Add and Subtract Fractions Using a Common Unit | Grade 6 Math | LevelKite

You already know how to count. Adding and subtracting fractions is counting too, as long as you count the right things.

A fraction measures an amount against a whole, the amount that counts as 1. Its two numbers do different jobs. The denominator (the bottom number) tells you how many equal parts make the whole, so it names the size of each part. The numerator (the top number) counts how many of those parts you have.

So $\frac{3}{8}$ of a mile means 3 parts, and each part is one eighth of a mile. You can treat eighths of a mile as a unit: a fixed size that you count, just like minutes or meters.

Counting is simple when the unit matches. For example, 3 minutes and 2 minutes make 5 minutes. In the same way, 3 eighths of a mile and 2 eighths of a mile make 5 eighths of a mile. Only the count changes. The size of an eighth of a mile stays the same.

\frac{3}{8} + \frac{2}{8} = \frac{3 + 2}{8} = \frac{5}{8}

Key concept

Count same-size parts of the same whole

Adding or subtracting fractions means counting parts. You can count parts together only when the fractions refer to the same whole and the parts are the same size.

Like denominators are equal denominators. For fractions of the same whole with like denominators, add or subtract the numerators and keep the denominator, because the size of each part does not change.

In the formula, $a$ and $b$ are the numbers of parts, and $c$ is the number of equal parts in one whole, so $c$ is not $0$ . When you subtract in this lesson, $a$ is at least $b$ .

Unlike denominators are different denominators. For the same whole, they mean different part sizes, so you first rename the fractions so that both count same-size parts of that whole.

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \qquad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}

Worked example

Like denominators: add and subtract

You walk $\frac{5}{8}$ mile before lunch and $\frac{7}{8}$ mile after lunch. How far do you walk in all? How much farther do you walk after lunch than before lunch?

Name the whole and the unit. Both distances are fractions of the same whole, 1 mile, and both have the denominator 8. So both count parts of the same size: eighths of a mile.
$\frac{5}{8} = 5 \text{ eighths} \qquad \frac{7}{8} = 7 \text{ eighths}$
Add the counts to find the total. Together, 5 eighths and 7 eighths make 12 eighths. Keep the denominator 8, because joining the two distances does not change the size of an eighth of a mile.
$\frac{5}{8} + \frac{7}{8} = \frac{5 + 7}{8} = \frac{12}{8}$
Rename the total as a mixed number, a whole number plus a fraction. Since 8 eighths make 1 whole mile, 4 eighths are left over. Those 4 eighths of a mile are the same distance as 1 half of a mile, because 4 is half of 8.
$\frac{12}{8} = \frac{8}{8} + \frac{4}{8} = 1\frac{4}{8} = 1\frac{1}{2}$
Subtract the counts to find how much farther. Taking 5 eighths away from 7 eighths leaves 2 eighths. The denominator stays 8 for the same reason: the parts are still eighths of a mile.
$\frac{7}{8} - \frac{5}{8} = \frac{7 - 5}{8} = \frac{2}{8}$
Rename this result with larger parts. Divide the numerator and the denominator by 2. This joins the eighths in pairs to make fourths of a mile, so the distance does not change.
$\frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4}$

Answer

You walk $1\frac{1}{2}$ miles in all, and you walk $\frac{1}{4}$ mile farther after lunch than before lunch.

Before you find an exact answer, it helps to know about how big it should be. An estimate is a number close to the exact answer that you find with easier numbers. You can estimate a sum (the result of adding) or a difference (the result of subtracting). The symbol $\approx$ means is about equal to.

An estimate has limits. It can show that an answer is unreasonable, but a close estimate does not prove that an exact answer is correct.

To estimate with fractions, use benchmarks: easy numbers to compare with. For a fraction between 0 and 1, the benchmarks are $0$ , $\frac{1}{2}$ and $1$ . Look at the numerator:

If it is much smaller than the denominator, the fraction is close to $0$ .
If it is about half of the denominator, the fraction is close to $\frac{1}{2}$ .
If it is almost equal to the denominator, the fraction is close to $1$ .

For example, if both fractions refer to the same whole, $\frac{8}{9} + \frac{1}{10} \approx 1 + 0 = 1$ . For a mixed number, keep the whole number and round the fraction part to a benchmark.

A benchmark comparison is fair only when the fraction and the benchmark describe the same whole.

