Adding and Subtracting Unlike and Mixed Fractions

1. Why we need a common denominator

Adding like fractions is easy because the pieces are the same size. $\frac{2}{7}$ + $\frac{3}{7}$ means 2 sevenths plus 3 sevenths, which is 5 sevenths: $\frac{5}{7}$. The denominator only tells us the name (size) of the pieces, so it stays the same; we add only the numerators.

With unlike fractions such as $\frac{1}{2}$ + $\frac{1}{3}$ the pieces are different sizes. It is like adding 1 cedi and 1 pesewa — you cannot say you have "2" of anything until you change them to the same unit. For fractions, the common unit is a common denominator: a number that both denominators divide into. The best choice is the LCM of the denominators, because it keeps the numbers small.

▶ Notice that the answer 5/6 is bigger than 1/2 — of course, since we added something to 1/2.

2. Adding and subtracting two or three unlike fractions

1. 1Find the LCM of all the denominators.
2. 2Change each fraction to an equivalent fraction with the LCM as its denominator (divide the LCM by the old denominator, then multiply the top and bottom by that number).
3. 3Add or subtract the numerators. Keep the common denominator.
4. 4Simplify: reduce to lowest terms and change an improper fraction to a mixed number.

Three fractions: $\frac{1}{2}$ + $\frac{1}{3}$ + $\frac{1}{4}$. The LCM of 2, 3 and 4 is 12. So $\frac{1}{2}$ = $\frac{6}{12}$, $\frac{1}{3}$ = $\frac{4}{12}$ and $\frac{1}{4}$ = $\frac{3}{12}$. Adding: $\frac{6+4+3}{12}$ = $\frac{13}{12}$ = $1\frac{1}{12}$.

Subtraction works the same way. $\frac{5}{6}$ − $\frac{3}{8}$: the LCM of 6 and 8 is 24, so $\frac{20}{24}$ − $\frac{9}{24}$ = $\frac{11}{24}$. You can even mix the operations: $\frac{3}{4}$ + $\frac{1}{6}$ − $\frac{1}{3}$ = $\frac{9}{12}$ + $\frac{2}{12}$ − $\frac{4}{12}$ = $\frac{7}{12}$. Work from left to right.

3. Adding mixed numbers

A mixed number such as $2\frac{2}{5}$ really means 2 + $\frac{2}{5}$. So when adding mixed numbers we can add the whole numbers first, then add the fractions, and finally combine.

Method 1 (wholes, then fractions): $2\frac{2}{5}$ + $1\frac{2}{3}$ = (2 + 1) + ($\frac{2}{5}$ + $\frac{2}{3}$) = 3 + $\frac{6}{15}$ + $\frac{10}{15}$ = $3\frac{16}{15}$. But $\frac{16}{15}$ is more than a whole: $\frac{16}{15}$ = $1\frac{1}{15}$. So the answer is 3 + $1\frac{1}{15}$ = $4\frac{1}{15}$.

▶ When the fraction parts add up to more than one whole, carry the extra whole into the whole-number part.

Method 2 (improper fractions): change each mixed number to an improper fraction first. $2\frac{2}{5}$ = $\frac{12}{5}$ and $1\frac{2}{3}$ = $\frac{5}{3}$. Then $\frac{12}{5}$ + $\frac{5}{3}$ = $\frac{36}{15}$ + $\frac{25}{15}$ = $\frac{61}{15}$ = $4\frac{1}{15}$. Same answer! Method 1 keeps numbers small; Method 2 avoids regrouping. Use whichever you understand best, and use the other to check.

4. Subtracting mixed numbers and regrouping

To subtract, again work with wholes and fractions separately. $2\frac{4}{5}$ − $1\frac{2}{3}$ = (2 − 1) + ($\frac{4}{5}$ − $\frac{2}{3}$) = 1 + $\frac{12}{15}$ − $\frac{10}{15}$ = $1\frac{2}{15}$.

Sometimes the first fraction is smaller than the one you are taking away. In $3\frac{1}{4}$ − $1\frac{3}{4}$ you cannot take $\frac{3}{4}$ from $\frac{1}{4}$. So regroup: take 1 whole from the 3 and change it into quarters: $3\frac{1}{4}$ = 2 + $\frac{4}{4}$ + $\frac{1}{4}$ = $2\frac{5}{4}$. Now $2\frac{5}{4}$ − $1\frac{3}{4}$ = $1\frac{2}{4}$ = $1\frac{1}{2}$.

▶ Counting back across a whole number is exactly what regrouping does on paper.

Three mixed numbers: $3\frac{1}{2}$ + $2\frac{1}{3}$ − $1\frac{3}{4}$. Wholes: 3 + 2 − 1 = 4. Fractions over 12: $\frac{6}{12}$ + $\frac{4}{12}$ − $\frac{9}{12}$ = $\frac{1}{12}$. Answer: $4\frac{1}{12}$. If the fractions had come out negative, you would regroup 1 whole from the 4.

Worked examples

Try each one yourself first, then open the solution and compare every step.