What is a ratio and how do you find equivalent ratios?

A ratio compares two quantities, written 2:3, "2 to 3" or 2/3, and means that for every 2 of the first thing there are 3 of the second. To find equivalent ratios, multiply or divide both parts by the same number: 2:3, 4:6, 6:9 and 8:12 are all equivalent. Two ratios are equivalent when they simplify to the same ratio.

What is a ratio in math?

A ratio in math is a comparison of two quantities that says how much of one there is for each amount of the other. A bowl with 2 red beads for every 3 blue beads has a red-to-blue ratio of 2:3, which can also be written "2 to 3" or 2/3. The order of the numbers matches the order of the words.

How do you know if two ratios are equivalent?

Two ratios are equivalent when they simplify to the same ratio or when their cross products are equal. For 4:6 and 6:9, both simplify to 2:3, and the cross products also match: 4 × 9 = 36 and 6 × 6 = 36. For 3:4 and 5:6 the cross products are 18 and 20, so those ratios are not equivalent.

How do you simplify a ratio?

A ratio is simplified by dividing both of its parts by their greatest common factor. For 12:18 the greatest common factor is 6, so 12 ÷ 6 = 2 and 18 ÷ 6 = 3, and the simplest form is 2:3. A ratio is fully simplified when the only number that divides both parts is 1.

What is the difference between a ratio and a fraction?

A ratio compares two quantities, while a fraction names a part of one whole. A part-to-part ratio of 2 red to 3 blue is written 2:3, but the fraction of the beads that are red is 2/5, because the whole is 2 + 3 = 5. A part-to-whole ratio such as 2:5 carries the same information as the fraction 2/5.

Ratios: How to Simplify and Find Equivalent Ratios

A pancake recipe asks for 3 cups of flour to 2 cups of milk, your child's grade 6 homework asks for equivalent ratios, and the cleaner bottle says to dilute 3:1. Ratios turn up in all three, and they run on one rule: whatever you do to one part, you do to the other. Once that rule is solid, simplifying ratios, knowing how to find equivalent ratios and scaling a recipe all become the same move.

What is a ratio?

A ratio is a comparison of two quantities that says how much of one there is for each amount of the other. If a bowl holds 2 red beads for every 3 blue beads, the ratio of red to blue is 2:3.

The same ratio has three common spellings: 2:3 with a colon, "2 to 3" in words, and 2/3 as a fraction. All three mean "for every 2 of the first, there are 3 of the second", and the first number always belongs to the first thing named.

Common Core standard 6.RP.A.1 asks sixth graders to "understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities". Its own example is a bird house with a ratio of wings to beaks of 2:1, "because for every 2 wings there was 1 beak."

Part-to-part and part-to-whole ratios

A part-to-part ratio compares one part with another part, and a part-to-whole ratio compares one part with the total. In the bead bowl, red to blue is 2:3 (part to part), while red to all beads is 2:5 (part to whole), because 2 + 3 = 5 shares make the whole.

That difference decides the fraction. With 2 red for every 3 blue, red beads are 2/5 of the bowl and blue beads are 3/5, which is 40% and 60%. Reading 2:3 as "two thirds red" is the most common ratio error.

A tape diagram makes the shares visible. Draw 2 boxes for red and 3 for blue; if the bowl holds 10 beads, the 5 boxes share 10 equally, so each box is worth 2, red is 2 × 2 = 4 and blue is 3 × 2 = 6.

A tape diagram for the ratio 2 to 3: two red boxes and three blue boxes spanning a total of 10, each box worth 2, so 2 to 3 becomes 4 to 6.

How to simplify ratios and find equivalent ratios, step by step

Equivalent ratios are ratios that describe the same relationship, and you find them by multiplying or dividing both parts by the same number. Simplifying is the dividing direction, and scaling up is the multiplying direction.

