Unit 1 · Lesson 3 of 6

Fractions: add and subtract

About 20-25 minutes · 5 steps · one idea: same-size pieces first (the LCD)

You'll be able to: add and subtract fractions and mixed numbers, including negative ones, using the least common denominator.

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Try them from memory before you peek at anything.

\(\dfrac{3}{10} \div \dfrac{9}{4}\)

Flip the second: \(\dfrac{3}{10} \cdot \dfrac{4}{9}\). Cancel 3 and 9 (divide by 3): 1 and 3. Cancel 4 and 10 (divide by 2): 2 and 5. Multiply: \(\dfrac{1 \cdot 2}{5 \cdot 3} = \dfrac{2}{15}\).

\(-7 - (-12)\)

Subtracting \(-12\) means adding 12: \(-7 + 12\). Different signs: \(12 - 7 = 5\), and the bigger size is positive. Answer: \(5\).

Step 2 of 5The idea

You can only add pieces of the same size. For fractions, that means the denominators (bottoms) must match before you add or subtract.

Same denominator. Add or subtract the tops and keep the bottom: \(\dfrac{2}{7} + \dfrac{3}{7} = \dfrac{5}{7}\).

Different denominators. Find the LCD (least common denominator), the smallest number both bottoms divide into. Rewrite each fraction with that bottom by multiplying its top and bottom by the same number. Then add the tops.

Mixed numbers. Change to an improper fraction first: whole number times bottom, plus top, over the same bottom. \(2\tfrac{1}{3} = \dfrac{2 \cdot 3 + 1}{3} = \dfrac{7}{3}\).

Negatives. Put the sign on the top and use the rules from Lesson 2: \(-\dfrac{3}{5} + \dfrac{1}{5} = \dfrac{-3 + 1}{5} = -\dfrac{2}{5}\).

To find the LCD, list multiples of the bigger bottom until the smaller one divides in. For 6 and 8: 8, 16, 24. Since 6 goes into 24, the LCD is 24.

A negative mixed number is negative as a whole: \(-1\tfrac{1}{2}\) means \(-\dfrac{3}{2}\).

Why this shows up on the placement test: equations with fractions (Unit 2) and rational expressions (Unit 7) all start with this same LCD step.

Step 3 of 5Worked examples

Example 1: \(\dfrac{5}{6} - \dfrac{3}{8}\)

  1. Find the LCD of 6 and 8. Multiples of 8: 8, 16, 24. 6 goes into 24, so the LCD is 24.
  2. Rewrite: \(\dfrac{5}{6} = \dfrac{5 \cdot 4}{6 \cdot 4} = \dfrac{20}{24}\) and \(\dfrac{3}{8} = \dfrac{3 \cdot 3}{8 \cdot 3} = \dfrac{9}{24}\). (Multiplying top and bottom by the same number does not change the value.)
  3. Subtract the tops: \(\dfrac{20 - 9}{24} = \dfrac{11}{24}\). 11 and 24 share no factor, so it is already in lowest terms.

Example 2: \(2\tfrac{1}{4} + \left(-\dfrac{5}{6}\right)\)

  1. Change the mixed number: \(2\tfrac{1}{4} = \dfrac{2 \cdot 4 + 1}{4} = \dfrac{9}{4}\).
  2. The LCD of 4 and 6 is 12: \(\dfrac{9}{4} = \dfrac{27}{12}\) and \(-\dfrac{5}{6} = -\dfrac{10}{12}\).
  3. Add the tops: \(\dfrac{27 + (-10)}{12} = \dfrac{17}{12}\), which is \(1\tfrac{5}{12}\) as a mixed number. (Both forms are correct. Algebra usually keeps the improper fraction.)

Step 4 of 5Practice

Type each answer as a fraction, like 7/12 or -5/4, and press Check. If you miss twice, open the worked solution. Then mark how it went.

\(\dfrac{2}{9} + \dfrac{5}{9}\)

Same bottom, so add the tops: \(\dfrac{2 + 5}{9} = \dfrac{7}{9}\).

\(\dfrac{3}{4} + \dfrac{1}{6}\)

The LCD of 4 and 6 is 12. \(\dfrac{3}{4} = \dfrac{9}{12}\) and \(\dfrac{1}{6} = \dfrac{2}{12}\). Add: \(\dfrac{9 + 2}{12} = \dfrac{11}{12}\).

\(\dfrac{7}{10} - \dfrac{4}{15}\)

Multiples of 15: 15, 30. 10 goes into 30, so the LCD is 30. \(\dfrac{7}{10} = \dfrac{21}{30}\) and \(\dfrac{4}{15} = \dfrac{8}{30}\). Subtract: \(\dfrac{21 - 8}{30} = \dfrac{13}{30}\).

\(-\dfrac{2}{3} + \dfrac{1}{4}\)

The LCD is 12. \(-\dfrac{2}{3} = -\dfrac{8}{12}\) and \(\dfrac{1}{4} = \dfrac{3}{12}\). Add the tops: \(-8 + 3 = -5\). Answer: \(-\dfrac{5}{12}\).

\(-\dfrac{3}{8} - \dfrac{5}{6}\)

The LCD of 8 and 6 is 24. \(-\dfrac{3}{8} = -\dfrac{9}{24}\) and \(\dfrac{5}{6} = \dfrac{20}{24}\). Tops: \(-9 - 20 = -29\). Answer: \(-\dfrac{29}{24}\).

\(1\tfrac{1}{2} + 2\tfrac{2}{3}\)

Change both: \(1\tfrac{1}{2} = \dfrac{3}{2}\) and \(2\tfrac{2}{3} = \dfrac{8}{3}\). The LCD is 6: \(\dfrac{9}{6} + \dfrac{16}{6} = \dfrac{25}{6}\) (that is \(4\tfrac{1}{6}\)).

Which is equal to \(\dfrac{1}{2} + \dfrac{2}{5}\)?

  1. \(\dfrac{9}{10}\)
  2. \(\dfrac{3}{7}\)
  3. \(\dfrac{3}{10}\)
  4. \(\dfrac{1}{5}\)

The LCD is 10: \(\dfrac{5}{10} + \dfrac{4}{10} = \dfrac{9}{10}\). Answer: (a).

(b) adds the tops and the bottoms. (c) adds the tops but forgets to rewrite them over 10. (d) multiplies instead of adding.

You have \(5\tfrac{1}{2}\) feet of ribbon and cut off \(1\tfrac{3}{4}\) feet. How many feet are left?

Subtract: \(5\tfrac{1}{2} - 1\tfrac{3}{4} = \dfrac{11}{2} - \dfrac{7}{4}\). The LCD is 4: \(\dfrac{22}{4} - \dfrac{7}{4} = \dfrac{15}{4}\) feet, which is \(3\tfrac{3}{4}\) feet.

Step 5 of 5Wrap-up

You can now add and subtract any fractions: change mixed numbers to improper fractions, rewrite over the LCD, then combine the tops and keep the bottom.

Read or watch next: OpenStax Elementary Algebra 2e, 1.6 Add and Subtract Fractions (free textbook, with more practice at the end of the section).

Also good: Khan Academy: Adding and subtracting fractions with unlike denominators (short videos). Quick rules: Number skills cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "How do I find the LCD quickly when the denominators are big, like 18 and 24?" It knows this course and can explain things another way.