Add, Subtract and Multiply Fractions | Grade 7 Math | LevelKite

Start with what a fraction tells you. A fraction such as $\frac{3}{8}$ describes equal parts of one whole. The 8 is the denominator: it tells you how many equal parts make the whole, so it names the size of each part (eighths). The 3 is the numerator: it tells you how many of those parts you have. So $\frac{3}{8}$ means 3 parts, each of size $\frac{1}{8}$ .

Every rule in this lesson grows from those two meanings. You will add and subtract fractions, and then multiply fractions and mixed numbers, which are numbers like $2\frac{1}{4}$ that join a whole number and a fraction.

Throughout, every numerator is a whole number (0, 1, 2, 3, and so on) and every denominator is a whole number greater than 0, so no fraction here is negative. A denominator is never 0, because you cannot split a whole into 0 equal parts. This lesson shows every step with its actual numbers, so you can check each one yourself.

When two fractions have the same denominator, their parts are the same size, so you can count the parts together. For example, $\frac{5}{8} + \frac{2}{8}$ is 5 eighths and 2 more eighths, which makes 7 eighths: $\frac{7}{8}$ . Taking 2 eighths away from 5 eighths leaves 3 eighths, so $\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$ .

Only the count of parts changes. Each part is still an eighth, so the denominator stays 8. The result of adding is the sum, and the result of subtracting is the difference.

Here is the adding rule in symbols. Let $a$ and $b$ stand for the two numerators, which are whole numbers. Let $n$ stand for the denominator they share, a whole number greater than 0.

\frac{a}{n} + \frac{b}{n} = \frac{a + b}{n}

Subtraction follows the same pattern: $\frac{a}{n} - \frac{b}{n} = \frac{a - b}{n}$ . In this lesson $a$ is at least as large as $b$ , so the difference is never negative.

When the denominators differ, the parts are different sizes, so you cannot count them together yet. First rename both fractions so they have a common denominator, which means the same denominator. Use the least common denominator (LCD): the smallest whole number greater than 0 that both denominators divide with no remainder. You already know how to find it.

Renaming keeps a fraction's value. Take $\frac{3}{4}$ and the denominator 12. Each fourth splits into 3 twelfths, because $4 \times 3 = 12$ . So 3 fourths become $3 \times 3 = 9$ twelfths: $\frac{3}{4} = \frac{9}{12}$ . You multiplied the numerator and the denominator by the same whole number greater than 0. That makes 3 times as many parts, each one third as large, so the amount is unchanged.

Write a final answer in simplest form: the numerator and denominator share no common factor (a whole number that divides both with no remainder) greater than 1.

Worked example

Adding fractions with different denominators

Find $\frac{3}{8} + \frac{5}{12}$ .

The denominators 8 and 12 are different, so the parts are different sizes. Find the least common denominator. It must be a number that 12 divides, so count by 12s: 12, 24, 36. Dividing 12 by 8 leaves a remainder, but 24 divided by 8 is exactly 3. So the LCD is 24.
$24 = 8 \times 3 = 12 \times 2$
Rename three eighths in twenty-fourths. Each eighth is 3 twenty-fourths, because 8 times 3 is 24. So multiply the numerator and the denominator by 3. The value stays the same.
$\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$
Rename five twelfths in twenty-fourths. Each twelfth is 2 twenty-fourths, because 12 times 2 is 24. So multiply the numerator and the denominator by 2.
$\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}$
Both fractions now count twenty-fourths, so the parts are the same size. Add the numerators and keep the denominator 24.
$\frac{9}{24} + \frac{10}{24} = \frac{9 + 10}{24} = \frac{19}{24}$
Check the form and the size. 19 and 24 share no common factor greater than 1, so the answer is in simplest form. Each fraction you added is less than one half, because 3 is less than half of 8 and 5 is less than half of 12. Two amounts that are each less than one half add to less than 1, and 19 twenty-fourths is less than 1.
$\frac{19}{24} < \frac{24}{24} = 1$

Answer

The sum of $\frac{3}{8}$ and $\frac{5}{12}$ is $\frac{19}{24}$ .

