Solving Proportions

What is a Proportion?

A proportion is an equation stating that two ratios are equal.

General form: a/b = c/d

Read as: "a is to b as c is to d"

Example: 2/3 = 4/6

This is a true proportion because both ratios equal 2/3 when simplified.

Example 2: 3/5 = 6/10

True proportion: 3/5 = 6/10 = 0.6

Example 3: 1/2 ≠ 2/3

NOT a proportion: 0.5 ≠ 0.67

Parts of a Proportion

In a/b = c/d:

Means: b and c (middle terms) Extremes: a and d (outer terms)

Example: 2/3 = 8/12

Extremes: 2 and 12 Means: 3 and 8

Cross Products Property: In a true proportion, the product of the means equals the product of the extremes.

a/b = c/d → ad = bc

Example: 2/3 = 8/12

Extremes: 2 × 12 = 24 Means: 3 × 8 = 24 ✓

Testing if Ratios Form a Proportion

Method 1: Simplify both ratios

Example 1: Does 6/9 = 10/15?

6/9 = 2/3 (divide by 3) 10/15 = 2/3 (divide by 5)

Yes, both equal 2/3! ✓

Example 2: Does 4/7 = 12/21?

4/7 is already simplified 12/21 = 4/7 (divide by 3)

Yes, proportion! ✓

Method 2: Cross multiply

If ad = bc, then a/b = c/d

Example: Does 3/4 = 9/12?

Cross products: 3 × 12 = 36 and 4 × 9 = 36

Equal products → Yes! ✓

Example 2: Does 5/6 = 7/8?

Cross products: 5 × 8 = 40 and 6 × 7 = 42

Not equal → No! ✗

Solving Proportions with One Variable

Use cross multiplication!

Step 1: Cross multiply Step 2: Solve the resulting equation

Example 1: x/5 = 3/15

Cross multiply: 15x = 5 × 3 15x = 15 x = 1

Check: 1/5 = 3/15? Both equal 1/5 ✓

Example 2: 4/x = 2/7

Cross multiply: 4 × 7 = 2x 28 = 2x x = 14

Check: 4/14 = 2/7? Both equal 2/7 ✓

Example 3: 6/8 = x/12

Cross multiply: 6 × 12 = 8x 72 = 8x x = 9

Check: 6/8 = 9/12? Both equal 3/4 ✓

Example 4: 5/x = 15/27

Cross multiply: 5 × 27 = 15x 135 = 15x x = 9

Proportions with Variable in Different Positions

Variable in numerator (x/b = c/d):

Example: x/7 = 4/14

Cross multiply: 14x = 28 x = 2

Variable in denominator (a/x = c/d):

Example: 6/x = 3/5

Cross multiply: 6 × 5 = 3x 30 = 3x x = 10

Variable on right side (a/b = x/d):

Example: 2/5 = x/20

Cross multiply: 2 × 20 = 5x 40 = 5x x = 8

Variable in extreme right (a/b = c/x):

Example: 3/4 = 9/x

Cross multiply: 3x = 36 x = 12

Proportions with Larger Numbers

Example 1: x/35 = 12/15

Cross multiply: 15x = 35 × 12 15x = 420 x = 28

Example 2: 18/24 = 27/x

Cross multiply: 18x = 24 × 27 18x = 648 x = 36

Example 3: 45/x = 9/16

Cross multiply: 45 × 16 = 9x 720 = 9x x = 80

Proportions with Decimals

Example 1: 0.5/x = 2/8

Cross multiply: 0.5 × 8 = 2x 4 = 2x x = 2

Example 2: x/3 = 1.5/4.5

Cross multiply: 4.5x = 3 × 1.5 4.5x = 4.5 x = 1

Tip: Can convert decimals to fractions first

1.5 = 3/2, so x/3 = (3/2)/(9/2)

Or just work with decimals!

Proportions with Fractions

Example 1: (1/2)/x = (1/4)/(1/3)

Cross multiply: (1/2) × (1/3) = x × (1/4) 1/6 = x/4 x = 4/6 = 2/3

Example 2: x/(2/3) = 6/4

Cross multiply: 4x = 6 × (2/3) 4x = 4 x = 1

Tip: May be easier to clear fractions first by multiplying!

Word Problems with Proportions

Step 1: Identify the two ratios Step 2: Set up proportion Step 3: Cross multiply and solve Step 4: Check answer in context

Example 1: Recipe Scaling

Recipe for 4 servings uses 3 cups flour. How much for 10 servings?

Set up: 3 cups / 4 servings = x cups / 10 servings

Cross multiply: 3 × 10 = 4x 30 = 4x x = 7.5 cups

Example 2: Map Scale

Map scale: 2 inches = 50 miles. How many miles for 7 inches?

Set up: 2 in / 50 mi = 7 in / x mi

Cross multiply: 2x = 350 x = 175 miles

Example 3: Unit Price

5 pounds of apples cost 8 dollars. How much for 8 pounds?

Set up: 5 lb / 8 dollars = 8 lb / x dollars

Wait! This is backwards. Rewrite:

8 dollars / 5 lb = x dollars / 8 lb

Cross multiply: 8 × 8 = 5x 64 = 5x x = 12.80 dollars

Example 4: Similar Figures

Two similar triangles. Small triangle base is 6 cm, height is 4 cm. Large triangle base is 15 cm. Find height.

