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5th Grade · 5.NF.A.2

Fraction word problems in 5th grade

Fraction word problems are short stories in which the numbers are fractions and your child decides whether to add or subtract. In 5th grade, the fractions in one story often have different denominators, such as thirds and fourths. Your child learns to draw the story, write an equation and check the answer with a quick estimate.

What the standard says

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

5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

What it means, in plain words

Most stories in this grade ask for a total or for what is left. Adding fractions word problems ask how much there is in all, such as the part of a garden planted with two vegetables. Subtraction stories ask how much is left or how much more one amount is. All the fractions in one story must refer to the same whole: the same garden, the same pitcher or the same class.

Drawing the story is the first step in fraction word problems. A visual fraction model is a picture of the fractions, such as a bar cut into equal parts or a number line. The picture shows whether the story needs addition or subtraction. Then your child writes an equation, finds a common denominator and solves it. Fraction addition word problems in 5th grade also include mixed numbers, such as 1121\frac{1}{2} liters.

The last step is a check with benchmark fractions. A benchmark fraction is an easy number to compare with, such as 0, 12\frac{1}{2} or 1. If both parts of a sum are less than 12\frac{1}{2}, the total must be less than 1. A sum that comes out smaller than one of its parts shows a mistake at a glance.

Your child should be able to:

  • Decide whether a story needs addition or subtraction
  • Draw a bar or a number line that shows the fractions in a story
  • Write and solve an equation with unlike denominators or mixed numbers
  • Use 0, 12\frac{1}{2} and 1 to estimate and to spot an answer that cannot be right

Example

In a garden, 25\frac{2}{5} of the beds have tomatoes and 14\frac{1}{4} have peppers. What fraction of the beds have tomatoes or peppers, and what fraction is left for other plants?

  1. Add the two parts. Fifths and fourths both fit into twentieths: 25=820\frac{2}{5} = \frac{8}{20} and 14=520\frac{1}{4} = \frac{5}{20}.
  2. 820+520=1320\frac{8}{20} + \frac{5}{20} = \frac{13}{20} of the beds have tomatoes or peppers.
  3. The whole garden is 2020\frac{20}{20}, so 2020−1320=720\frac{20}{20} - \frac{13}{20} = \frac{7}{20} is left.
  4. Check: 25\frac{2}{5} and 14\frac{1}{4} are both less than 12\frac{1}{2}, so their total must be less than 1.
  5. So thirteen-twentieths of the beds have tomatoes or peppers, and seven-twentieths are left for other plants.

Example

A pitcher holds 1121\frac{1}{2} liters of lemonade, and 23\frac{2}{3} liter is poured into glasses. How much lemonade is left in the pitcher?

  1. Estimate first: 23\frac{2}{3} is a bit more than 12\frac{1}{2}, so a bit less than 1 liter will be left.
  2. Write the mixed number as a fraction: 112=321\frac{1}{2} = \frac{3}{2}.
  3. Use sixths: 32−23=96−46=56\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}.
  4. 56\frac{5}{6} is a bit less than 1, which matches the estimate.
  5. So five-sixths of a liter of lemonade is left.

Where children usually get stuck

  • A story says that 12\frac{1}{2} of a class walks to school and 13\frac{1}{3} rides a bus. Asked what fraction comes another way, your child answers 56\frac{5}{6}.

    The child adds the two fractions because they are the only numbers in the story. The question asks for the part that is left: 1−56=161 - \frac{5}{6} = \frac{1}{6}.

    What helps. Ask your child to draw a bar for the whole class and shade the two groups. The question is about the part that is not shaded.

  • A jug has 2 liters of water. First 12\frac{1}{2} liter is poured out, then 34\frac{3}{4} liter more. Your child answers 1121\frac{1}{2} liters.

    The child solved the first step and stopped. A story with two steps needs a plan before any arithmetic.

    What helps. Draw a bar for the 2 liters and cross off each amount in turn. What is left is 34\frac{3}{4} liter.

  • Your child cannot tell whether 47+59\frac{4}{7} + \frac{5}{9} is more or less than 1 without working it all out.

    The child sees each fraction as two separate numbers, not as one amount. Without a sense of size, an estimate is impossible.

    What helps. Compare each fraction with 12\frac{1}{2}. Half of 7 is 3.5, so 47\frac{4}{7} is more than a half. Half of 9 is 4.5, so 59\frac{5}{9} is more than a half too. Two amounts above a half add up to more than 1.

Try four practice questions

Real 5th 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
    A student wrote the equation below.
    23+34=12\displaystyle \frac{2}{3}+\frac{3}{4}=\frac{1}{2}
    Use the benchmark line below to compare fractions to 0, 12\frac{1}{2}, and 1.
    0
    12\displaystyle \frac{1}{2}
    1
    Select the comparison that proves the result cannot be correct.
    A
    23+34>112<1\displaystyle \begin{array}{l} \frac{2}{3}+\frac{3}{4}>1 \\[0.5em] \frac{1}{2}<1 \end{array}
    B
    23+34<012<1\displaystyle \begin{array}{l} \frac{2}{3}+\frac{3}{4}<0 \\[0.5em] \frac{1}{2}<1 \end{array}
    C
    23+34<112<1\displaystyle \begin{array}{l} \frac{2}{3}+\frac{3}{4}<1 \\[0.5em] \frac{1}{2}<1 \end{array}
    D
    23+34>112>1\displaystyle \begin{array}{l} \frac{2}{3}+\frac{3}{4}>1 \\[0.5em] \frac{1}{2}>1 \end{array}
    Show the answer and how to solve it

    The correct comparison is the one that pairs 23+34>1\frac{2}{3}+\frac{3}{4}>1 with 12<1\frac{1}{2}<1: the sum is greater than 1, but the claimed result is less than 1.