Write the ratio in the order of the words. "Flour to milk" is 3:2, and "milk to flour" is 2:3.
Check that both parts use the same unit. Thirty minutes to 2 hours becomes 30:120 once both are in minutes.
Find the greatest common factor of the two parts. For 12:18 the factors of 12 are 1, 2, 3, 4, 6 and 12, the factors of 18 are 1, 2, 3, 6, 9 and 18, and the largest one they share is 6.
Divide both parts by that factor to simplify. 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 12:18 simplifies to 2:3.
Multiply both parts by the same number to scale up. 2:3 times 2 is 4:6, times 3 is 6:9 and times 4 is 8:12, and every one of them is equivalent to 2:3.
Check the result. Simplify your answer again; an equivalent ratio always comes back to the same simplest form.

Play free in your browserMultiples RushGrade 3–5

How to determine if ratios are equivalent

Two ratios are equivalent when they simplify to the same ratio, and a second test, cross-multiplication, gives the same verdict in one line. For a:b and c:d, multiply a × d and b × c; if the two products match, the ratios are equivalent.

Take 4:6 and 6:9. Simplifying gives 2:3 both times, and the cross products agree: 4 × 9 = 36 and 6 × 6 = 36. Now take 3:4 and 5:6, where 3 × 6 = 18 and 4 × 5 = 20, so they are not equivalent even though each part grew by 2.

Cross-multiplication is fast, but it hides the reason it works. The What Works Clearinghouse practice guide on fractions instruction recommends that teachers "develop students' conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure".

Ratio vs fraction vs percent

A ratio, a fraction and a percent can describe the same bowl of beads, but each answers a different question. The ratio compares the parts, the fraction names one part of the whole, and the percent scales that fraction to 100.

Form	Bead bowl (2 red, 3 blue)	What it compares	Best for	Watch out for
Ratio, part to part	2:3 red to blue	One part with another part	Recipes, mixes, scale drawings	Reading it as 2/3 of the whole
Ratio, part to whole	2:5 red to all	One part with the total	Probability, shares of a group	Forgetting to add the parts first
Fraction	2/5 of the beads are red	One part of one whole	Measuring, sharing, calculating	Using the part-to-part numbers
Percent	40% red	A part per 100	Comparing groups of different sizes	Rounding before the last step

Turning a ratio into a percent

A part-to-part ratio becomes a percent in two moves: add the parts to get the whole, then divide the part you want by that whole. For 2:3, the whole is 5 shares, red is 2/5, and 2 ÷ 5 = 0.4, which is 40%.

Standard 6.RP.A.3c asks students to "find a percent of a quantity as a rate per 100", which is a ratio with a second part fixed at 100. The guide on how to calculate percentages takes that step further, and the explainer on fractions for parents covers the part-of-a-whole side.

Play free in your browserPercentagesGrade 5+

Ratio tables, unit rates and scaling a recipe

A ratio table lists equivalent ratios in rows or columns, so you can read off any amount you need. Grade 6 uses it as the main tool: standard 6.RP.A.3 names "tables of equivalent ratios, tape diagrams, double number line diagrams, or equations" as the ways to solve ratio problems.

Start the table with the original ratio, then multiply both parts by 2, 3 and 4. For 2:3 that gives 4:6, 6:9 and 8:12, and every column still divides to 2/3.

A ratio table with the columns 2 to 3, 4 to 6, 6 to 9 and 8 to 12, with arrows marked times 2, times 3 and times 4 from the first column, and each column written as a fraction equal to 2 thirds.

A unit rate is the ratio scaled to one of the second quantity. If 4 notebooks cost $6, the unit rate is $6 ÷ 4 = $1.50 per notebook, and 10 notebooks cost 10 × $1.50 = $15. The standards use a lawn example: 4 lawns in 7 hours means 35 hours, which is 5 × 7, covers 5 × 4 = 20 lawns.

Scaling a recipe

Scaling a recipe means finding the multiplier from the old batch to the new one and applying it to every ingredient. A recipe with 3 cups of flour to 2 cups of milk makes 12 pancakes; for 18 pancakes the multiplier is 18 ÷ 12 = 1.5.