Rounding to benchmarks, with every amount measured against the same whole

Number	Compare numerator and denominator	Closest benchmark
$\frac{1}{9}$	1 is much smaller than 9	$0$
$\frac{4}{7}$	Half of 7 is 3.5, and 4 is close to 3.5	$\frac{1}{2}$
$\frac{9}{10}$	9 is almost equal to 10	$1$
$2\frac{5}{12}$	Half of 12 is 6, and 5 is close to 6	$2\frac{1}{2}$
$4\frac{1}{10}$	1 is much smaller than 10	$4$
$6\frac{8}{9}$	8 is almost equal to 9, so the fraction part is close to 1	$6 + 1 = 7$

Worked example

Estimate with benchmarks

One bag of soil weighs $3\frac{7}{8}$ pounds. Another bag weighs $1\frac{4}{9}$ pounds. Estimate the total weight of the two bags. Then estimate how much heavier the first bag is.

Check the whole. Both weights are measured in pounds, so both mixed numbers refer to the same whole, 1 pound. That makes it fair to compare them with benchmarks and to combine them.
Round the first weight. The numerator 7 is almost equal to the denominator 8, so 7 eighths of a pound is close to 1 pound. Add that 1 to the whole number 3.
$3\frac{7}{8} \approx 3 + 1 = 4$
Round the second weight. Half of 9 is 4.5, and the numerator 4 is close to 4.5, so 4 ninths of a pound is close to 1 half of a pound. Keep the whole number 1.
$1\frac{4}{9} \approx 1 + \frac{1}{2} = 1\frac{1}{2}$
Estimate the sum by adding the rounded numbers. First add the whole numbers, 4 and 1, to get 5. Then include the 1 half.
$4 + 1\frac{1}{2} = 4 + 1 + \frac{1}{2} = 5\frac{1}{2}$
Estimate the difference by subtracting the rounded numbers. Take away 1 to get 3, then take away the 1 half.
$4 - 1\frac{1}{2} = 4 - 1 - \frac{1}{2} = 3 - \frac{1}{2} = 2\frac{1}{2}$

Answer

The two bags weigh about $5\frac{1}{2}$ pounds in all, and the first bag is about $2\frac{1}{2}$ pounds heavier.

When two fractions of the same whole have unlike denominators, they count parts of different sizes. Take $\frac{1}{2} + \frac{3}{8}$ , where both fractions refer to the same whole. One counts halves and the other counts eighths. Together, 1 half and 3 eighths are not 4 halves, and they are not 4 eighths, so you cannot just add the counts.

You need a common unit: one part size that can name both amounts. Here eighths work. Cut each half of the whole into 4 equal pieces. The whole now has $2 \times 4 = 8$ equal parts, and your 1 half has become $1 \times 4 = 4$ of them. The amount has not changed: there are 4 times as many parts, and each part is one fourth as large. So $\frac{1}{2}$ and $\frac{4}{8}$ are equivalent fractions, fractions that name the same amount of the same whole. Now both fractions count eighths of that whole, and you can add the counts.

\frac{1}{2} + \frac{3}{8} = \frac{1 \times 4}{2 \times 4} + \frac{3}{8} = \frac{4}{8} + \frac{3}{8} = \frac{4 + 3}{8} = \frac{7}{8}

In $\frac{1}{2} + \frac{3}{8}$ , only one fraction needed a new name. Often you must rename both. A common denominator is a denominator that both fractions can share. It is a multiple of each denominator: a number you get when you multiply that denominator by 1, 2, 3 and so on. The smallest one is the least common denominator. Any common denominator gives a correct result. The least one keeps the numbers small.

For fractions of the same whole with unlike denominators:

Estimate the answer with benchmarks.
Find a common denominator.
Rename each fraction as an equivalent fraction with that denominator. Multiply its numerator and its denominator by the same whole number greater than 0, so the amount stays the same.
Add or subtract the numerators and keep the common denominator.
Compare the result with your estimate. If they are far apart, look for a mistake.

Renaming changes only the names of the amounts. The parts are the same size because both fractions refer to the same whole and now have the same denominator.

Worked example

Unlike denominators: add

A bucket holds $\frac{5}{6}$ gallon of water. You pour in $\frac{5}{8}$ gallon more. How much water is in the bucket now?

