A ratio compares two scoops of sugar to three scoops of water. We write it 2:3 and read it "two to three." Its single-number value is the fraction 23.
Up to now, a number told you how much of one thing you had: three apples, half a pizza, 0.25 of a dollar. A ratio answers a different question — not "how much?" but "how does this compare to that?" When a recipe says two scoops of sugar for every three scoops of water, the important fact is not the two or the three alone; it is the relationship between them. Keep that relationship and you can make a tiny cup or a giant barrel of the same sweet drink. Ratios are how we hold a comparison still while the actual amounts grow or shrink.
By the end of this lesson you will be able to do five things: say what a ratio means and write it correctly; see how a ratio is built from fractions and division, so you can find its value; turn a comparison of two different kinds of thing into a rate and a unit price; simplify any ratio to lowest whole-number terms; and stretch a ratio to compare three or more quantities at once. One steady color habit runs through every figure and widget: the first quantity (the antecedent, the "part") is amber, the second quantity (the consequent, the "whole," the "out of 100") is blue, and a third quantity is purple.
4.1.1 What a ratio means
A ratio is a "so-many to so-many" comparison of two quantities. Picture making sugar water with 2 scoops of sugar and 3 scoops of water. The ratio of sugar to water is 2:3, which we read aloud as "2 to 3." You can also write it with the word "to" (2 to 3) or as a fraction (23) — three faces of the same comparison. The little colon is just shorthand for the word "to."
The same picture gives two different ratios. Part-to-part compares the groups to each other (2:3). Part-to-whole compares one group to the total of 5 scoops (2:5). Always say which one you mean.
That figure hides a trap worth naming out loud. Part-to-part compares the two groups directly: sugar to water is 2:3. Part-to-whole compares one group to the total of everything: since there are 2 + 3 = 5 scoops in all, the ratio of sugar to the whole mix is 2:5. Both are correct ratios about the same drink — they just answer different questions, so you must be clear about which comparison you are making.
Worked example — reading and writing ratios
A fruit bowl has 4 apples and 6 oranges.
(a) Apples to oranges? Compare the two groups: 4:6, read "four to six." (This is part-to-part.)
(b) Oranges to apples? Now the orange count comes first: 6:4. The order follows the words.
(c) Apples to all the fruit? The whole is 4 + 6 = 10 pieces, so apples to total is 4:10 (part-to-whole).
Order matters: 2:3 is not 3:2
A ratio is a comparison with a direction, so the order of the numbers carries meaning. 2:3 sugar-to-water makes a drink that is mostly water; flip it to 3:2 and you get a much sweeter drink. Always write the quantities in the same order as the words that describe them: "sugar to water" means the sugar number comes first.
🎮 Try itMix a ratio of sugar to water
Set the scoops of sugar and water. Watch the ratio appear in every form — "to", colon, the part-to-whole share, and the simplest form.
Sugar scoops2
Water scoops3
4.1.2 How ratios relate to fractions and division
Here is the bridge that makes ratios feel familiar: the symbol a:b is another way of writing a ÷ b. And from your fraction work you already know that a ÷ b is exactly the fraction ab. So a ratio and a fraction are close cousins — the colon and the fraction bar are doing the same job of comparing the top amount to the bottom amount.
One comparison, four faces: the ratio 3:4 is the division 3 ÷ 4, which is the fraction 34, whose single-number value is 0.75.
This gives us a useful split. The ratioa:b is a way of comparing — it keeps both quantities in view. The value of that ratio is a single number, the quotient a ÷ b, which lives at one point on the number line. The ratio 3:4 has value 34 = 0.75. Two ratios that look different can share the same value: 3:4 and 6:8 both have value 0.75, which is precisely why they describe the same mix — a fact we will lean on hard in Lesson 4.2.
Worked example — value of 6:8
Find the value of the ratio 6:8.
The value is the quotient 6 ÷ 8 = 68 = 0.75. So 6:8 has the very same value as 3:4 — they are equal ratios wearing different clothes.
Ratio versus value
A ratio compares two quantities and keeps both numbers; its value collapses them into one number, a ÷ b = ab. Same value means the same comparison. Watch the direction, though: the value of a:b is ab, while the value of b:a is ba — flipping the order flips the fraction.
🎮 Try itSee the three faces of a ratio
Pick a and b. The widget shows a:b as a fraction, as a division, and as a decimal value rounded to two places.
a (first)3
b (second)4
4.1.3 Rates and unit price
So far our two quantities have been the same kind of thing — scoops and scoops, apples and oranges. A rate is a special ratio that compares two different kinds of quantity, each with its own unit. "$12 for 3 kilograms" compares dollars to kilograms. "120 miles in 2 hours" compares miles to hours. Because the units differ, a rate almost always carries a little word — per — to remind you what is being compared to what.
