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Proportional
Scaling

Learn to grow and shrink amounts by the same factor to solve real-world ratio problems with confidence.

ISEE Lower Level

What Is a Ratio?

A ratio compares two amounts. If a recipe uses 2 cups of flour for every 3 cups of sugar, the ratio of flour to sugar is \( 2 : 3 \).

2 cups flour : 3 cups sugar

When Ratios Stay Equal

A proportion says two ratios are equal. If we keep the same recipe but make a bigger batch, the ratio must stay the same.

2 : 3 = 4 : 6 small batch double batch

What Is Scaling?

Scaling means multiplying both parts of a ratio by the same number. This grows or shrinks the amounts while keeping the ratio equal.

2 : 3 multiply both by 3 6 : 9

The Scaling Rule

\( a : b \;=\; (a \times n) : (b \times n) \)
 same number n same number n

The same idea works for shrinking. To make a ratio smaller, divide both parts by the same number: \( 6 : 9 \div 3 = 2 : 3 \).

Finding the Scale Factor

The scale factor is the number you multiply by. To find it, ask: "What do I multiply the first amount by to get the new amount?"

4
\( \times 5 \)
 20
Scale factor = \( 20 \div 4 = 5 \)

Scaling Up vs. Scaling Down

Scaling Up
You multiply both parts. The numbers get bigger. Example: \( 3:4 \to 9:12 \) by \( \times 3 \).
Scaling Down
You divide both parts. The numbers get smaller. Example: \( 9:12 \to 3:4 \) by \( \div 3 \).

How to Solve a Scaling Problem

1
Write the ratio. Find the two amounts that are being compared.
2
Find the scale factor. Divide the new known amount by the matching old amount.
3
Multiply. Multiply the unknown part by the same scale factor.
4
Check. Make sure both ratios are equal.

Example 1: Apples

3 apples cost $2. How much do 9 apples cost?

3 apples : $2
\( \times 3 \)
 9 apples : ?
\( 9 \div 3 = 3 \), so \( \$2 \times 3 = \$6 \)

Example 2: Recipe

A recipe needs 2 eggs for every 5 pancakes. How many eggs are needed for 20 pancakes?

2 eggs : 5
\( \times 4 \)
 ? : 20
\( 20 \div 5 = 4 \), so \( 2 \times 4 = 8 \) eggs

Example 3: Scaling Down

A big jug mixes 12 cups water with 8 cups juice. To make a small batch, you use 3 cups water. How much juice?

12 water : 8
\( \div 4 \)
 3 water : ?
\( 12 \div 3 = 4 \), so \( 8 \div 4 = 2 \) cups juice

Common Mistakes

✕ Adding instead of multiplying
Going from \( 2:3 \) to \( 5:6 \) by adding 3 is wrong. You must multiply both parts.
✕ Scaling only one part
If you double the first amount, you must double the second amount too.
✕ Using the wrong factor
Find the scale factor from the part you know, not by guessing a number.

Quick Summary

Scaling Up Scaling Down
OperationMultiplyDivide
Numbers getBiggerSmaller
RuleSame factor for bothSame factor for both
Ratio staysEqualEqual

Let's Practice!

Answer the next 5 questions using the scaling steps you learned. Remember: find the scale factor, then multiply or divide both parts.

Question 1 of 5

If 4 pencils cost $2, how much do 12 pencils cost?

Question 2 of 5

A recipe uses 3 cups flour for every 2 cups sugar. If you use 9 cups flour, how much sugar do you need?

Question 3 of 5

A drink mix uses 10 cups water with 4 cups syrup. For a smaller batch you use 5 cups water. How much syrup?

Question 4 of 5

The ratio of cats to dogs at a shelter is \( 2:5 \). If there are 8 cats, how many dogs are there?

Question 5 of 5

A car travels 60 miles in 1 hour at a steady speed. How far does it travel in 4 hours?

You Did It!

You now know how to use proportional scaling to solve ratio problems. Remember the golden rule: do the same thing to both parts.