Check the whole and estimate. Both amounts are fractions of the same whole, 1 gallon. The first amount, 5 sixths, is close to 1, because 5 is almost 6. The second, 5 eighths, is close to 1 half, because half of 8 is 4 and 5 is close to 4. The sum should be near 1 and 1 half gallons.
$1 + \frac{1}{2} = 1\frac{1}{2}$
Find a common unit. Sixths and eighths of a gallon are different sizes. List multiples of each denominator until one matches: 6, 12, 18, 24 and 8, 16, 24. The least common denominator is 24, so count in twenty-fourths of a gallon.
$6 \times 4 = 24 \qquad 8 \times 3 = 24$
Rename 5 sixths. Cut each sixth of a gallon into 4 equal pieces, because 6 times 4 is 24. Multiply the numerator and the denominator by 4. The amount stays the same: there are 4 times as many parts, and each part is one fourth as large.
$\frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}$
Rename 5 eighths the same way. Cut each eighth of a gallon into 3 equal pieces, because 8 times 3 is 24. Multiply the numerator and the denominator by 3.
$\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$
Add the counts. Both fractions now count twenty-fourths of a gallon, so the parts are the same size. Together, 20 twenty-fourths and 15 twenty-fourths make 35 twenty-fourths.
$\frac{20}{24} + \frac{15}{24} = \frac{20 + 15}{24} = \frac{35}{24}$
Rename and check. Since 24 twenty-fourths make 1 gallon, 11 twenty-fourths are left over. No whole number greater than 1 divides both 11 and 24, so the fraction part stays as it is. It is close to 1 half, because 11 is close to half of 24. So the result is near the estimate and is reasonable.
$\frac{35}{24} = \frac{24}{24} + \frac{11}{24} = 1\frac{11}{24}$

Answer

The bucket now holds $1\frac{11}{24}$ gallons of water, which is close to the estimate of $1\frac{1}{2}$ gallons.

Worked example

Unlike denominators: subtract

Find $\frac{11}{12} - \frac{7}{15}$ . Both fractions refer to the same whole.

Estimate first. Both fractions refer to the same whole, so you can compare each one with a benchmark. Since 11 is almost 12, 11 twelfths is close to 1. Half of 15 is 7.5, so 7 fifteenths is close to 1 half. The difference should be near 1 half.
$1 - \frac{1}{2} = \frac{1}{2}$
Find the least common denominator. Multiples of 12 are 12, 24, 36, 48, 60. Multiples of 15 are 15, 30, 45, 60. The first match is 60, so the common unit is sixtieths of the whole.
$12 \times 5 = 60 \qquad 15 \times 4 = 60$
Rename both fractions in sixtieths. For 11 twelfths, multiply the numerator and the denominator by 5. For 7 fifteenths, multiply both by 4. Each fraction keeps its amount, because you multiply its numerator and its denominator by the same number.
$\frac{11}{12} = \frac{11 \times 5}{12 \times 5} = \frac{55}{60} \qquad \frac{7}{15} = \frac{7 \times 4}{15 \times 4} = \frac{28}{60}$
Subtract the counts. Both fractions now count sixtieths of the same whole, so the parts are the same size. Taking 28 sixtieths away from 55 sixtieths leaves 27 sixtieths.
$\frac{55}{60} - \frac{28}{60} = \frac{55 - 28}{60} = \frac{27}{60}$
Rename with larger parts and check. Both 27 and 60 are multiples of 3, so divide the numerator and the denominator by 3. This joins the sixtieths in threes to make twentieths, so the amount does not change. Since 9 is close to half of 20, the result is near the estimate and is reasonable.
$\frac{27}{60} = \frac{27 \div 3}{60 \div 3} = \frac{9}{20}$

Answer

The difference is $\frac{9}{20}$ of the whole, which is close to the estimate of $\frac{1}{2}$ .

Takeaway. Adding and subtracting fractions is counting. You can count parts together only when the fractions refer to the same whole and the parts are the same size. With like denominators, add or subtract the numerators and keep the denominator. With unlike denominators, first rename the fractions so that they share a common denominator and both count same-size parts of that whole. Estimate with the benchmarks $0$ , $\frac{1}{2}$ and $1$ to see whether an answer is reasonable.

Next, you will complete fraction addition and subtraction sentences and add and subtract mixed numbers.