A unit rate squeezes a rate down to "per one." You divide so the second quantity becomes exactly 1. From "$12 for 3 kg," divide both by 3 to get the unit rate $4 per 1 kg — written $4/kg and read "four dollars per kilogram." That is the price of a single kilogram, the unit price. The same move turns "120 miles in 2 hours" into 60 miles per hour, and "90 words in 3 minutes" into 30 words per minute.
To find a unit price, divide the price by the quantity so the bottom becomes 1. "$12 for 3 kg" becomes "$4 per kg."
Unit prices have a everyday superpower: they let you settle the better buy. Two packages with different prices and different sizes are impossible to compare head-to-head — until you bring both down to the price of a single unit. Then it is just two numbers, and the smaller per-unit price is the better deal.
Worked example — which is the better buy?
Brand A: $12 for 3 kg of rice. Brand B: $10 for 2 kg of rice.
Brand A unit price: $12 ÷ 3 = $4.00 / kg.
Brand B unit price: $10 ÷ 2 = $5.00 / kg.
Per kilogram, A costs $4 and B costs $5, so Brand A is the better buy — even though its sticker price is higher, you pay less for each kilogram you take home.
🎮 Try itFind the better buy
Set the price and quantity of two products. The widget computes each unit price and crowns the better buy — the one that costs less per unit.
A · $12units3
B · $10units2
4.1.4 Simplifying ratios to lowest whole-number terms
Just like fractions, ratios come in families that all mean the same thing, and we usually write the simplest member. To simplify a ratio, divide both terms by their greatest common factor (GCF) — the exact move you used to reduce a fraction. Take 8:12. The GCF of 8 and 12 is 4, so divide both by 4: 8:12(÷4)2:3. A ratio is in lowest terms when the only number that divides both is 1.
Bundle the 8 amber blocks into 2 groups of 4 and the 12 blue blocks into 3 groups of 4. Counting bundles instead of blocks gives the simplest ratio 2:3.
What if the terms are not whole numbers? First clear them to whole numbers, then reduce. If a ratio has decimals, multiply both terms by a power of ten: 1.5:2(×2)3:4. If a ratio has fractions, multiply both by a common denominator: 12:13(×6)3:2. Multiplying both terms by the same number never changes the comparison — it is the equal-ratios rule running in reverse.
Worked example — three quick simplifications
15 : 25. GCF of 15 and 25 is 5. Divide both by 5: 15:25 → 3:5.
1.2 : 0.8. Multiply both by 10 to clear the decimals: 12:8. GCF is 4, so divide by 4: 3:2.
½ : ⅓. Multiply both by 6 (a common denominator): 3:2. Already lowest terms, so 3:2.
Divide both terms, never just one
The comparison only stays the same if you do the same thing to both terms. Halving only the first term turns 8:12 into 4:12 — a completely different ratio. And don't stop early: 4:6 still shares a factor of 2, so it is not yet in lowest terms (the answer is 2:3).
🎮 Try itSimplify a ratio step by step
Build a ratio, read off its GCF, then tap ÷ GCF to reduce both terms. The widget keeps going until the only common factor is 1 — that's lowest terms.
First term8
Second term12
4.1.5 Continued ratios: comparing three or more
Nothing says a ratio has to stop at two quantities. A continued ratio compares three or more amounts at once, written with extra colons: red:yellow:blue = a:b:c. It says, all in one breath, how the three groups compare to each other. To honor our color habit, the first term is amber, the second blue, and the third purple.
You simplify a continued ratio the same way as a two-term ratio, but now you divide all the terms by a factor they all share. Mixing paint with 4 parts red, 6 parts yellow, and 8 parts blue gives 4:6:8. All three are even, sharing a factor of 2, so divide each by 2: 2:3:4. Now 2, 3, and 4 share no factor bigger than 1, so that is lowest terms — the recipe for a smaller batch of the exact same color.
Three bars of length 4, 6, and 8. Group every 2 parts and count the groups: 2:3:4, the same proportions in lowest terms.
Worked example — simplify 9 : 12 : 15
A trail mix uses 9 scoops of peanuts, 12 of raisins, and 15 of chocolate, so the ratio is 9:12:15.
What divides all three? 9, 12, and 15 are each a multiple of 3, and nothing larger divides all three. Divide each by 3: 3:4:5. Since 3, 4, and 5 share no common factor above 1, that is lowest terms.
🎮 Try itSimplify a three-color ratio
Set the parts of red, yellow, and blue. The widget finds the factor all three share and shows the simplified ratio with proportional bars.