Flour becomes 3 × 1.5 = 4.5 cups and milk becomes 2 × 1.5 = 3 cups, so the new batch is 4.5:3. That ratio simplifies back to 3:2, which confirms the batter will taste the same.

The same move mixes paint. For a green made from 2 parts yellow to 1 part blue, a 9-liter batch has 2 + 1 = 3 parts, each part is 9 ÷ 3 = 3 liters, and you need 6 liters of yellow and 3 liters of blue. The guide to decimals for parents helps when a multiplier such as 1.5 or 0.75 turns up.

How to check that ratios have landed

Five quick checks show whether a child or an adult understands ratios.

Ratio language: Ask what 3:1 means for a 3:1 juice-to-water mix. "Three cups of juice for every cup of water" is the answer to listen for.
Simplest form: Give 15:25. The answer is 3:5, found by dividing both parts by 5.
An equivalent ratio: Ask for a ratio equivalent to 2:3 with 12 as the first part. The answer is 12:18, because 12 ÷ 2 = 6 and 3 × 6 = 18.
A part-to-whole question: With 12 boys and 15 girls in a class, ask what fraction are girls. The answer is 15/27, which simplifies to 5/9.
A rate: Ask how much 7 items cost at 4 for $6. The answer is $10.50, from a unit rate of $1.50.

A five-minute test

A five-minute test of three mixed questions (simplify, equivalence, recipe) shows more than a long worksheet of one type, as long as you ask for the working.

The national curriculum in England expects the same skills at a similar age. The mathematics programmes of study ask Year 6 pupils to "solve problems involving the relative sizes of 2 quantities", and key stage 3 adds "use ratio notation, including reduction to simplest form".

Multiplication facts carry every one of these checks, and the multiplication worksheet for 6th graders gives printable practice with an answer key.

Common ratio mistakes and edge cases

Four mistakes cause most wrong answers on ratio problems, and each has a one-line fix.

Adding instead of multiplying: Turning 2:3 into 4:5 by adding 2 to each part changes the mix, because 2/3 is about 0.67 and 4/5 is 0.8. Multiply both parts by the same number: 2:3 becomes 4:6.
Swapping the order: Boys to girls 12:15 simplifies to 4:5, and girls to boys is 5:4. Read the question again and write the first-named quantity first.
Reading part to part as part to whole: A 2:3 red-to-blue mix is 2/5 red, and calling it 2/3 red overstates the red share. Add the parts to find the whole before you write a fraction.
Mixing units: 30 minutes to 2 hours is 30:120 = 1:4, and writing 30:2 makes the minutes look larger than the hours. Convert both parts to the same unit before you simplify.

Edge case: dilution labels and map scales

A label that says "dilute 3:1" uses ratio language, so read which part comes first before you pour. If the label means 3 parts water to 1 part concentrate, a 2-liter bucket has 3 + 1 = 4 parts, each part is 2,000 ÷ 4 = 500 ml, and you add 1,500 ml of water to 500 ml of concentrate, which makes the concentrate 25% of the mix.

A map scale is a ratio with both parts in the same unit. On a 1:25,000 map, 4 cm on paper stands for 4 × 25,000 = 100,000 cm on the ground, which is 1 km.

Your ratio checklist and next steps

Order: write the ratio in the order of the words.
Units: put both parts in the same unit first.
Simplify: divide both parts by the greatest common factor.
Scale: multiply or divide both parts by the same number, never add.
Check: simplify both ratios, or compare the cross products.
Whole: add the parts before you turn a ratio into a fraction or a percent.

Adults rebuilding their number sense can start with the guide on how to get better at math as an adult or the math fluency game for adults. Parents of sixth graders will find grade-level practice on the math game for 6th graders page.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty that raises the level only after the easier facts come out right, and daily practice in 33 languages. Twenty-two of its games play free in the browser with no sign-up.