6th Grade · 6.RP.A.2

Unit rate in 6th grade

A unit rate tells you how much of one thing goes with 1 of another, such as 60 miles per hour or 2 dollars per pound. In 6th grade, your child learns that every ratio has a unit rate and finds it by dividing. Your child also learns rate language, the words "per" and "for each".

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.RP.A.2 Understand the concept of a unit rate $\frac{a}{b}$ associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.

What it means, in plain words

Rates compare two quantities measured in different units, such as miles and hours or dollars and pounds. A rate such as "120 miles in 2 hours" is a ratio, $120:2$ . A unit rate gives the amount for 1 of the second quantity: 60 miles per hour. The word "per" means "for each 1".

To find a unit rate, divide the first amount by the second. For a ratio $a:b$ , the unit rate is $\frac{a}{b}$ , which is $a \div b$ . Every ratio gives two unit rates, one for each order. If 3 pounds of apples cost 6 dollars, the price is 2 dollars per pound, and 1 dollar buys $\frac{1}{2}$ pound of apples.

Unit rates in 6th grade can be fractions or decimals, such as $\frac{2}{5}$ cup of blue paint per cup of white paint. Your child should not expect a whole number every time. Saying what the unit rate means, with both units, is part of the answer.

Your child should be able to:

Find a unit rate by dividing, including when the answer is a fraction or a decimal
Use rate language such as "per" and "for each" to say what a unit rate means
Find both unit rates for a ratio and pick the one the question asks for
Use unit rates to compare prices, such as the cost of one item in two different packs

Example

A pack of 6 juice boxes costs 9 dollars. What is the price per juice box?

Price per juice box means dollars for 1 juice box, so divide the dollars by the juice boxes.
$9 \div 6 = 1.5$
The unit rate is 1.50 dollars per juice box, or 1 dollar and 50 cents for each one.

Example

A paint mix uses 2 cups of blue paint for every 5 cups of white paint. How much blue paint is there per cup of white paint?

Blue paint per cup of white paint means blue divided by white: $2 \div 5 = \frac{2}{5}$ .
So there is $\frac{2}{5}$ cup of blue paint for each cup of white paint.
The other unit rate goes the other way: $5 \div 2 = 2\frac{1}{2}$ cups of white paint for each cup of blue paint.

