A proportion is an equation that says two ratios are equal, such as 12 = 36. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.
A proportion says two ratios name the same rate. Scaling both parts by 3 keeps the ratio, and the cross products match: 3 × 12 = 4 × 9.
Formal definition and notation
A proportion is an equation of the form ab = cd, with b ≠ 0 and d ≠ 0, read a is to b as c is to d and also written a : b = c : d. It is true exactly when the cross products are equal: ad = bc. Two quantities are in a proportional relationship when every pair of matching values has the same ratio; then y = kx for a constant of proportionalityk, and the graph is a straight line through the origin.
Symbol
Read it as
Meaning
ab = cd
a is to b as c is to d
the ratio of a to b equals the ratio of c to d
a : b = c : d
a to b equals c to d
the same proportion written with colons
ad = bc
a d equals b c
the cross products, equal in every true proportion
Key facts
Two ways to test a proportion: compare cross products, or simplify both ratios. 68 and 912 both simplify to 34.
Keep the same units in the same position: lapsminutes = lapsminutes.
Recipes, maps, scale drawings, unit prices and percents (part over whole = p100) are all proportion problems.
In a proportional relationship, a table of values has the same ratio in every row, and the graph is a straight line through (0, 0).
Not every increasing pattern is proportional: a taxi that charges 3 dollars plus 2 dollars per mile is not, because 1 mile costs 5 dollars but 2 miles cost 7.
Cross products
Cross products
ab = cd exactly when a × d = b × c
a, d
the numerator of the first ratio and the denominator of the second
b, c
the denominator of the first ratio and the numerator of the second
To find a missing value, set the cross products equal and solve: 34 = x20 gives 4x = 60, so x = 15.
Try it: build equal ratios
Interactive
Set how many cups of juice and water go in one batch. Every row of the table is the first row times the same number, so any two rows form a proportion; the graph shows them on one line through 0.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Proportion examples, step by step
Example 1: Test two ratios
Warm-up
Do 410 and 615 form a proportion?
Cross products: 4 × 15 = 60 and 10 × 6 = 60.
They are equal, so the ratios are equal.
Check by simplifying: both ratios simplify to 25.
Answer: Yes. 410 = 615 is a true proportion.
Example 2: Solve for a missing value
Core
Solve 58 = x56.
Cross multiply: 8x = 5 × 56.
5 × 56 = 280, so 8x = 280.
Divide by 8: x = 35.
Check: 56 ÷ 8 = 7 and 5 × 7 = 35.
Answer: x = 35.
Example 3: A map scale
Test-style
On a map, 2 cm represents 15 km. Two towns are 9 cm apart on the map. What is the actual distance between them?
Write both ratios as map cm over real km: 215 = 9x.
Cross multiply: 2x = 15 × 9 = 135.
Divide by 2: x = 67.5.
Check the size: 9 cm is 4.5 times 2 cm, and 4.5 × 15 = 67.5.
Answer: The towns are 67.5 km apart.
Common mistakes
✗ Mistake3 laps take 4 minutes. How long do 9 laps take? 34 = x9.
✓ Instead3 laps4 min = 9 lapsx min, so 3x = 36 and x = 12 minutes.
Both ratios must compare quantities in the same order. Laps go on top in both, minutes on the bottom in both.
✗ Mistake23 = x + 19, so 2 × 9 = 3 × x + 1.
✓ Instead18 = 3(x + 1) = 3x + 3, so x = 5.
Multiply the whole numerator by the other denominator. Put parentheses around x + 1.
✗ Mistake34 = 56, because you add 2 to the top and the bottom.
✓ Instead3 × 6 = 18 and 4 × 5 = 20, so these ratios are not equal.
Equal ratios come from multiplying or dividing both parts by the same number, not from adding.
A unit rate is a ratio with 1 in the second place, like 0.75 laps per minute. In a proportional relationship every ratio simplifies to that same unit rate.
Quick check: 3 questions
Question 1 of 3
Which ratio forms a proportion with 35?
Show answer and explanation
Answer: B.915 is 35 with both parts times 3, and 3 × 15 = 5 × 9 = 45. Choices A and D add the same number to both parts, and C flips the ratio.
Question 2 of 3
Solve 7x = 2112.