Worked example

Subtracting, then simplifying

Find $\frac{9}{10} - \frac{1}{6}$ .

The denominators differ, so find the LCD of 10 and 6. Count by 10s: 10, 20, 30. Dividing 10 or 20 by 6 leaves a remainder, but 30 divided by 6 is exactly 5. So the LCD is 30.
$30 = 10 \times 3 = 6 \times 5$
Rename both fractions in thirtieths. Each tenth is 3 thirtieths, and each sixth is 5 thirtieths. So multiply the numerator and the denominator of nine tenths by 3, and of one sixth by 5. Each fraction keeps its value.
$\frac{9}{10} - \frac{1}{6} = \frac{9 \times 3}{10 \times 3} - \frac{1 \times 5}{6 \times 5} = \frac{27}{30} - \frac{5}{30}$
Both fractions now count thirtieths. Subtract the numerators and keep the denominator 30. Because 27 is greater than 5, the difference is positive.
$\frac{27}{30} - \frac{5}{30} = \frac{27 - 5}{30} = \frac{22}{30}$
Write the answer in simplest form. 22 and 30 share the common factor 2, so divide both by 2. This is renaming in reverse: half as many parts, each twice as large, so the value stays the same. 11 and 15 share no common factor greater than 1, so you can stop.
$\frac{22}{30} = \frac{22 \div 2}{30 \div 2} = \frac{11}{15}$
Check by adding back. If the difference is correct, adding one sixth to it returns nine tenths. Use thirtieths again: 22 thirtieths plus 5 thirtieths is 27 thirtieths, and dividing 27 and 30 by 3 gives nine tenths.
$\frac{22}{30} + \frac{5}{30} = \frac{27}{30} = \frac{9}{10}$

Answer

The difference between $\frac{9}{10}$ and $\frac{1}{6}$ is $\frac{11}{15}$ .

Now for multiplication. Multiplying by a fraction means taking that fraction of an amount. So $\frac{2}{3} \times \frac{4}{5}$ asks for two thirds of four fifths. The result of multiplying is the product.

Work it out in parts. To take thirds of $\frac{4}{5}$ , split each fifth into 3 equal pieces. The whole now has $5 \times 3 = 15$ pieces, so each piece is $\frac{1}{15}$ , and the 4 fifths become $4 \times 3 = 12$ pieces: $\frac{4}{5} = \frac{12}{15}$ . One third of 12 fifteenths is $12 \div 3 = 4$ fifteenths. Two thirds is twice as much: $2 \times 4 = 8$ fifteenths. So $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ .

Compare that with the starting numbers. The 8 is $2 \times 4$ , the numerators multiplied. The 15 is $3 \times 5$ , the denominators multiplied. Multiplying the denominators gives the new part size, fifteenths. Multiplying the numerators counts how many of those parts you keep. You do not need a common denominator to multiply.

In the rule below, $p$ and $r$ stand for the numerators (whole numbers), and $q$ and $s$ stand for the denominators (whole numbers greater than 0).

\frac{p}{q} \times \frac{r}{s} = \frac{p \times r}{q \times s}

The same two fractions under three operations

Problem	Common denominator needed?	Work	Answer
$\frac{3}{4} + \frac{1}{6}$	Yes. With LCD 12: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ , $\frac{1 \times 2}{6 \times 2} = \frac{2}{12}$	$\frac{9}{12} + \frac{2}{12} = \frac{9 + 2}{12}$	$\frac{11}{12}$
$\frac{3}{4} - \frac{1}{6}$	Yes. The same renaming: $\frac{9}{12}$ and $\frac{2}{12}$	$\frac{9}{12} - \frac{2}{12} = \frac{9 - 2}{12}$	$\frac{7}{12}$
$\frac{3}{4} \times \frac{1}{6}$	No. The denominators multiply instead.	$\frac{3 \times 1}{4 \times 6} = \frac{3}{24} = \frac{3 \div 3}{24 \div 3}$	$\frac{1}{8}$

Key concept

Same-size parts to add or subtract; a part of a part to multiply

A denominator names the size of the parts, and a numerator counts them. That one idea explains both rules.

