What is the difference between a ratio and a rate?

A ratio compares any two quantities by division, and the two quantities can share a unit, like 3 cups of flour to 2 cups of sugar. A rate compares two quantities with different units, like 120 miles in 3 hours. Every rate is a ratio, but a ratio is only called a rate when the units differ.

How do I know which number to divide by when finding a unit rate?

Read the words after "per." That quantity becomes 1, so it goes in the denominator. For dollars per pound, divide dollars by pounds. Then check whether your answer is a reasonable size: if 5 pounds cost 7 dollars, one pound must cost between 1 and 7 dollars.

Can a unit rate be a decimal or a fraction?

Yes. Unit rates such as 1.5 dollars per muffin, 8.4 minutes per mile, and 2/3 muffin per dollar are all correct. Only round when the answer is money, and then round to the nearest cent. Otherwise keep the exact decimal or fraction.

Why does the bigger package sometimes cost more per unit?

Pricing is set by the store, not by math, so bulk is not automatically cheaper. Sale prices, coupons, and packaging costs can make a smaller size cheaper per ounce. That is why you compute the unit price for both options instead of assuming the larger one wins.

M7MATH-3.1

Ratios & Unit Rates

Learn how to tell a ratio from a rate, divide to find any unit rate, and use unit prices to decide which of two offers is the better deal.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Ratios & Unit Rates, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

When you compare 3 cups of flour to 2 cups of sugar, you are using a ratio. When you say a car went 120 miles in 3 hours, you are using a rate. These two ideas look almost the same on paper, and that is exactly why students mix them up. The tool that makes rates genuinely useful is the unit rate — the amount that goes with exactly one of something, like miles per hour or dollars per ounce.

In this lesson you will learn how to spot the difference between a ratio and a rate, how to divide to get a unit rate (and which number goes on the bottom), and how to use unit rates to settle real arguments like "which bag of pretzels is actually the better buy?" These skills come back constantly in this unit, because a unit rate is the first step toward finding a constant of proportionality later on.

Ratios, Rates, and Unit Rates

A ratio compares two quantities by division. You can write the ratio of 3 cups of flour to 2 cups of sugar as

3:2

, as "3 to 2", or as the fraction

\frac{3}{2}

. Notice that both quantities here are measured in cups — the same unit.

A rate is a special kind of ratio that compares two quantities measured in different units: 120 miles in 3 hours, 7 dollars for 5 pounds, 45 pages in 3 minutes. Because the units differ, you almost always read a rate with the word "per": miles per hour, dollars per pound, pages per minute.

A unit rate is a rate written so the second quantity is exactly

1

. The rate 120 miles in 3 hours becomes the unit rate 40 miles per 1 hour, usually shortened to 40 miles per hour.

Example	Same or different units?	Name
3 cups flour to 2 cups sugar	same	ratio
8 boys to 10 girls	same	ratio
120 miles in 3 hours	different	rate
40 miles in 1 hour	different, second is 1	unit rate
7 dollars for 5 pounds	different	rate

One caution: every rate is a ratio, but not every ratio is a rate. So if a question asks "is this a ratio?" the answer for a rate is yes. The sharper question is whether it is a rate, and that depends on whether the units are different. A second caution: a unit rate is a single number with a unit attached, such as 40 miles per hour. It is not written as "40 to 1 miles per hour."

Computing a Unit Rate by Dividing

To find a unit rate, divide the first quantity by the second quantity. The number you divide by is the one that becomes 1.

Suppose 5 pounds of apples cost 7 dollars. To find dollars per pound, divide dollars by pounds:

\frac{7 \text{ dollars}}{5 \text{ pounds}} = 1.4 \text{ dollars per pound}

The words "per pound" tell you pounds goes in the denominator, so pounds is what gets reduced to 1. This is the single most common place students go wrong: they divide

5 \div 7 \approx 0.71

and write "0.71 dollars per pound." Checking the size of the answer catches this. If 5 pounds cost 7 dollars, one pound must cost less than 7 dollars but more than 1 dollar, so 1.4 is reasonable and 0.71 is not.

A reliable habit is to carry the units through the division and label the answer. Write the fraction with units, divide, then say the result out loud: "one point four dollars for every one pound."

