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Rates and unit rates

Three liters from a water cooler fill twelve cups. How many cups does one liter fill, and how much water goes into one cup?

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Why this exists

Marek watches boxes roll past on a factory conveyor belt. In 6 minutes he counts 90 boxes. The ratio 90 : 6 is true, but the question people ask about a belt is how fast it runs, and the clearest answer is how many boxes pass in exactly one minute: 15. With that single number any stretch of time is easy. Ten minutes carry ten lots of 15 boxes.

The same need turns up wherever two different kinds of thing are tied together: cups poured for each liter in a water cooler, bottles packed into each crate at a bottle bank. A comparison of two amounts measured in different units is a rate, and bringing it down to exactly one of the second amount gives the unit rate, the most useful number in the comparison.

Definition

Day 1: rates and the unit rate

A ratio compares two amounts. When the two amounts are measured in different units, such as boxes and minutes, cups and liters, or bottles and crates, the ratio is called a rate. The units travel with the numbers: “90 boxes in 6 minutes”, “25 cups for every 5 liters”. A rate can also be written as a fraction with its units,

\frac{90 \text{ boxes}}{6 \text{ minutes}}

The word per means “for each” or “for every”, and is often written as a slash, as in boxes/minute.

The unit rate is the amount of the first quantity that goes with exactly 1 of the second. For the belt it is 15 boxes per minute, short for 15 boxes for every 1 minute. It is found by dividing the first amount by the second:

90 \div 6 = 15 \text{ boxes per minute}

Day 2: two unit rates in one ratio

A rate can be read in either order. “12 cups for 3 liters” is also “3 liters for 12 cups”, so the same ratio gives two unit rates:

  • cups per liter: 12 \div 3 = 4 cups for each liter;
  • liters per cup: \displaystyle 3 \div 12 = \frac{1}{4} liter for each cup.

Both describe the same cooler. The useful one is the unit rate that is per the quantity you already know. The amounts may also be fractions, such as \displaystyle \frac{3}{4} liter for 3 cups; the unit rate is still found by dividing, now with division of fractions.

Intuition

Put the belt on a The double number line (Figure 1): minutes on top, boxes below, 6 minutes above 90 boxes. Cutting the gap from 0 to 6 minutes into 6 equal parts cuts the gap from 0 to 90 boxes into 6 equal parts too, and the first part lines up 1 minute with 15 boxes. Every other mark is a number of copies of that small step: 4 minutes is 4 copies, so 4 \times 15 = 60 boxes. In a ratio table the same pair is the row whose minutes entry is 1.

0
0
1
15
2
30
3
45
4
60
5
75
6
90
\text{time (min)}
\text{boxes}
Figure 1Marek’s belt: 90 boxes in 6 minutes. The mark that lines up with 1 minute reads 15 boxes, the unit rate.

Division is the right move because sharing 90 boxes equally among 6 minutes is exactly what 90 \div 6 means.

The two unit rates of one ratio sit in one picture of a cooler (Figure 2). Each column is one liter, cut into the cups it fills. Read down a column and one liter holds 4 cups. Look at one cell and one cup is one of four equal parts of a liter, \displaystyle \frac{1}{4} liter. Same picture, two questions: how many cups in one liter, and how much of a liter in one cup.

1\text{ liter}
1\text{ liter}
1\text{ liter}
4\text{ cups} = 1\text{ liter}
1\text{ cup} = \tfrac{1}{4}\text{ liter}
Figure 2Kenji’s cooler: 3 liters fill 12 cups. The shaded liter holds 4 cups; the shaded cup is \tfrac{1}{4} of a liter.

Results

The unit rate is a division. The unit rate of a rate is the first amount divided by the second, measured in units of the first per one of the second.

Using a unit rate. An amount of the second quantity goes with that many copies of the unit rate, so multiply. On the belt, 10 minutes carry 15 \times 10 = 150 boxes. To go the other way, divide by the unit rate: 240 boxes take 240 \div 15 = 16 minutes.

Two unit rates, one ratio. The two unit rates of a ratio are reciprocals: each is the other flipped over, and their product is 1. For the cooler,

4 \times \frac{1}{4} = 1

Choosing. Multiply by the unit rate that is per the quantity you know. Knowing liters, use cups per liter; knowing cups, use liters per cup.

Why it is true

Dividing both amounts of a ratio by the same number gives an equivalent ratio (Scaling a ratio up or down). Divide both amounts of 90 : 6 by 6, the second amount:

90 : 6 = (90 \div 6) : (6 \div 6) = 15 : 1

The second amount is now 1, so the first is the amount for exactly one: the unit rate, found by dividing the first amount by the second. Scaling 15 : 1 up by 10 gives 150 : 10, which is why multiplying the unit rate by an amount of the second quantity gives the matching first amount.

For the other unit rate, divide both amounts of 6 : 90 (minutes to boxes) by 90. That gives \displaystyle \frac{6}{90} : 1, and \displaystyle \frac{6}{90} = \frac{1}{15}, so each box takes \displaystyle \frac{1}{15} minute. Written as fractions, the two unit rates are \displaystyle \frac{90}{6} and \displaystyle \frac{6}{90}, one the other turned upside down, and

\frac{90}{6} \times \frac{6}{90} = \frac{540}{540} = 1

In words: 15 boxes pass in a minute and each takes \displaystyle \frac{1}{15} minute, so 15 of them take \displaystyle 15 \times \frac{1}{15} = 1 minute, back where we started.

Worked examples

Examples 1 to 3 go with the first day, 4 to 7 with the second.

Example 1

Boxes per minute

Marek counts 90 boxes passing on the belt in 6 minutes. Write the rate, find the unit rate, and find how many boxes the belt carries in 10 minutes.

Solution

Boxes and minutes are different units, so this is a rate: 90 boxes per 6 minutes. The unit rate is boxes for one minute, so divide the boxes by the minutes:

90 \div 6 = 15 \text{ boxes per minute}

Each minute carries 15 boxes, so 10 minutes carry 10 lots of 15: 15 \times 10 = 150 boxes.

Check. 150 \div 10 = 15, the same unit rate, so 150 : 10 is equivalent to 90 : 6.

Example 2

A unit rate that is not whole

Lorna’s water cooler at the gym fills 15 small cups from 2 liters. How many cups does one liter fill, and how many full cups do 5 liters fill?

Solution

Cups per liter means cups divided by liters:

15 \div 2 = 7.5 \text{ cups per liter}

A unit rate need not be a whole number: one liter fills 7 cups and half of another. For 5 liters, take 5 lots of 7.5: 5 \times 7.5 = 37.5 cups. Only full cups count, so 5 liters fill 37 full cups with half a cup of water left over.

Check. 4 liters is twice 2 liters, so it fills 30 cups; one more liter adds 7.5, giving 37.5.

Example 3

Using a unit rate both ways

At the bottle bank Nadine packs 168 bottles into 7 crates, the same number in each. How many bottles go in each crate? How many bottles do 12 crates hold? How many crates does she need for 600 bottles?

Solution

Bottles per crate is bottles divided by crates: 168 \div 7 = 24 bottles per crate.

Twelve crates hold 12 lots of 24: 12 \times 24 = 288 bottles.

For 600 bottles the bottles are known and the crates are not. Each crate takes 24 bottles, so the question is how many 24s make 600:

600 \div 24 = 25 \text{ crates}

Check. 25 \times 24 = 600. The last part divided by the unit rate; Example 5 shows how the other unit rate turns that division into a multiplication.

Example 4

Two unit rates from one cooler

Kenji’s office cooler fills 12 cups from 3 liters (Figure 2). Find the cups per liter and the liters per cup, and show that their product is 1. How much water do 30 cups take?

