A 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.
In Spanish
razón
Also called
comparison, part-to-part ratio, part-to-whole ratio
How to say it
RAY-shee-oh
Grade level
6-8
Grade 6Grade 7Grade 8CCSS 6.RP.A.1GEDSATACT
Ratio in a picture
One ratio, three ways to write it: 3 to 5, 3 : 5 or 35. Swapping the order or comparing with the whole gives a different ratio. Scaling every box by the same amount keeps the ratio.
Formal definition and notation
For quantities a and b, the ratio of a to b is written a : b, a to b, or ab when b ≠ 0; the value ab tells how many times as large a is as b. Two ratios are equivalent when one is the other with both terms multiplied by the same nonzero number, like 3 : 5 and 9 : 15. A ratio of two quantities in different units, such as 120 miles to 2 hours, is a rate, and the rate for one unit is a unit rate.
Symbol
Read it as
Meaning
a : b
a to b
the ratio of a to b, with a named first
ab
a to b, or a over b
the same ratio as a fraction; its value is a ÷ b
2 : 3 : 5
two to three to five
a ratio of three quantities, such as cement, sand and gravel in a mix
Key facts
Part to part: boys to girls, 3 : 5. Part to whole: boys to all students, 3 : 8, which also says 38 of the class are boys.
Order matters: 3 : 5 (boys to girls) is not the same as 5 : 3 (girls to boys).
Simplify a ratio like a fraction: 12 : 18 = 2 : 3 after dividing both terms by 6.
Use the same unit for both quantities: 30 minutes to 2 hours is 30 : 120 = 1 : 4, not 30 : 2.
A ratio table lists equivalent ratios, and its points lie on a straight line through (0, 0).
Equivalent ratios
Equivalent ratios
a : b = (a × k) : (b × k) for any number k ≠ 0
a : b
the starting ratio
k
the scale: multiply (or divide) both terms by it
To simplify, divide both terms by their greatest common factor: 12 : 18 = 2 : 3. To find a missing amount, scale: if 3 : 5 and there are 9 boys, there are 15 girls.
Try it: build a ratio table
Interactive
Set how many red and blue beads go in one group. The table multiplies both numbers by the same amount, so every row is an equivalent ratio, and the readout gives the simplest form.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Ratio examples, step by step
Example 1: Write and simplify
Warm-up
A class has 12 girls and 16 boys. Write the ratio of girls to boys in simplest form.
A fraction is a number that names part of one whole. A ratio can compare two separate amounts: 3 : 5 boys to girls does not mean three fifths of anything.
A unit rate is a ratio rewritten with 1 as the second number, usually with two different units: 120 miles in 2 hours is 60 miles per 1 hour.
Quick check: 3 questions
Question 1 of 3
A bowl holds 9 apples and 6 oranges. What is the ratio of oranges to apples in simplest form?
Show answer and explanation
Answer: B. Oranges first: 6 : 9, and dividing by 3 gives 2 : 3. Choice A reverses the order, and C and D compare oranges with all the fruit instead of with apples.
Question 2 of 3
Which ratio is equivalent to 4 : 7?
Show answer and explanation
Answer: C.12 : 21 is 4 : 7 with both terms times 3. Choice A adds 4 to both terms, B reverses the order, and D compares 4 with the total of 11.
Question 3 of 3
A recipe uses flour and sugar in the ratio 5 : 2. How much flour goes with 3 cups of sugar?
Show answer and explanation
Answer: A. Sugar goes from 2 to 3, a scale of 1.5, so flour is 5 × 1.5 = 7.5 cups. Choice B adds 1 cup to each side instead of scaling, C sets the ratio up backward, and D doubles the flour.
Ratio: frequently asked questions
What is the difference between a ratio and a fraction?
A fraction names part of one whole, like 38 of a pizza. A ratio compares two quantities, which can be two parts (3 boys to 5 girls) or a part and the whole (3 boys to 8 students). Ratios can be written as fractions, and part-to-whole ratios act just like fractions, but part-to-part ratios do not.
How do you simplify a ratio?
Divide both terms by their greatest common factor, just as you simplify a fraction. For 18 : 24 the GCF is 6, so the ratio simplifies to 3 : 4. If the terms are decimals or fractions, first multiply both terms by the same number to make them whole: 1.5 : 2 = 3 : 4.
Can a ratio compare more than two numbers?
Yes. A ratio like 2 : 3 : 5 compares three quantities, such as cups of red, yellow and blue paint. It works the same way: multiply every term by the same number to scale it. With 30 cups in all, there are 2 + 3 + 5 = 10 parts of 3 cups each: 6, 9 and 15 cups.
NumberA 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 2MileA 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 6Unit 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 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 6Simplest FormSimplest form is the way of writing a fraction so its numerator and denominator have no common factor other than 1. To get there, divide the top and bottom by their greatest common factor: 68 = 6 ÷ 28 ÷ 2 = 34. The value stays the same; only the numbers get smaller. It is also called lowest terms.Number System · Grade 4, Grade 5, Grade 6Greatest Common FactorThe greatest common factor (GCF) is the largest factor shared by two or more numbers: the largest number that divides into both without a remainder. The GCF of 18 and 24 is 6. Find it by listing factors or by multiplying the prime factors the numbers share. The GCF is how you simplify fractions in one step.Number System · Grade 5, Grade 6, Grade 7FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5ProportionA 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.Ratios and Proportions · Grade 6, Grade 7, Grade 8MultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5, Grade 6Constant 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 1Scale 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 1Similar 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 7PercentA 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 7