Show answer and explanation
Answer: D. Cross multiply: 21x = 7 × 12 = 84, so x = 4. Choice A sets up the ratio upside down, B subtracts instead of scaling, and C is the scale factor 21 ÷ 7, not x.
Question 3 of 3
5 notebooks cost 8 dollars. At the same rate, how much do 20 notebooks cost?
Show answer and explanation
Answer: A.58 = 20x gives 5x = 160, so x = 32. Choice B adds 15 instead of multiplying by 4, C sets the ratio up upside down, and D uses 2 dollars per notebook.
Proportion: frequently asked questions
How do you solve a proportion?
Write the two ratios with matching units in matching places, then set the cross products equal. For 46 = x27: 6x = 4 × 27 = 108, so x = 18. You can also scale: 6 times 4.5 is 27, so 4 times 4.5 is 18. Check that the answer makes sense in the story.
What is the difference between a ratio and a proportion?
A ratio compares two quantities, such as 2 cups of juice to 3 cups of water, written 2 : 3 or 23. A proportion is a statement that two ratios are equal, such as 23 = 812. A ratio is like a single fraction; a proportion is like an equation.
What is a proportional relationship?
It is a relationship between two quantities where the ratio yx is the same for every pair of values, so y = kx. Total cost at a fixed price per item is proportional: 3 items at 4 dollars cost 12, and 5 cost 20. Its graph is a straight line through the origin.
Proportional RelationshipA proportional relationship is a relationship between two quantities in which one is always the same multiple of the other, so y = kx for a fixed number k. The ratio yx is the same for every pair, and the graph is a straight line through the origin. Example: at 3 dollars per pound, cost = 3 × pounds.Ratios and Proportions · Grade 7, Grade 8, Algebra 1RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8Constant of ProportionalityThe constant of proportionality is the constant ratio between two proportional quantities, k = yx. It is the k in y = kx and stays the same for every pair of values, which makes the graph a straight line through the origin. If 3 notebooks cost 9 dollars, k = 3 dollars per notebook.Ratios and Proportions · Grade 7, Grade 8, Algebra 1OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Scale DrawingA scale drawing is a drawing of a real object in which every length is multiplied by the same number, so it has the same shape but a different size. Its scale, such as 1 cm = 2 m, links a length on the drawing to the actual length. Maps, floor plans and blueprints are scale drawings.Ratios and Proportions · Grade 7Unit PriceA unit price is the cost of one unit of something, found by dividing the total price by the number of units. For example, 18 dollars for 6 pounds is 18 ÷ 6 = 3 dollars per pound. It is the fairest way to compare products sold in different package sizes: the lower unit price costs less per unit.Ratios and Proportions · Grade 6, Grade 7, Grade 8MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Unit RateA unit rate is a rate that compares a quantity to one unit of another quantity, such as 50 miles per hour or 4 dollars per notebook. To find it, divide the first quantity by the second: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. The word per means for each one.Ratios and Proportions · Grade 6, Grade 7, Grade 8Equivalent FractionsEquivalent fractions are fractions that name the same value, like 12, 24 and 36. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Number System · Grade 3, Grade 4, Grade 5Scale FactorA scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 12 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.Ratios and Proportions · Grade 7, Grade 8, Algebra 1PercentA percent is a number out of 100. Percent means per hundred, so 25% means 25 out of 100, or 25100 = 0.25. To find a percent of a quantity, change the percent to a decimal and multiply: 20% of 80 is 0.20 × 80 = 16. The symbol for percent is %.Number System · Grade 5, Grade 6, Grade 7Similar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8, Algebra 1RateA rate is a ratio that compares two quantities measured in different units, such as 150 miles in 3 hours or 12 dollars for 4 pounds. The word “per” signals a rate. Dividing gives the unit rate, the amount for 1 unit: 150 miles in 3 hours is 50 miles per hour.Ratios and Proportions · Grade 6, Grade 7Cross MultiplicationCross multiplication is a shortcut for solving a proportion: if ab = cd, then ad = bc. Multiply each numerator by the denominator on the other side and set the two products equal. To solve x6 = 1015, write 15x = 60, so x = 4. It works only when one fraction equals one fraction.Ratios and Proportions · Grade 7, Grade 8, Algebra 1