Rate given	Question	Division	Unit rate
120 miles in 3 hours	miles per hour	$120 \div 3$	40 miles per hour
45 pages in 3 minutes	pages per minute	$45 \div 3$	15 pages per minute
6 dollars for 4 muffins	dollars per muffin	$6 \div 4$	1.5 dollars per muffin
12 laps in 8 minutes	laps per minute	$12 \div 8$	1.5 laps per minute

Unit rates are not always whole numbers, and they do not have to be. A decimal like 1.5 or a fraction like

\frac{3}{2}

is a perfectly good unit rate. Round only when the problem is about money, and then round to the nearest cent.

Two Unit Rates from the Same Pair

Every pair of quantities gives you two different unit rates, depending on which quantity you set to 1. They are reciprocals of each other.

Take 6 dollars for 4 muffins.

Dollars per muffin:

6 \div 4 = 1.5

, so each muffin costs 1 dollar and 50 cents.

Muffins per dollar:

4 \div 6 = \frac{2}{3}

, so one dollar buys

\frac{2}{3}

of a muffin.

Both statements describe the same deal. Neither one is wrong — but they answer different questions, and they point in opposite directions when you compare offers. With dollars per muffin, smaller is the better buy because you are paying less for each item. With muffins per dollar, larger is the better buy because your dollar gets you more.

This is where careless comparisons fall apart. If you compute dollars per ounce for one snack bag and ounces per dollar for the other, the two numbers cannot be compared at all — they are not the same kind of measurement. Before you compare, make sure both unit rates have the same unit and the same order.

The phrase after "per" always names the quantity that equals 1. So "dollars per ounce" means one ounce, and "ounces per dollar" means one dollar. Reading the phrase backward — the quantity before "per" goes on top — sets up the division correctly every time:

\text{dollars per ounce} = \frac{\text{dollars}}{\text{ounces}}, \qquad \text{ounces per dollar} = \frac{\text{ounces}}{\text{dollars}}

Using Unit Rates to Compare Two Offers

Stores rarely package things in matching amounts, which is exactly why unit rates exist. A 12 ounce bottle for 1 dollar and 92 cents and a 20 ounce bottle for 3 dollars and 4 cents cannot be compared by price alone or by size alone. Convert each to the same unit rate and the comparison becomes a single number-to-number check.

The unit price is the cost of one unit of whatever is being sold — one ounce, one pound, one pencil. Compute it for both offers, then choose the smaller unit price for the better deal.

Bottle A:

1.92 \div 12 = 0.16

, or 16 cents per ounce.

Bottle B:

3.04 \div 20 = 0.152

, or about 15.2 cents per ounce.

Bottle B costs less per ounce, so it is the better value even though its sticker price is higher. Bigger packages are usually cheaper per unit, but not always — that is why you compute instead of guessing.

The same reasoning applies to speeds, wages, and rates of work. Which runner is faster: one who runs 3 miles in 24 minutes or one who runs 5 miles in 42 minutes? Minutes per mile gives 8 and 8.4, so the first runner is faster because fewer minutes per mile means more speed.

Two things to watch. First, units must match before you divide. Comparing cents per ounce to dollars per ounce, or minutes per mile to hours per mile, produces nonsense. Second, be clear about which direction is "better" for the rate you chose. For cost per item and time per task, lower is better. For items per dollar and distance per hour, higher is better. Naming the unit rate in full — "cents per ounce" — keeps that straight.

Key terms

Ratio.: A comparison of two quantities by division, written as $a:b$ , "a to b", or $\frac{a}{b}$ . The quantities may share the same unit.
Rate.: A ratio that compares two quantities measured in different units, such as 120 miles in 3 hours or 7 dollars for 5 pounds.
Unit rate.: A rate in which the second quantity is exactly 1, found by dividing the first quantity by the second, such as 40 miles per hour.
Unit price.: A unit rate for cost: the price of one unit of a product, such as cents per ounce or dollars per pound.
Per.: A word meaning "for each one of." The quantity named after "per" is the one set equal to 1 and placed in the denominator.
Equivalent ratios.: Two ratios that describe the same relationship because both numbers were multiplied or divided by the same nonzero number, such as $120:3$ and $40:1$ .
Reciprocal unit rates.: The two unit rates you get from one pair of quantities by choosing which quantity equals 1; their product is 1, as with 1.5 dollars per muffin and $\frac{2}{3}$ muffin per dollar.

Worked example

A store sells orange juice in two sizes: a 64 ounce jug for 4 dollars and 48 cents, and a 96 ounce jug for 6 dollars and 24 cents. Which jug is the better buy? Also state how many ounces each jug gives you per dollar.

