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Using cross multiplication to solve proportions
Learn step-by-step with practice exercises built right in.
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
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.
Solve for x: x/5 = 12/15
Step 1: Use cross multiplication: In a proportion a/b = c/d, we have ad = bc
Step 2: Apply cross multiplication: x · 15 = 5 · 12 15x = 60
Step 3: Solve for x: x = 60/15 x = 4
Step 4: Check by substituting back: 4/5 = 12/15 4/5 = 4/5 ✓ (both simplify to 4/5)
Answer: x = 4
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a/b = c/d → ad = bc
Example: 2/3 = 8/12
Extremes: 2 × 12 = 24 Means: 3 × 8 = 24 ✓
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! ✗
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
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
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
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!
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!
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
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
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
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
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!)
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
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!)
Setting up proportion incorrectly Make sure corresponding parts align!
Cross multiplying wrong a/b = c/d → ad = bc (not ab = cd!)
Arithmetic errors Double-check multiplication and division
Not simplifying x = 20/5 → x = 4 (don't stop at 20/5!)
Units confusion Keep same units on same sides of equation
Forgetting to check Substitute back to verify!
Inverse vs direct confusion More workers → less time (flip ratio!)
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
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
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
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!
Solve:
Use cross multiplication:
Check: → both equal ✓
Answer:
Solve: 3/8 = x/24
Step 1: Cross multiply: 3 · 24 = 8 · x 72 = 8x
Step 2: Solve for x: x = 72/8 x = 9
Step 3: Check: 3/8 = 9/24 3/8 = 3/8 ✓ (9/24 simplifies to 3/8 by dividing by 3)
Answer: x = 9
Solve:
Cross multiply:
A recipe calls for 2 cups of flour for every 3 cups of sugar. If you use 8 cups of flour, how much sugar do you need?
Step 1: Set up a proportion: Let x = cups of sugar needed
flour/sugar = flour/sugar 2/3 = 8/x
Step 2: Cross multiply: 2 · x = 3 · 8 2x = 24
Step 3: Solve for x: x = 24/2 x = 12
Step 4: Check the ratio: Original ratio: 2:3 New ratio: 8:12 = 2:3 ✓ (divide both by 4)
Answer: 12 cups of sugar
Solve: (x-2)/4 = 6/8
Step 1: Simplify the right side first: 6/8 = 3/4
So we have: (x-2)/4 = 3/4
Step 2: Cross multiply: 4(x - 2) = 4 · 3 4(x - 2) = 12
Step 3: Divide both sides by 4: x - 2 = 3
Step 4: Solve for x: x = 3 + 2 x = 5
Step 5: Check: (5-2)/4 = 3/4 ✓ 3/4 = 3/4 ✓
Answer: x = 5
A recipe calls for 2 cups of flour for every 3 cups of sugar. If you use 8 cups of flour, how many cups of sugar do you need?
Set up a proportion:
Cross multiply:
Answer: 12 cups of sugar
On a map, 2.5 inches represents 75 actual miles. If two cities are 6 inches apart on the map, what is the actual distance between them?
Step 1: Set up a proportion: Let x = actual distance in miles
map distance/actual distance = map distance/actual distance 2.5/75 = 6/x
Step 2: Cross multiply: 2.5 · x = 75 · 6 2.5x = 450
Step 3: Solve for x: x = 450/2.5 x = 180
Step 4: Check using unit rate: 75 miles ÷ 2.5 inches = 30 miles per inch 6 inches × 30 miles/inch = 180 miles ✓
Step 5: Interpret: The scale is 1 inch = 30 miles 6 inches on the map = 180 actual miles
Answer: 180 miles
Subtract 6:
Divide by 3:
Answer: