A Rate for One Unit | Grade 6 Math | LevelKite

You already know how to make equivalent ratios in a ratio table. Now you will use that skill to answer one question: how much goes with just one?

A hose fills a pool steadily, so every minute adds the same amount of water. It adds $12$ liters in $4$ minutes. The ratio of liters to minutes is $12$ to $4$ . A ratio like this, between two quantities that each have a unit, is called a rate.

How many liters does the hose add in $1$ minute? The answer is a unit rate. A unit rate tells you how much of the first quantity goes with exactly $1$ unit of the second quantity.

Equivalent ratios for the hose

Liters of water	Minutes	Change from the first row
$12$	$4$	none: this is the given ratio
$6$	$2$	both divided by $2$
$3$	$1$	both divided by $4$

All three rows describe the same steady flow, so they are equivalent ratios. You know why from ratio tables: dividing both quantities by the same nonzero number keeps the ratio equivalent. Here, cutting the $4$ minutes into $4$ equal parts cuts the $12$ liters into $4$ equal parts too.

The last row is the useful one, because its minutes equal $1$ :

minutes: $4 \div 4 = 1$
liters: $12 \div 4 = 3$

So the unit rate is $3$ liters per minute. The word per means for each one. You can also say $3$ liters for each minute or $3$ liters for every minute.

The unit after per is the unit you have exactly $1$ of. Call it the reference unit. Here the reference unit is the minute.

Key concept

A unit rate is the amount for one unit

In a ratio of $a$ to $b$ , let $a$ be the amount of the first quantity and $b$ the amount of the second quantity. The unit rate is the amount of the first quantity for exactly $1$ unit of the second quantity.

To find it, divide $a$ by $b$ . In this division, $a$ is the dividend, $b$ is the divisor and the answer is the quotient. The divisor $b$ is never $0$ , because dividing by $0$ is not defined.

Why this works: dividing $a$ by $b$ is the same as dividing both parts of the ratio by the nonzero number $b$ . That keeps the ratio equivalent and turns $b$ into $1$ .

The unit rate is the full quotient. It may be a whole number, a fraction or a decimal, and you write it with no remainder.

\text{unit rate} = a \div b = \frac{a}{b}, \quad b \neq 0

Worked example

A whole-number unit rate

A machine fills jars at a steady pace. It fills $45$ jars in $9$ minutes. What is the unit rate in jars per minute?

Name the quantities. The question asks for jars per minute, so the reference unit is the minute. The first quantity is 45 jars and the second quantity is 9 minutes. The divisor 9 is not 0, so you can divide.
$a = 45 \text{ jars}, \quad b = 9 \text{ minutes}$
Divide both quantities by 9. Dividing both by the same nonzero number keeps the ratio equivalent, and it turns 9 minutes into 1 minute.
$45 \div 9 = 5, \qquad 9 \div 9 = 1$
Check with multiplication, which undoes division. At 5 jars in each of 9 minutes, the machine fills 5 times 9 jars in all. The quotient 5 is exact, so the product must equal the dividend, 45.
$5 \times 9 = 45$

Answer

The unit rate is $5$ jars per minute. The machine fills $5$ jars for each $1$ minute.

Worked example

A unit rate less than 1

A potter uses $3$ kilograms of clay to make $4$ identical bowls. How many kilograms of clay does the potter use per bowl?

Name the quantities. The question asks for kilograms per bowl, so the reference unit is the bowl. The first quantity is 3 kilograms and the second quantity is 4 bowls. The divisor 4 is not 0.
$a = 3 \text{ kilograms}, \quad b = 4 \text{ bowls}$
Divide 3 by 4. A fraction is another way to write a division, so 3 divided by 4 is three fourths. Clay can be split, so the 4 bowls share all 3 kilograms and nothing is left over. The quotient has no remainder.
$3 \div 4 = \frac{3}{4}$
Write the fraction as a decimal. Multiply the numerator and the denominator by 25 to make hundredths. Multiplying both by the same nonzero number keeps the fraction equal.
$\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75$
Check with multiplication. The quotient is exact, so three fourths of a kilogram for each of 4 bowls must give back the dividend, 3 kilograms.
$\frac{3}{4} \times 4 = \frac{12}{4} = 3$

Answer

The unit rate is $\frac{3}{4}$ kilogram of clay per bowl, which is $0.75$ kilogram for each $1$ bowl.