Where children usually get stuck

For the paint mix, your child divides $5 \div 2$ and says there are $2\frac{1}{2}$ cups of blue paint per cup of white paint.
In earlier grades, division mostly went from the bigger number to the smaller one. The child picks that order without checking which amount comes before "per".
What helps. Say the units out loud: blue per white means blue divided by white. Then ask: with less blue paint than white paint in the mix, can each cup of white get more than 1 cup of blue?
Your child says a 12-pack of juice boxes for 15 dollars costs more than a 6-pack for 9 dollars, so the 6-pack is the better deal.
The child compares the total prices. A fair comparison needs the price of one juice box in each pack.
What helps. Work out the price per juice box for both packs: 1.25 dollars in the 12-pack and 1.50 dollars in the 6-pack. Each box is cheaper in the bigger pack.
Your child finds $9 \div 6 = 1.5$ and writes the answer as "1.5 juice boxes".
The division is right, but the child lost track of what the number counts. In 1.5 dollars per juice box, the 1.5 counts dollars.
What helps. Ask your child to finish every answer with a full rate: "___ dollars per ___". Saying both units shows what the number means.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
A pasta recipe uses $\frac{1}{2}$ cup of olive oil for 2 servings. Lena wants to find cups of olive oil per 1 serving. Her work is shown below.
$\displaystyle \frac{1}{2} \times 2$
$\displaystyle =$
$\displaystyle 1$
Which statement best describes Lena's calculation?
A
She reversed the quantities and found servings per cup instead of cups per serving.
B
She treated dividing by $\frac{1}{2}$ as the same as dividing by 2.
C
She multiplied instead of dividing to get a rate for 1 serving.
D
There is no mistake.
Show the answer and how to solve it
The correct choice says Lena multiplied instead of dividing to get a rate for 1 serving.
1. 1The $\frac{1}{2}$ cup of olive oil is for 2 servings. Divide that amount into 2 equal shares to find cups of olive oil per 1 serving.
2. 2The correct setup divides the oil by the number of servings.
3. 3Lena's line shows $\frac{1}{2} \times 2 = 1$ . She multiplied the oil by the servings instead of dividing, so her result is not cups per serving.
4. 4The correct choice says she multiplied instead of dividing to get a rate for 1 serving.
Common mistake. A common mistake is accepting the result because $\frac{1}{2} \times 2 = 1$ is correct arithmetic. In a unit-rate problem the correct operation is division: $\frac{1}{2} \div 2 = \frac{1}{4}$ .
The idea underneath. In unit-rate problems, the quantity named before per is the quantity that must be divided by the other quantity.
2On level
A bread recipe uses the ratio below.
$\displaystyle \frac{2}{3}\text{ cup of flour for }4\text{ bags}$
Enter the number of cups of flour for 1 bag. Type the number only, with no units. You may write a fraction.
Loading...
Show the answer and how to solve it
Answer. \frac{1}{6}
The answer is $\frac{1}{6}$ .
1. 1Look at the ratio. It shows $\frac{2}{3}$ cup of flour for $4$ bags.
2. 2To find the amount for $1$ bag, divide $\frac{2}{3}$ by $4$ .
3. 3Dividing by $4$ is the same as multiplying by $\frac{1}{4}$ . Simplify $\frac{2}{12}$ by dividing numerator and denominator by $2$ to get $\frac{1}{6}$ .
Common mistake. A common mistake is multiplying $\frac{2}{3}$ by $4$ and getting $\frac{8}{3}$ . For the rate per $1$ bag, divide by $4$ instead of multiplying.
The idea underneath. A rate for $1$ is found by dividing the first quantity by the second quantity in the original ratio.
3On level
Look at the double number line below. It shows a reading challenge at a constant pace, and Sofia is trying to write the number of pages in 1 minute.
$\displaystyle \text{Sofia's sentence: "The unit rate is 5 minutes per page."}$
Select the best description of Sofia's mistake.
A
It uses the correct number for the requested rate, but reverses the units.
B
It gives a correct rate for 1 page, instead of for 1 minute.
C
It repeats a pair from the diagram (like 20 pages in 4 minutes) instead of giving the amount for just 1 minute.
D
It gives a rate in pages per 1 minute, but the number is wrong.
Show the answer and how to solve it
The answer is the choice explaining that the number 5 is right but the labels are backward. The double number line shows 20 ÷ 4 = 5 pages per minute, not 5 minutes per page.
1. 1On the double number line, 4 minutes lines up with 20 pages. Use that pair to find the pages in 1 minute.
2. 2Divide both numbers in that pair by 4.
3. 3So the correct unit rate sentence is 5 pages per 1 minute.
4. 4Sofia's sentence says 5 minutes per page. The number 5 is correct, but it is attached to minutes — it should be attached to pages.
5. 5So the best description is that Sofia used the correct number (5) but mixed the units — the rate is 5 pages per minute, not 5 minutes per page.
Common mistake. A common mistake is checking only the number 5 and ignoring the unit labels. A unit-rate sentence needs both the correct quotient (5) and the correct label order (pages per minute, not minutes per page).
The idea underneath. The word per tells which quantity is being made 1, so the number and its units must stay together; swapping the labels changes the meaning even when the number is right.
4Stretch
A car travels 20 miles on 16 gallons of gas. Emma's work for the miles per gallon is shown below.
$\displaystyle 16 \div 20 = 0.8$
Select the statement that best explains Emma's mistake.
A
She divided in the wrong order.
B
She used multiplication instead of division.
C
She stopped after simplifying the ratio.
D
She used one starting number as the answer.
Show the answer and how to solve it
Emma's mistake is that she divided in the wrong order. The correct miles per gallon is $20 \div 16 = 1.25$ , not $16 \div 20 = 0.8$ .
1. 1Look at Emma's work. Miles per gallon means the number of miles for every 1 gallon, so the miles should be divided by the gallons: $20 \div 16$ .
2. 2The correct calculation is:
3. 3Emma computed $16 \div 20 = 0.8$ instead. She switched the positions of 20 and 16 in the division. Her result of 0.8 represents gallons per mile, not miles per gallon.
4. 4The correct statement is the one that says she divided in the wrong order. The correct miles per gallon is $20 \div 16 = 1.25$ , not $16 \div 20 = 0.8$ .
Common mistake. A common mistake is reversing the two quantities because both 20 and 16 appear in the same situation. For miles per gallon, the miles go first: divide 20 by 16 to get 1.25.
The idea underneath. In unit-rate problems, the order of division matters because the first quantity's unit is the unit that stays in the answer.