Solution

Cups per liter: 12 \div 3 = 4 cups. Liters per cup: \displaystyle 3 \div 12 = \frac{3}{12} = \frac{1}{4} liter, which is what each cell of Figure 2 shows. Their product is

4 \times \frac{1}{4} = \frac{4}{4} = 1

which says four cups of a quarter liter each make one liter.

For 30 cups the cups are known, so use liters per cup: \displaystyle 30 \times \frac{1}{4} = \frac{30}{4} = 7.5 liters.

Check. 7.5 liters at 4 cups per liter fill 7.5 \times 4 = 30 cups.

Example 5

Choosing the useful unit rate

Silas packs 72 bottles into 6 crates, the same number in each. How many bottles do 15 crates hold? How many crates does he need for 150 bottles?

Solution

The ratio gives two unit rates: 72 \div 6 = 12 bottles per crate, and \displaystyle 6 \div 72 = \frac{6}{72} = \frac{1}{12} crate per bottle.

For 15 crates the crates are known, so use bottles per crate: 15 \times 12 = 180 bottles.

For 150 bottles the bottles are known, so use crates per bottle:

150 \times \frac{1}{12} = \frac{150}{12} = 12.5 \text{ crates}

That is 12 full crates and half of another, so Silas needs 13 crates. Multiplying by \displaystyle \frac{1}{12} is the same as dividing by 12, since dividing by a number is multiplying by its reciprocal, so 150 \div 12 gives the same 12.5.

Example 6

A fraction of a minute

On another belt Petra counts 12 boxes in \displaystyle \frac{2}{3} minute. Find the boxes per minute and the minutes per box.

Solution

Think in thirds of a minute. \displaystyle \frac{2}{3} minute is two thirds, so one third carries 12 \div 2 = 6 boxes, and a whole minute, three thirds, carries 3 \times 6 = 18 boxes. Division gives the same:

12 \div \frac{2}{3} = 12 \times \frac{3}{2} = \frac{36}{2} = 18 \text{ boxes per minute}

Minutes per box divides the other way:

\frac{2}{3} \div 12 = \frac{2}{3} \times \frac{1}{12} = \frac{2}{36} = \frac{1}{18} \text{ minute}

Check. \displaystyle 18 \times \frac{1}{18} = 1, as two unit rates of one ratio must give.

Example 7

One rate feeding another

At a bottling plant a belt carries 15 boxes per minute, and each box holds 24 bottles. How many bottles does the belt carry per minute? How long does it take to carry 1800 bottles?

Solution

Look at a single minute. In it the belt carries 15 boxes, each holding 24 bottles, so that minute carries 15 lots of 24:

15 \times 24 = 360 \text{ bottles per minute}

The new unit rate is bottles per minute, built from boxes per minute and bottles per box. For 1800 bottles the bottles are known: 1800 \div 360 = 5 minutes.

Check. In 5 minutes the belt carries 5 \times 15 = 75 boxes, and 75 \times 24 = 1800 bottles.

Boundaries

The rate must be steady. A unit rate describes every part of the stretch only if the rate is the same throughout. If Marek’s belt stops for a few minutes, 90 boxes in 6 minutes still divides to 15, but that is an average, not what happens in each minute.

Nothing is per zero. The unit rate divides by the second amount, so that amount cannot be 0: “5 boxes in 0 minutes” has no boxes per minute. A belt that carries 0 boxes in 5 minutes has 0 boxes per minute, but no minutes per box, since that would divide by 0.

Counts come whole. Working with a unit rate can give a fraction of something that only comes whole: 37.5 cups means 37 full cups, and 12.5 crates means 13 crates are needed. The situation decides whether to round down or up.

A unit rate needs its units. “4” alone could be 4 cups per liter or 4 liters per cup, which describe very different coolers. The units say which amount is per which.

The units must match the question. A rate in boxes per minute does not answer a question in hours until the units are changed, which is the work of changing the units of a rate.