To add or subtract fractions, the parts must be the same size. Rename the fractions with a common denominator, then add or subtract the numerators and keep that denominator.
To multiply fractions, you take a part of a part. Multiply the numerators and multiply the denominators. You do not need a common denominator.

These rules hold whenever no denominator is 0.

A mixed number such as $2\frac{1}{4}$ means $2 + \frac{1}{4}$ : a whole number plus a fraction less than 1. The multiplication rule needs one numerator and one denominator for each number you multiply. So first rewrite every mixed number as an improper fraction, a fraction whose numerator is greater than or equal to its denominator.

To rewrite $2\frac{1}{4}$ , rename the 2 wholes as fourths. Each whole is 4 fourths, so 2 wholes are $2 \times 4 = 8$ fourths. Add the 1 extra fourth: $2\frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}$ . Only the way of writing changes, not the value. This is the same first step you take to find a mixed number's reciprocal, the number you multiply it by to get 1.

A whole number fits the rule too. Write it over 1, as in $3 = \frac{3}{1}$ , because 3 wholes are 3 parts of size 1. After you multiply, write the product in simplest form. If it is an improper fraction, you can write it as a whole number or a mixed number.

Worked example

Multiplying two mixed numbers

One batch of bread dough needs $1\frac{2}{3}$ cups of flour. How many cups of flour do you need for $2\frac{1}{4}$ batches?

Each batch needs the same amount of flour, so multiply the number of batches by the cups of flour in one batch. You need 2 and one fourth times 1 and two thirds cups.
$2\frac{1}{4} \times 1\frac{2}{3}$
Rewrite each mixed number as an improper fraction so you can use the multiplication rule. 2 wholes are 8 fourths, and 1 more fourth makes 9 fourths. 1 whole is 3 thirds, and 2 more thirds make 5 thirds.
$2\frac{1}{4} = \frac{8 + 1}{4} = \frac{9}{4} \quad \text{and} \quad 1\frac{2}{3} = \frac{3 + 2}{3} = \frac{5}{3}$
Use the multiplication rule: multiply the numerators and multiply the denominators. The new parts are twelfths, because 4 times 3 is 12, and you have 9 times 5 of them. You do not need a common denominator.
$\frac{9}{4} \times \frac{5}{3} = \frac{9 \times 5}{4 \times 3} = \frac{45}{12}$
Write the product in simplest form. 45 and 12 share the common factor 3, so divide both by 3, which keeps the value. Then 15 and 4 share no common factor greater than 1.
$\frac{45}{12} = \frac{45 \div 3}{12 \div 3} = \frac{15}{4}$
Write the improper fraction as a mixed number. 4 fourths make 1 whole, so 12 fourths make 3 wholes. That leaves 15 minus 12, which is 3 fourths.
$\frac{15}{4} = \frac{12}{4} + \frac{3}{4} = 3 + \frac{3}{4} = 3\frac{3}{4}$
Check the size. The number of batches is between 2 and 3, and the cups of flour in one batch are between 1 and 2. All of these numbers are positive, so the product must be between 2 times 1 and 3 times 2, which means between 2 and 6. The answer fits.
$2 \times 1 < 3\frac{3}{4} < 3 \times 2$

Answer

You need $3\frac{3}{4}$ cups of flour for $2\frac{1}{4}$ batches.

Before you combine two fractions, ask what the operation does to the parts. Adding and subtracting count parts together or take parts away, so the parts must first be the same size: rename with a common denominator, then add or subtract only the numerators. Multiplying takes a part of a part, so you multiply the numerators together and the denominators together, after you rewrite each mixed number as an improper fraction.

Next, you will use common denominators again to add and subtract mixed numbers.