Step 1: Decide what unit rate to use. The two jugs hold different amounts, so compare cost per ounce. The phrase "cents per ounce" tells you to divide cost by ounces.

Step 2: Find the unit price of the small jug. Work in cents to keep the arithmetic clean: 4 dollars and 48 cents is 448 cents.

448 \div 64 = 7

The small jug costs 7 cents per ounce.

Step 3: Find the unit price of the large jug. 6 dollars and 24 cents is 624 cents.

624 \div 96 = 6.5

The large jug costs 6.5 cents per ounce.

Step 4: Compare. For cost per ounce, smaller is better, and

6.5 < 7

. The 96 ounce jug is the better buy by half a cent per ounce.

Step 5: Now find the reciprocal unit rate, ounces per dollar. Divide ounces by dollars.

Small jug:

64 \div 4.48 \approx 14.3

ounces per dollar.

Large jug:

96 \div 6.24 \approx 15.4

ounces per dollar.

Step 6: Check that both answers agree. For ounces per dollar, larger is better, and 15.4 is greater than 14.3, so the large jug wins again. The two unit rates point to the same conclusion, which confirms the work. If they had disagreed, that would signal a division done in the wrong order.

Practice questions

A pack of 5 notebooks sells for 6 dollars and 25 cents. What is the unit price?

0.80 dollars per notebook
1.25 dollars per notebook
1.25 notebooks per dollar
31.25 dollars per notebook

Answer: 1.25 dollars per notebook

"Dollars per notebook" means notebooks goes in the denominator, so divide

6.25 \div 5 = 1.25

. The answer 0.80 comes from dividing backward (

5 \div 6.25

), which would be notebooks per dollar, not dollars per notebook. The third option has the right number but the wrong unit — one dollar does not buy 1.25 notebooks when each notebook costs more than a dollar. The last option comes from multiplying instead of dividing.

Jamal earns 42 dollars for 6 hours of yard work. Priya earns 66 dollars for 11 hours of yard work. Whose job pays better, and by how much per hour? Show the unit rate for each.

Answer: Jamal's job pays better, at 7 dollars per hour compared with Priya's 6 dollars per hour — 1 dollar more per hour.

Dollars per hour means dividing dollars by hours. For Jamal,

42 \div 6 = 7

dollars per hour. For Priya,

66 \div 11 = 6

dollars per hour. Since these are earnings, a larger rate is better, so Jamal's job pays more, and

7 - 6 = 1

dollar more per hour. Notice that Priya earns more money in total; comparing total pay alone would give the wrong conclusion because she also works far more hours. That is exactly the situation unit rates are built for.

A printer produces 45 pages in 3 minutes at a steady rate. At that rate, how many pages will it print in 8 minutes?

120 pages
135 pages
15 pages
24 pages

Answer: 120 pages

First find the unit rate in pages per minute:

45 \div 3 = 15

pages per minute. Then multiply by 8 minutes:

15 \times 8 = 120

pages. The choice 15 is the unit rate itself, not the answer to the question asked, and 135 comes from multiplying

45 \times 3

. Finding the unit rate first and then scaling up is the strategy you will use again when you write equations for proportional relationships.

FAQ

What is the difference between a ratio and a rate?: A ratio compares any two quantities by division, and the two quantities can share a unit, like 3 cups of flour to 2 cups of sugar. A rate compares two quantities with different units, like 120 miles in 3 hours. Every rate is a ratio, but a ratio is only called a rate when the units differ.
How do I know which number to divide by when finding a unit rate?: Read the words after "per." That quantity becomes 1, so it goes in the denominator. For dollars per pound, divide dollars by pounds. Then check whether your answer is a reasonable size: if 5 pounds cost 7 dollars, one pound must cost between 1 and 7 dollars.
Can a unit rate be a decimal or a fraction?: Yes. Unit rates such as 1.5 dollars per muffin, 8.4 minutes per mile, and $\frac{2}{3}$ muffin per dollar are all correct. Only round when the answer is money, and then round to the nearest cent. Otherwise keep the exact decimal or fraction.
Why does the bigger package sometimes cost more per unit?: Pricing is set by the store, not by math, so bulk is not automatically cheaper. Sale prices, coupons, and packaging costs can make a smaller size cheaper per ounce. That is why you compute the unit price for both options instead of assuming the larger one wins.

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