Where this goes next

A unit rate turns two amounts into one number, and that makes rates easy to compare: the price of one item picks the better buy between two packet sizes, and speed is distance per unit of time. Changing a rate’s units, from boxes per minute to boxes per hour, uses factors that equal one. When both amounts are fractions, as in \displaystyle \frac{1}{2} km in \displaystyle \frac{1}{4} hour, the unit rate is a fraction divided by a fraction, which Unit rates when the amounts are fractions takes on. Later the unit rate becomes the The constant of proportionality and proportional tables, the single number every input is multiplied by, and it is the quickest way to solve a proportion from words.

Exercises

Day 1: finding and using a unit rate

  1. A belt carries 84 boxes in 7 minutes. Find the unit rate in boxes per minute.
  2. Nadine packs 126 bottles into 6 crates, the same number in each. How many bottles go in each crate?
  3. A water cooler fills 22 cups from 4 liters. How many cups does one liter fill?
  4. Which of these are rates? (a) 3 green crates to 5 brown crates; (b) 45 boxes in 3 minutes; (c) 14 cups for 2 liters; (d) 6 full cups to 2 empty cups.
  5. A belt carries 13 boxes per minute. How many boxes does it carry in 25 minutes?
  6. At 13 boxes per minute, how long does the belt take to carry 156 boxes?
  7. A belt carries 210 boxes in 15 minutes. Find the unit rate, then the number of boxes it carries in 40 minutes.
  8. A cooler fills 30 cups from 8 liters. Find the cups per liter, and the number of full cups that 12 liters fill.
  9. At a bottle bank 9 crates hold 198 bottles, the same number in each. How many crates are needed for 352 bottles?
  10. A ratio table for a cooler pairs 2 liters with 9 cups and 6 liters with 27 cups. What goes in the row for 1 liter, and in the row for 7 liters?

Day 2: two unit rates, and fractions

  1. A cooler fills 35 cups from 5 liters. Find both unit rates.
  2. Silas packs 98 bottles into 7 crates, the same number in each. Find both unit rates.
  3. A belt carries 48 boxes in 3 minutes. Find both unit rates.
  4. A cooler fills 8 cups from 3 liters. Find both unit rates, then the water that 20 cups take.
  5. Crates at a bottle bank hold 16 bottles each. Which unit rate answers “how many crates for 100 bottles?”, and how many crates are needed?
  6. Half a liter fills 3 cups. How many cups does one liter fill?
  7. A belt carries 14 boxes in \displaystyle \frac{2}{3} minute. Find the boxes per minute, then the number of boxes in 4 minutes.
  8. A cooler fills 12 cups from \displaystyle 1\frac{1}{2} liters. Find both unit rates.
  9. Three quarters of a crate holds 15 bottles. How many bottles does a full crate hold?
  10. A belt carries 5 boxes every \displaystyle \frac{1}{4} minute. Find the boxes per minute and the minutes per box, then how long 150 boxes take.
  11. A belt carries 18 boxes per minute, and each box holds 12 bottles. How many bottles does it carry per minute, and how long does it take to carry 1080 bottles?

