similar Triangles

Triangle \( ABC \) is similar to triangle \( DEF \). If \( AB = 6 \), \( BC = 9 \), and \( DE = 10 \), find \( EF \).

\( BC = 9 \)
\( AB = 6 \)
 B C A ~
\( EF = ? \)
\( DE = 10 \)
 E F D

Example 3: Recipe Scaling

A recipe for 8 cookies needs 3 cups of flour. How much flour do you need for 20 cookies?

8 cookies 3 cups flour
\(\times ?\)
 20 cookies ? cups flour

Example 4: Model Building

A model car is built at a scale of 1 : 24. The real car is 15 feet long and 6 feet wide. Find the model's length and width in inches.

1
Scale factor: \( k = \frac{1}{24} \). Every real foot becomes \( \frac{1}{24} \) of a foot in the model.
2
Model length: \( \frac{15}{24} = 0.625 \) feet \( = 0.625 \times 12 = \) 7.5 inches.
3
Model width: \( \frac{6}{24} = 0.25 \) feet \( = 0.25 \times 12 = \) 3 inches.

Common Mistakes

✕ Adding Instead of Multiplying
Adding a constant to each side does not produce a similar figure. Scaling requires multiplying by the same factor.
✕ Mismatching Sides
Pairing the shortest side of one figure with the longest side of another leads to a wrong proportion. Always match corresponding sides.
✕ Forgetting Units
If the original is in feet and the answer needs inches, you must convert. A correct number in the wrong unit is still a wrong answer.

Scaling Summary

Concept Formula When to Use
Scale Factor \( k = \frac{\text{new}}{\text{original}} \) Find how much bigger/smaller
Proportion \( \frac{a}{b} = \frac{c}{d} \) Two pairs of corresponding values
Cross-Multiply \( ad = bc \) Solve for an unknown in a proportion
Similar Figures \( \frac{s_1}{s_2} = \frac{s_3}{s_4} = k \) All side ratios are equal

Time to Practice

You've learned the key ideas behind proportional scaling. Now test your skills with 5 questions covering maps, similar figures, recipes, and models.

Set Up
Identify corresponding parts and write a proportion.
Solve
Cross-multiply and divide to find the unknown.
Check
Verify your ratios are equal and units make sense.

Question 1 of 5

On a map, 3 inches represent 120 miles. How many miles do 5 inches represent?

Question 2 of 5

Two similar rectangles have widths of 4 cm and 10 cm. If the smaller rectangle's length is 6 cm, what is the larger rectangle's length?

Question 3 of 5

A recipe for 12 muffins requires 2 cups of sugar. How many cups of sugar are needed for 30 muffins?

Question 4 of 5

A model train is built at a 1 : 48 scale. If the real train car is 60 feet long, how long is the model in inches?

Question 5 of 5

Triangle \( PQR \) is similar to triangle \( STU \). If \( PQ = 8 \), \( QR = 12 \), \( PR = 14 \), and \( ST = 20 \), what is the perimeter of triangle \( STU \)?