5th Grade · 5.NF.A.1

Adding fractions with unlike denominators in 5th grade

Adding fractions with unlike denominators means adding fractions whose bottom numbers are different, such as $\frac{1}{2} + \frac{2}{5}$ . In 5th grade, your child learns to rewrite the fractions so that they have the same denominator, then add or subtract. The same method works for mixed numbers, such as $2\frac{2}{3}$ .

What the standard says

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

5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

What it means, in plain words

A fraction has two parts. The denominator, the bottom number, says how many equal parts the whole is cut into. The numerator, the top number, says how many of those parts there are. Fractions with unlike denominators are made of parts of different sizes, so they cannot be added directly. Halves and fifths are different sizes, just as inches and feet are.

Adding fractions with unlike denominators starts with equivalent fractions, which are fractions that name the same amount, such as $\frac{1}{2}$ and $\frac{5}{10}$ . Your child finds a common denominator, a number that both denominators divide into, and rewrites each fraction with it. Multiplying the two denominators always gives one. Then only the numerators are added. Subtracting fractions with unlike denominators uses the same steps: $\frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}$ .

Adding mixed numbers means adding numbers such as $2\frac{2}{3}$ , which is 2 and two-thirds. Your child adds the whole numbers, adds the fractions, then tidies up. If the fractions add up to more than 1, the extra whole moves into the whole-number part. When subtracting, a whole may need to be split into fraction parts first.

Your child should be able to:

Find a common denominator for two fractions
Rewrite a fraction as an equivalent fraction with a new denominator
Add and subtract fractions with unlike denominators, including answers greater than 1
Add and subtract mixed numbers, regrouping when needed

Example

What is $\frac{1}{2} + \frac{2}{5}$ ?

Halves and fifths are different sizes. Both 2 and 5 divide into 10, so use tenths.
$\frac{1}{2} = \frac{5}{10}$ and $\frac{2}{5} = \frac{4}{10}$ .
Add the numerators: $\frac{5}{10} + \frac{4}{10} = \frac{9}{10}$ .
So one-half plus two-fifths is nine-tenths.

Example

What is $2\frac{2}{3} + 1\frac{1}{2}$ ?

Add the whole numbers: $2 + 1 = 3$ .
Add the fractions using sixths: $\frac{2}{3} + \frac{1}{2} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}$ .
$\frac{7}{6}$ is more than 1. It equals $1\frac{1}{6}$ .
Combine the parts: $3 + 1\frac{1}{6} = 4\frac{1}{6}$ .
So 2 and two-thirds plus 1 and one-half is 4 and one-sixth.

Where children usually get stuck

Your child writes $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$ .
The child adds the tops and the bottoms as if they were four separate whole numbers. The denominator names the size of the parts, so it is not added.
What helps. Ask whether a sum can be smaller than one of the fractions you started with. $\frac{2}{5}$ is less than $\frac{1}{2}$ , so it cannot be right. The correct sum is $\frac{5}{6}$ .
Your child rewrites $\frac{2}{3}$ as $\frac{2}{6}$ to get a denominator of 6.
Only the denominator was changed. To keep the amount the same, the numerator and the denominator must both be multiplied by the same number.
What helps. Fold a strip of paper into thirds and shade two. Then fold each third in half. The shading now covers 4 of 6 parts, so $\frac{2}{3} = \frac{4}{6}$ .
For $5\frac{1}{6} - 2\frac{1}{2}$ , your child writes $3\frac{2}{6}$ .
In sixths, the problem reads $5\frac{1}{6} - 2\frac{3}{6}$ . The child cannot take 3 sixths from 1 sixth, so the child takes 1 from 3 instead.
What helps. Show that one whole can be split into $\frac{6}{6}$ , so $5\frac{1}{6} = 4\frac{7}{6}$ . Then $4\frac{7}{6} - 2\frac{3}{6} = 2\frac{4}{6}$ , which is $2\frac{2}{3}$ .

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.