Multiplication undoes division, so you can check a unit rate by multiplying it by the divisor, which is never $0$ . What you should see depends on the kind of quotient:

Exact quotient. The product equals the dividend: $\text{quotient} \times \text{divisor} = \text{dividend}$ .
Rounded quotient. Rounding changes the quotient a little, so the product is close to the dividend but not equal to it. This check is approximate, but it still catches big mistakes.

One situation is different. Suppose you share whole things that cannot be split, such as marbles, and every share must be a whole number. If some are left over, the answer is a whole-number quotient and a remainder, the amount left over. That answer is not a unit rate, because the leftovers are not shared. Check it with $\text{divisor} \times \text{whole quotient} + \text{remainder} = \text{dividend}$ .

Keep these answers apart. A fractional share has no remainder. A whole-number share with leftovers has no fraction. The table shows one example of each case.

Four answers and how to check each one

Situation	Answer	Check
$12$ liters poured equally into $4$ jugs	$12 \div 4 = 3$ liters per jug. Exact whole number, no remainder.	$3 \times 4 = 12$
$14$ liters poured equally into $4$ jugs	$14 \div 4 = 3.5$ liters per jug. Exact fractional share, no remainder.	$3.5 \times 4 = 14$
$20$ liters poured equally into $3$ jugs	$20 \div 3 \approx 6.67$ liters per jug. Rounded to the nearest hundredth.	$6.67 \times 3 = 20.01$ , which is close to $20$
$14$ marbles shared equally into $4$ bags	$3$ marbles in each bag, remainder $2$ . Whole-number share with leftovers.	$4 \times 3 + 2 = 14$

Worked example

A unit rate you have to round

A tank drains steadily. It loses $10$ liters of water in $3$ hours. Find the unit rate in liters per hour, first as an exact fraction and then as a decimal rounded to the nearest hundredth.

The question asks for liters per hour, so the reference unit is the hour. Divide the 10 liters by the 3 hours. A fraction is another way to write a division. The divisor 3 is not 0.
$10 \div 3 = \frac{10}{3}$
Write the exact quotient as a mixed number. Split 10 into 9 plus 1, because 9 is the largest multiple of 3 that fits in 10. Water can be split, so the last 1 liter is shared too. It becomes one third of a liter for each hour, not a remainder.
$\frac{10}{3} = \frac{9}{3} + \frac{1}{3} = 3 + \frac{1}{3} = 3\tfrac{1}{3}$
Write the quotient as a decimal. The whole part is 3. For the one third, divide 1 by 3. Ten tenths divided by 3 is 3 tenths with 1 tenth left. Ten hundredths divided by 3 is 3 hundredths with 1 hundredth left. The same step repeats forever, so you must round. The thousandths digit is 3, which is less than 5, so the hundredths digit stays 3.
$3\tfrac{1}{3} = 3.333\ldots \approx 3.33$
Check the exact fraction. An exact quotient times the divisor equals the dividend.
$\frac{10}{3} \times 3 = \frac{30}{3} = 10$
Check the rounded decimal. A rounded quotient gives a product that is close to the dividend but not equal to it, so this check is approximate. The small gap comes from rounding, not from a mistake.
$3.33 \times 3 = 9.99 \approx 10$

Answer

The unit rate is exactly $\frac{10}{3}$ liters per hour, which is $3\frac{1}{3}$ liters per hour. Rounded to the nearest hundredth, it is about $3.33$ liters per hour.

A unit rate tells you how much of one quantity goes with exactly $1$ unit of another quantity. To find it for a ratio of $a$ to $b$ , divide $a$ by $b$ , where $b$ is not $0$ . The unit after per names the divisor.

Write the full quotient with no remainder, then multiply it by the divisor to check. An exact quotient gives back the dividend, and a rounded quotient comes close to it. If you share whole things that cannot be split and some are left over, give a whole-number quotient and a remainder instead, and check divisor times whole quotient plus remainder.

Next, you will use unit rates to find unit prices and to calculate speed, distance or time.