Set up: 6/4 = 15/x

Cross multiply: 6x = 60 x = 10 cm

Rate Problems Using Proportions

Example 1: Speed

Car travels 120 miles in 2 hours. How far in 5 hours (at same speed)?

Set up: 120 mi / 2 h = x mi / 5 h

Cross multiply: 120 × 5 = 2x 600 = 2x x = 300 miles

Example 2: Work Rate

3 workers take 8 hours to paint a house. How long for 6 workers?

This is INVERSE proportion (more workers, less time):

Set up: 3 workers / 6 workers = x hours / 8 hours

Cross multiply: 3 × 8 = 6x 24 = 6x x = 4 hours

Or flip: 6 workers / 3 workers = 8 hours / x hours 6x = 24, x = 4

Example 3: Typing

Type 240 words in 4 minutes. How many words in 10 minutes?

Set up: 240 words / 4 min = x words / 10 min

Cross multiply: 240 × 10 = 4x 2400 = 4x x = 600 words

Percent Problems as Proportions

Form: part/whole = percent/100

Example 1: What is 30% of 80?

x/80 = 30/100

Cross multiply: 100x = 2400 x = 24

Example 2: 15 is what percent of 60?

15/60 = x/100

Cross multiply: 60x = 1500 x = 25%

Example 3: 12 is 40% of what number?

12/x = 40/100

Cross multiply: 40x = 1200 x = 30

Similar Figures and Proportions

Corresponding sides of similar figures are proportional

Example 1: Similar Rectangles

Small rectangle: 4 by 6 Large rectangle: x by 15

Set up: 4/6 = x/15

Cross multiply: 6x = 60 x = 10

Example 2: Similar Triangles

Triangle 1 sides: 3, 4, 5 Triangle 2 sides: 9, x, y

Find x (corresponds to 4): 3/4 = 9/x 3x = 36 x = 12

Find y (corresponds to 5): 3/5 = 9/y 3y = 45 y = 15

Proportions in Geometry

Example 1: Angle Bisector Theorem

Angle bisector divides opposite side proportionally

Example 2: Golden Ratio

a/b = (a+b)/a ≈ 1.618

Example 3: Trigonometric Ratios

In similar right triangles, ratios of sides are equal (foundation of trig!)

Converting Units with Proportions

Example 1: Feet to inches

5 feet = ? inches

Set up: 1 ft / 12 in = 5 ft / x in

Cross multiply: x = 60 inches

Example 2: Dollars to cents

25 dollars = ? cents

1 dollar / 100 cents = 25 dollars / x cents

x = 2,500 cents

Example 3: Metric conversion

1 kilometer = 1000 meters 5.5 km = ? meters

1 km / 1000 m = 5.5 km / x m

x = 5,500 meters

Setting Up Proportions Correctly

Key: Make sure units match on each side!

Correct: miles/hours = miles/hours

WRONG: miles/hours = hours/miles

Example: Car goes 150 miles in 3 hours. How far in 5 hours?

Correct: 150 mi / 3 h = x mi / 5 h

Also correct: 3 h / 150 mi = 5 h / x mi

WRONG: 150 mi / 3 h = 5 h / x mi (units don't match!)

Common Mistakes to Avoid

1. Setting up proportion incorrectly Make sure corresponding parts align!

2. Cross multiplying wrong a/b = c/d → ad = bc (not ab = cd!)

3. Arithmetic errors Double-check multiplication and division

4. Not simplifying x = 20/5 → x = 4 (don't stop at 20/5!)

5. Units confusion Keep same units on same sides of equation

6. Forgetting to check Substitute back to verify!

7. Inverse vs direct confusion More workers → less time (flip ratio!)

Advanced Proportion Problems

Example 1: Three-way proportion

If a:b = 2:3 and b:c = 4:5, find a:c

Make b the same in both: a:b = 2:3 = 8:12 b:c = 4:5 = 12:15

So a:c = 8:15

Example 2: Continued proportion

a/b = b/c (b is geometric mean)

If a = 4 and c = 9: 4/b = b/9 b² = 36 b = 6

Example 3: Extended proportion

a/b = c/d = e/f = k (common ratio)

Then a = bk, c = dk, e = fk

Real-World Applications

Cooking: Scale recipes up or down

Construction: Scale drawings, blueprints

Finance: Exchange rates, tax calculations

Medicine: Dosage calculations

Geography: Map scales, distances

Shopping: Unit prices, best deals

Art: Scaling images, maintaining proportions

Science: Dilutions, mixtures, conversions

Sports: Statistics, batting averages

Quick Reference

Proportion: a/b = c/d

Cross Products: ad = bc

To solve: Cross multiply, then solve equation

To test: Check if ad = bc

Setting up: Make sure units correspond

Means: Middle terms (b and c)

Extremes: Outer terms (a and d)

Word problems: Identify two equal ratios

Practice Strategy

• Start with simple numerical proportions
• Practice cross multiplication
• Work on identifying ratios from word problems
• Set up proportions carefully (units matter!)
• Check answers by substituting back
• Distinguish direct vs inverse relationships
• Practice with real-world contexts
• Use proportions for percent problems
• Apply to similar figures in geometry
• Master unit conversions
• Draw pictures to visualize relationships
• Verify answers make sense in context
• Practice mental estimation first
• Work backwards from answer to check

Proportions are one of the most practical tools in algebra. They appear everywhere from cooking to construction, medicine to map reading. Master proportions and you'll solve countless real-world problems with confidence!