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7th Grade · 7.RP.A.1

Unit rates with fractions in 7th grade

Unit rates with fractions answer the question "how much for one?" when the amounts are fractions. A unit rate is the amount of one quantity for exactly 1 of another, such as miles per hour. In 7th grade, your child divides one fraction by another to find these rates, such as how much of a wall a painter covers in one hour.

What the standard says

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

7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

What it means, in plain words

A rate compares two quantities, such as miles and hours. The unit rate is the amount for exactly 1 of the second quantity: 30 miles in 2 hours is a unit rate of 15 miles per hour. In 6th grade the numbers were mostly whole. In 7th grade they can be fractions, and the two units can be the same kind, such as cups of one thing per cup of another.

To find unit rates with fractions, your child writes the first amount over the second and divides. With fractions this makes complex fractions, a fraction with a fraction on top, on the bottom or both. Each one is a division. Dividing by a fraction is the same as multiplying by its reciprocal, the fraction turned upside down, so dividing by 14\frac{1}{4} is the same as multiplying by 4.

Order matters. Walls per hour and hours per wall use the same two numbers in opposite order, and they answer different questions. The quantity after the word "per" goes on the bottom.

Your child should be able to:

  • Write a rate with fractions as a complex fraction
  • Divide fractions to find the amount for exactly 1 unit
  • Choose which quantity goes on the bottom from the word "per"
  • Find unit rates for lengths, areas and other quantities, in the same or different units

Example

A painter covers 34\frac{3}{4} of a wall in 12\frac{1}{2} hour. How much wall does the painter cover per hour, and how long does one whole wall take?

  1. Walls per hour: 3412=34×21=64=112\frac{\frac{3}{4}}{\frac{1}{2}} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = 1\frac{1}{2}
  2. Hours per wall: 1234=12×43=46=23\frac{\frac{1}{2}}{\frac{3}{4}} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}
  3. The painter covers 1121\frac{1}{2} walls per hour, and one whole wall takes 23\frac{2}{3} of an hour, which is 40 minutes.

Example

A robot mower cuts 25\frac{2}{5} of an acre in 23\frac{2}{3} of an hour. How many acres does it cut per hour?

  1. Acres per hour puts acres on top: 2523\frac{\frac{2}{5}}{\frac{2}{3}}
  2. 25×32=610=35\frac{2}{5} \times \frac{3}{2} = \frac{6}{10} = \frac{3}{5}
  3. The mower cuts 35\frac{3}{5} of an acre per hour.

Where children usually get stuck

  • Asked for walls per hour, your child works out 12÷34\frac{1}{2} \div \frac{3}{4} and answers 23\frac{2}{3}.

    The child put the numbers in the order they came in the sentence, hours first. Walls per hour needs the walls on top.

    What helps. Read "per" as "for each". Walls per hour means walls for each hour, so the hours go on the bottom.

  • A runner covers 6 miles in 1 hour 20 minutes. Your child writes the time as 1.20 hours and gets 5 miles per hour.

    Minutes are out of 60, not 100. Twenty minutes is 2060=13\frac{20}{60} = \frac{1}{3} of an hour, so the time is 1131\frac{1}{3} hours and the rate is 6÷43=4126 \div \frac{4}{3} = 4\frac{1}{2} miles per hour.

    What helps. Before any rate problem with time, ask your child to write the minutes as a fraction of an hour.

  • Your child stops at 2523\frac{\frac{2}{5}}{\frac{2}{3}} and does not know what to do next.

    A fraction stacked on a fraction looks like a new kind of number. It is a division written in a different way.

    What helps. Have your child read the long fraction bar out loud as "divided by", then rewrite the problem as a division on one line.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    Which choice shows the simplified value of the complex fraction below?
    3412\displaystyle \dfrac{\frac{3}{4}}{\frac{1}{2}}
    A
    32\displaystyle \frac{3}{2}
    B
    23\displaystyle \frac{2}{3}
    C
    38\displaystyle \frac{3}{8}
    D
    64\displaystyle \frac{6}{4}
    Show the answer and how to solve it

    The correct choice is 32\frac{3}{2} because 34÷12=34×21=64\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4}, and 64\frac{6}{4} simplifies to 32\frac{3}{2}.

    1. 1The complex fraction means 34\frac{3}{4} divided by 12\frac{1}{2}.
    2. 2Multiply the top and bottom of the complex fraction by 21\frac{2}{1}. This keeps its value the same and makes the bottom 12×21=1\frac{1}{2} \times \frac{2}{1} = 1. Dividing by 11 leaves 34×21\frac{3}{4} \times \frac{2}{1}.
    3. 3Multiply across to get 64\frac{6}{4}, then divide both 66 and 44 by 22 to simplify to 32\frac{3}{2}.

    Common mistake. A common mistake is multiplying 34\frac{3}{4} and 12\frac{1}{2} straight across to get 38\frac{3}{8}. Because 12\frac{1}{2} is the divisor, it must be flipped to 21\frac{2}{1} before multiplying.

    The idea underneath. A complex fraction is just one fraction divided by another, so the important move is to flip only the divisor and then multiply.