Reasoning

  1. Using 168 bottles in 7 crates, explain why dividing the first amount of a rate by the second gives the amount for exactly one of the second.
  2. A cooler fills 12 cups from 3 liters. One description says “4 cups per liter”, another says “4 liters per cup”. Which is right, and how can you tell without dividing?
  3. Using 98 bottles in 7 crates, explain why the two unit rates of one ratio always multiply to 1.
Answers
  1. 12 boxes per minute. 84 \div 7 = 12.
  2. 21 bottles. 126 \div 6 = 21.
  3. 5.5 cups. 22 \div 4 = 5.5.
  4. (b) and (c). Each compares amounts in different units (boxes with minutes, cups with liters). (a) compares crates with crates and (d) cups with cups, so they are ratios but not rates.
  5. 325 boxes. 13 \times 25 = 325.
  6. 12 minutes. 156 \div 13 = 12.
  7. 14 boxes per minute; 560 boxes. 210 \div 15 = 14, and 14 \times 40 = 560.
  8. 3.75 cups per liter; 45 full cups. 30 \div 8 = 3.75, and 12 \times 3.75 = 45.
  9. 16 crates. 198 \div 9 = 22 bottles per crate, and 352 \div 22 = 16.
  10. 4.5 cups and 31.5 cups. 9 \div 2 = 4.5 (and 27 \div 6 = 4.5 agrees), and 7 \times 4.5 = 31.5.
  11. 7 cups per liter and \displaystyle \frac{1}{7} liter per cup. 35 \div 5 = 7 and \displaystyle 5 \div 35 = \frac{5}{35} = \frac{1}{7}.
  12. 14 bottles per crate and \displaystyle \frac{1}{14} crate per bottle. 98 \div 7 = 14 and \displaystyle 7 \div 98 = \frac{7}{98} = \frac{1}{14}.
  13. 16 boxes per minute and \displaystyle \frac{1}{16} minute per box. 48 \div 3 = 16 and \displaystyle 3 \div 48 = \frac{1}{16}.
  14. \displaystyle 2\frac{2}{3} cups per liter and \displaystyle \frac{3}{8} liter per cup; 7.5 liters. \displaystyle 8 \div 3 = \frac{8}{3}, \displaystyle 3 \div 8 = \frac{3}{8}, and \displaystyle 20 \times \frac{3}{8} = \frac{60}{8} = 7.5.
  15. Crates per bottle, \displaystyle \frac{1}{16}, since the bottles are known; 7 crates. \displaystyle 100 \times \frac{1}{16} = 6.25, so 6 full crates and a quarter of another, which still needs a crate.
  16. 6 cups. \displaystyle 3 \div \frac{1}{2} = 3 \times 2 = 6.
  17. 21 boxes per minute; 84 boxes. \displaystyle 14 \div \frac{2}{3} = 14 \times \frac{3}{2} = 21, and 21 \times 4 = 84.
  18. 8 cups per liter and \displaystyle \frac{1}{8} liter per cup. \displaystyle 12 \div \frac{3}{2} = 12 \times \frac{2}{3} = 8, and \displaystyle \frac{3}{2} \div 12 = \frac{3}{24} = \frac{1}{8}.
  19. 20 bottles. One quarter of a crate holds 15 \div 3 = 5, so four quarters hold 20; or \displaystyle 15 \div \frac{3}{4} = 15 \times \frac{4}{3} = 20.
  20. 20 boxes per minute; \displaystyle \frac{1}{20} minute per box; 7.5 minutes. \displaystyle 5 \div \frac{1}{4} = 20, \displaystyle \frac{1}{4} \div 5 = \frac{1}{20}, and \displaystyle 150 \times \frac{1}{20} = 7.5.
  21. 216 bottles per minute; 5 minutes. 18 \times 12 = 216, and 1080 \div 216 = 5.
  22. Dividing both amounts of 168 : 7 by 7 gives the equivalent ratio 24 : 1, because dividing both parts of a ratio by the same number keeps it the same. The second amount is now 1 crate, so 24 is the number of bottles for exactly one crate, and it was found as 168 \div 7.
  23. “4 cups per liter”. A liter is more water than a cup, so one liter fills several cups and one cup takes only part of a liter. “4 liters per cup” would put more water in one cup than the whole 3 liters.
  24. The unit rates are 98 \div 7 = 14 bottles per crate and \displaystyle 7 \div 98 = \frac{1}{14} crate per bottle. As fractions they are \displaystyle \frac{98}{7} and \displaystyle \frac{7}{98}, one the other flipped, and their product is \displaystyle \frac{686}{686} = 1. In words: 14 bottles fill a crate and each takes \displaystyle \frac{1}{14} of a crate, so the 14 together take \displaystyle 14 \times \frac{1}{14} = 1 crate. Any ratio gives two fractions that are flips of each other, so the product is always 1.
kg1128 · updated 2026-10-12 · CC-BY-SA-4.0