    1. 1Focus on the claimed equation and use the benchmark line.
    2. 2Both addends are greater than 12\frac{1}{2}.
    3. 3So their sum must be greater than 1.
    4. 4But the claimed result 12\frac{1}{2} is less than 1. The same value cannot be both greater than 1 and less than 1, so the comparison that pairs 23+34>1\frac{2}{3}+\frac{3}{4}>1 with 12<1\frac{1}{2}<1 disproves the claim.

    Common mistake. A common mistake is choosing a pair with only one true comparison. The correct comparison must place both the expression and the claimed result correctly against the same benchmark.

    The idea underneath. Benchmark comparisons can disprove an answer without finding the exact sum or difference.

  2. 2On level
    A piece of rope is 34\frac{3}{4} yard long, and the marked point on the number line below shows that starting length.
    Mia uses 38\frac{3}{8} yard. Click the point that shows how much rope is left.
    Tap or click the number line to place your point. Drag it to move, or tap it again to remove.
    013/4
    Show the answer and how to solve it

    The answer is 38\frac{3}{8} because 34\frac{3}{4} is 68\frac{6}{8} and 68−38=38\frac{6}{8} - \frac{3}{8} = \frac{3}{8}.

    1. 1The marked point shows the starting amount, 34\frac{3}{4} yard. Since 38\frac{3}{8} yard is used, you need to find the amount left.
    2. 2Change 34\frac{3}{4} to eighths so both fractions use the same size parts.
    3. 3Now subtract the eighths.
    4. 4The line is divided into 88 equal parts, so 38\frac{3}{8} is the third small tick after 00. That is the point to click.

    Common mistake. A common mistake is clicking the marked point at 34\frac{3}{4} without subtracting. Because Mia uses some rope, the amount left must be less than 34\frac{3}{4}, so move left 38\frac{3}{8} to reach 38\frac{3}{8}.

    The idea underneath. On a fraction number line, once both amounts are named with the same denominator, the numerator tells which equal-part tick to click.

  3. 3On level
    The recipe below lists the amounts for one batch of oatmeal.
    Recipe

    Honey Oatmeal

    • 112cupmilk
    • 13cupoats
    • 14cuphoney
    Emma mixes the milk and oats from the recipe in a bowl. Then she removes from the mixture an amount equal to the recipe's honey. Select the exact amount left.
    A
    1712\displaystyle 1\frac{7}{12}
    B
    156\displaystyle 1\frac{5}{6}
    C
    112\displaystyle 1\frac{1}{2}
    D
    114\displaystyle 1\frac{1}{4}
    Show the answer and how to solve it

    The answer is 17121\frac{7}{12} cup because 1121\frac{1}{2} cups plus 13\frac{1}{3} cup makes 1561\frac{5}{6} cups, and removing 14\frac{1}{4} cup leaves 17121\frac{7}{12} cup.

    1. 1Read the three amounts from the recipe: 1121\frac{1}{2} cup milk, 13\frac{1}{3} cup oats, and 14\frac{1}{4} cup honey.
    2. 2Add the milk and oats. Rename both fractions with the common denominator 6.
    3. 3Subtract the honey amount from that total. Rename with the common denominator 12.
    4. 4The exact amount left is 17121\frac{7}{12} cup, so choose the option that shows 17121\frac{7}{12}.

    Common mistake. A common mistake is stopping after the first step and choosing 1561\frac{5}{6}. This problem has a second step — you still need to subtract 14\frac{1}{4} cup of honey.

    The idea underneath. In a two-step fraction story, finish the first total, then rename the fractions with a common denominator before the second operation.

  4. 4Stretch
    Lena had 3143\frac{1}{4} yards of ribbon. She used 1231\frac{2}{3} yards to wrap gifts and then added 56\frac{5}{6} yard from a new spool. How much ribbon does she have now?
    Enter your exact answer as a fraction or mixed number.
    yards
    Answer format:
    Show the answer and how to solve it

    Answer. 5

    The answer is 25122\frac{5}{12} yards, which is also 2912\frac{29}{12} yards.

    1. 1Lena starts with 3143\frac{1}{4} yards. She used 1231\frac{2}{3} yards for wrapping, so that amount is taken away.
    2. 2Rewrite both fractions with the common denominator 12. Since 312<812\frac{3}{12} < \frac{8}{12}, rename 33123\frac{3}{12} as 215122\frac{15}{12} before subtracting.
    3. 3She added 56\frac{5}{6} yard from a new spool, so add that amount.
    4. 4Lena has 25122\frac{5}{12} yards of ribbon now.

    Common mistake. A common mistake is to subtract 312−812\frac{3}{12} - \frac{8}{12} directly. Since 312<812\frac{3}{12} < \frac{8}{12}, rename 33123\frac{3}{12} as 215122\frac{15}{12} first, because you need enough twelfths in the fractional part to subtract.

    The idea underneath. The story tells whether each amount is taken away or added back, and each fraction step works only after the pieces are renamed with a common denominator.