1Warm-up
Use the subtraction problem below to find the difference.
$\displaystyle 2\frac{3}{4} - 1\frac{1}{4}$
Click the point on the number line that shows the difference.
Tap or click the number line to place your point. Drag it to move, or tap it again to remove.
Show the answer and how to solve it
The difference is 1 and $\frac{2}{4}$ , so the correct point is two fourths past 1.
1. 1Look at the subtraction expression. Both fractions already have denominator 4, so you can subtract right away.
2. 2Subtract the whole-number parts and then the fraction parts.
3. 3The difference is 1 and $\frac{2}{4}$ . On the number line, find 1 and count 2 small ticks to the right. That is the point to click.
Common mistake. A common mistake is clicking 1 after subtracting only the whole-number parts. You also need to move $\frac{2}{4}$ more to the right, landing two small ticks past 1.
The idea underneath. Once the fractions are written with the line's denominator, the mixed number tells exactly how many equal parts past a whole number to click.
2On level
Two fractions are shown below.
$\displaystyle \frac{3}{4} \quad \text{and} \quad \frac{1}{6}$
Select the whole number that could be a common denominator for both fractions.
A
12
B
8
C
18
D
10
Show the answer and how to solve it
12 is the correct common denominator because 12 is a multiple of both 4 and 6.
1. 1Look at the denominators in the two fractions. They are 4 and 6.
2. 2A common denominator must be a multiple of both 4 and 6. Test 12.
3. 3The other choices do not work for both fractions.
Common mistake. A common mistake is choosing a number that works for only one denominator, such as 8 or 18. A common denominator must be divisible by both 4 and 6.
The idea underneath. A common denominator is any number that both denominators divide evenly, so testing divisibility tells you whether a choice works.
3On level
Both fractions are being rewritten with the same denominator. Type the missing numerator.
$\displaystyle \frac{3}{4} = 9/12$
$\displaystyle \frac{1}{3} =$
$\displaystyle /12$
Show the answer and how to solve it
Answer. 4
The missing numerator is $4$ because $\frac{1}{3}$ becomes $\frac{4}{12}$ when both the numerator and denominator are multiplied by $4$ .
1. 1Both rewritten fractions use denominator $12$ .
2. 2The incomplete fraction is $\frac{1}{3}$ . To change the denominator from $3$ to $12$ , multiply by $4$ .
3. 3Split each of the $3$ equal parts into $4$ smaller equal pieces. The whole now has $12$ pieces, and the $1$ original part becomes $4$ pieces. The amount stays the same.
4. 4The complete equivalent fraction is $\frac{1}{3} = \frac{4}{12}$ , so the missing numerator is $4$ .
Common mistake. A common mistake is typing $3$ because the other completed fraction used a factor of $3$ . For $\frac{1}{3}$ , the denominator $3$ must be multiplied by $4$ to make $12$ , so the numerator must also be multiplied by $4$ .
The idea underneath. Equivalent fractions are made by multiplying the numerator and denominator by the same number.
4Stretch
Find the exact sum of the fractions shown below.
$\displaystyle \frac{5}{8} + \frac{5}{6}$
Enter the exact sum as a fraction or mixed number.
Answer format:
Show the answer and how to solve it
Answer. 11
The exact sum is $1\frac{11}{24}$ because $\frac{5}{8}$ becomes $\frac{15}{24}$ and $\frac{5}{6}$ becomes $\frac{20}{24}$ , so $\frac{15}{24} + \frac{20}{24} = \frac{35}{24} = 1\frac{11}{24}$ .
1. 1Start with the two addends shown in the equation.
2. 2The least common denominator of 8 and 6 is 24.
3. 3Rename each fraction with denominator 24.
4. 4Now add the numerators because the denominators match.
5. 5The sum is greater than 1, so convert the improper fraction to a mixed number.
Common mistake. A common mistake after finding $\frac{35}{24}$ is writing $1\frac{35}{24}$ instead of converting correctly. Subtract $1 \times 24 = 24$ from $35$ to get the remainder $11$ , giving $1\frac{11}{24}$ .
The idea underneath. Fractions can be added only when they name equal-sized parts, so a common denominator lets you combine the numerators correctly.