Red4
Yellow6
Blue8
★ The big ideas, in one breath
A ratio is a "so-many to so-many" comparison of two quantities, written a:b and read "a to b" — and order matters, so 2:3 is not 3:2. Because a:b means a ÷ b, its value is the fraction ab. A rate compares two different kinds of quantity, and its unit rate — "per one" — gives the unit price that settles any better-buy question. You simplify a ratio by dividing both terms by their GCF (clearing decimals or fractions to whole numbers first), and a continued ratio like 4:6:8 simplifies the same way, dividing every term by a shared factor down to 2:3:4.
Coming up next — 4.2
You now know that 3:4 and 6:8 share a value. In Lesson 4.2 we turn that observation into a tool: a proportion is a statement that two ratios are equal, and "cross-multiplying" lets you solve for a missing quantity — scaling a recipe, a map, or a model with confidence.
✎ Exercises 4.1
Work each one out first, then open the answer to check your thinking.
A team has 5 forwards and 3 defenders. Write the ratio of forwards to defenders.
Show answer
5:3, read "five to three." The forwards come first because the question says "forwards to defenders."
A box holds 6 red and 4 blue marbles. Give (a) the part-to-part ratio of red to blue, and (b) the part-to-whole ratio of red to all the marbles.
Show answer
(a) red to blue = 6:4. (b) the whole is 6 + 4 = 10, so red to total = 6:10. (Part-to-part compares the two groups; part-to-whole compares one group to the total.)
Find the value of the ratio 3:5. Give it as a fraction and as a decimal.
Show answer
Value = 3 ÷ 5 = 35 = 0.6.
Brand X is $6 for 2 kg of flour; Brand Y is $9 for 3 kg. Find each unit price. Which is the better buy?
Show answer
X: $6 ÷ 2 = $3.00/kg. Y: $9 ÷ 3 = $3.00/kg. The unit prices are equal, so the two brands are the same value — pick either.
A car goes 150 miles in 3 hours. What is its speed as a unit rate (miles per hour)?
Show answer
150 ÷ 3 = 50 miles per hour. Dividing both quantities by 3 makes the time "per 1 hour."
Simplify each ratio to lowest terms: (a) 8:12, (b) 15:25.
Show answer
(a) GCF(8, 12) = 4, so 8:12 → 2:3. (b) GCF(15, 25) = 5, so 15:25 → 3:5.
Simplify 1.2:0.8 to whole numbers in lowest terms.
Show answer
Multiply both by 10 to clear the decimals: 12:8. GCF(12, 8) = 4, so divide by 4: 3:2.
Clear the fractions and simplify: 23:14.
Show answer
A common denominator of 3 and 4 is 12. Multiply both terms by 12: 23×12 = 8 and 14×12 = 3, giving 8:3. Since GCF(8, 3) = 1, it is already lowest terms: 8:3.
A paint mix is red : yellow : blue = 6 : 9 : 12. Simplify the continued ratio to lowest terms.
Show answer
6, 9, and 12 are all multiples of 3, and nothing larger divides all three. Divide each by 3: 2:3:4.
Challenge. A pancake recipe for 4 people uses flour to milk in the ratio 2:3, with 2 cups of flour. (a) How much milk? (b) To serve 8 people, how much flour and milk do you need, and what is the flour-to-milk ratio now?
Show answer
(a) The ratio 2:3 with 2 cups of flour means flour and milk move together; here flour is already 2 cups, so milk = 3 cups. (b) Doubling the recipe (4 → 8 people) doubles both amounts: 4 cups of flour and 6 cups of milk. The ratio is 4:6, which simplifies back to 2:3 — scaling a recipe never changes the ratio, only the amounts.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson introduces ratio and rate reasoning and is aligned to the U.S. Common Core standards for grades 6 and 7. It builds the concept of a ratio as a comparison of two quantities (6.RP.A.1), develops the unit rate associated with a ratio a:b — including unit price and constant speed (6.RP.A.2) — and applies ratio and rate reasoning to solve real-world problems such as better-buy comparisons and recipe scaling, computing unit rates along the way (6.RP.A.3 and 7.RP.A.1). The deliberate emphasis on part-to-part versus part-to-whole, on order ("2:3 is not 3:2"), and on the ratio-versus-value distinction targets the most common early misconceptions; the simplification work (dividing both terms by the GCF, after first clearing decimals and fractions to whole numbers) mirrors the equivalent-fraction skills students already own, making the move from fractions to ratios feel like a small, natural step rather than a new topic.
eastmath.com · Stage 4 · 4.1 From Fractions to Ratios · Intuition before notation
eastmath.com · 4.1 From Fractions to Ratios: Comparing Two Quantities · 4.1.3 Rates and unit price