  2. 2On level
    Look at the student's work below. Which mistake did the student make?
    Original
    1334\displaystyle \frac{\frac{1}{3}}{\frac{3}{4}}
    Step 1
    =13×34\displaystyle = \frac{1}{3} \times \frac{3}{4}
    Step 2
    =312\displaystyle = \frac{3}{12}
    Step 3
    =14\displaystyle = \frac{1}{4}
    A
    The student inverted the first fraction instead of the denominator fraction.
    B
    The student changed division to multiplication and left both fractions unchanged.
    C
    The student simplified the final fraction incorrectly.
    D
    In Step 1, the student swapped the top and bottom fractions of the whole complex fraction.
    Show the answer and how to solve it

    The correct description is that the student changed to multiplication but did not invert the denominator fraction. Dividing by 34\frac{3}{4} should be rewritten as multiplying by 43\frac{4}{3}.

    1. 1Start by checking the move from Original to Step 1. That is where division by a fraction should be rewritten correctly.
    2. 2When you divide by 34\frac{3}{4}, you multiply by its reciprocal, 43\frac{4}{3}.
    3. 3The student wrote 13×34\frac{1}{3} \times \frac{3}{4}. That changes division to multiplication, but it does not flip 34\frac{3}{4} to 43\frac{4}{3}.
    4. 4The simplification at the end is not the issue. 312\frac{3}{12} really does equal 14\frac{1}{4}, so the error is not in the final step.
    5. 5So the matching description is that the student changed to multiplication but did not invert the denominator fraction.

    Common mistake. A common mistake is remembering that a reciprocal is needed but not remembering which fraction to flip. Only the divisor 34\frac{3}{4} gets inverted, so 13÷34\frac{1}{3} \div \frac{3}{4} becomes 13×43\frac{1}{3} \times \frac{4}{3}.

    The idea underneath. Dividing by a fraction asks how many of those fractional parts fit into the amount, so the divisor is replaced by its reciprocal when you rewrite the division.

  3. 3On level
    Look at the recipe and Lena's expression below. She says it shows cups of butter per cup of honey; which statement best explains her work?
    Recipe

    Granola Mix

    • 34cuphoney
    • 23cupbutter
    34 cup honey23 cup butter\displaystyle \frac{\frac{3}{4}\text{ cup honey}}{\frac{2}{3}\text{ cup butter}}
    A
    The whole fraction is reversed. It gives cups of honey per cup of butter, not cups of butter per cup of honey.
    B
    The expression is correct.
    C
    The top ingredient amount was flipped. The honey amount should stay 34\frac{3}{4}, not 43\frac{4}{3}.
    D
    The bottom ingredient amount was flipped. The butter amount should stay 23\frac{2}{3}, not 32\frac{3}{2}.
    Show the answer and how to solve it

    The best explanation is: The whole fraction is reversed. Lena's expression represents cups of honey per cup of butter, not cups of butter per cup of honey.

    1. 1Use the recipe to find the two amounts: honey is 34\frac{3}{4} cup and butter is 23\frac{2}{3} cup.
    2. 2The words 'butter per honey' tell you the order: butter goes on top and honey goes on the bottom.
    3. 3Lena's expression puts honey on top and butter on the bottom instead — the opposite of what 'butter per honey' requires.
    4. 4Lena used the right two amounts, but the whole complex fraction is backward. Her expression represents cups of honey per cup of butter, not cups of butter per cup of honey.

    Common mistake. A common mistake is reading the ingredients top to bottom off the recipe card instead of following the words 'butter per honey.' In any rate written 'A per B,' A goes in the numerator and B goes in the denominator.

    The idea underneath. In a rate written 'A per B,' the words tell the fraction's structure: A is the numerator and B is the denominator.

  4. 4Stretch
    A cyclist's distance and time are shown in the table. What is the unit rate in miles per hour?
    QuantityAmount
    Distance4 1/6 miles
    Time2 1/2 hours
    A
    53\displaystyle \frac{5}{3}
    B
    35\displaystyle \frac{3}{5}
    C
    12512\displaystyle \frac{125}{12}
    D
    2\displaystyle 2
    Show the answer and how to solve it

    The answer is 53\frac{5}{3} because 4164\frac{1}{6} miles divided by 2122\frac{1}{2} hours equals 53\frac{5}{3} miles in 1 hour.

    1. 1Look at the table. Miles per hour means distance divided by time, so divide 4164\frac{1}{6} miles by 2122\frac{1}{2} hours.
    2. 2Rewrite both mixed numbers as improper fractions.
    3. 3Divide by multiplying by the reciprocal of the time fraction, then simplify by dividing numerator and denominator by 10.
    4. 4The unit rate is 53\frac{5}{3} miles per hour, so the first choice is correct.

    Common mistake. A common mistake is multiplying the two fractions instead of dividing, which gives 12512\frac{125}{12}. Dividing by a fraction means multiplying by its reciprocal.

    The idea underneath. In a unit rate, the unit named after per is the denominator, so miles per hour means miles divided by hours.