Grade by Grade
B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

Adding and Subtracting Fractions

B2.5 B2.6 A1About 45 minutes
Learning goals
  • I can add and subtract fractions by creating equivalent fractions with a common denominator.
  • I can use the LCM to find the lowest common denominator.
  • I can add and subtract mixed numbers and more than two fractions.
  • I can solve fraction problems in real situations.

Words to know

equivalent fractions
Fractions that name the same amount, like 3/4 and 9/12.
lowest common denominator (LCD)
The LCM of the denominators. The LCD of 3/4 and 1/6 is 12.
improper fraction
A fraction whose numerator is greater than or equal to its denominator, like 17/12.
mixed number
A whole number and a proper fraction together, like 1 5/12.
lowest terms
A fraction whose numerator and denominator have a GCF of 1, like 3/5.

Remember from Grade 6

In Grade 6 you added and subtracted fractions and mixed numbers with unlike denominators by making equivalent fractions first: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. You also learned to never add the denominators. Now you have two new tools: the LCM gives you the smallest common denominator, and the GCF simplifies your answer in one step.

Let's start with a trail

On a winter hike, Noor walks 3/4 km to the lookout and then 1/6 km to the frozen creek. How far does she walk?

Quarters and sixths are different-size pieces, so we can't count them together yet. The LCM of 4 and 6 is 12, so cut everything into twelfths: 3/4 = 9/12 and 1/6 = 2/12.

9/12 + 2/12 = 11/12 km.

A snowy forest trail with footprints leading toward a frozen creek, between evergreen and white birch trees.
Hiking legs like 3/4 km and 1/6 km use different-size pieces. Using the LCM as a common denominator lets you add them. AI-generated image
3/4 = 9/121/6 = 2/129/12 + 2/12 = 11/12
Fraction strips in twelfths. 3/4 is the same as 9/12 and 1/6 is the same as 2/12. Together they make 11/12.

The steps

  1. Find the lowest common denominator (LCD). It is the LCM of the denominators. (Multiplying the denominators always gives a common denominator, but often a larger one.)
  2. Make equivalent fractions. Multiply the numerator and denominator by the same number.
  3. Add or subtract the numerators. Keep the denominator.
  4. Simplify. Divide by the GCF. Change an improper fraction to a mixed number if that makes sense in the problem.

Mixed numbers: add or subtract the whole numbers and the fractions separately, or change everything to improper fractions. When you subtract and the first fraction is too small, regroup one whole.

Worked example

Example 1: Find the LCD with the LCM

Find 5/6 + 3/8.

  1. LCM of 6 and 8: 6 = 2 × 3, 8 = 2³. LCM = 2³ × 3 = 24.
  2. 5/6 = 20/24 (× 4) and 3/8 = 9/24 (× 3).
  3. 20/24 + 9/24 = 29/24.
  4. 29/24 = 1 5/24. (24 fits once into 29 with 5 left.)

5/6 + 3/8 = 29/24 = 1 5/24

Worked example

Example 2: Subtract mixed numbers with regrouping

Find 3 1/4 − 1 5/6.

  1. The LCD of 4 and 6 is 12. 3 1/4 = 3 3/12 and 1 5/6 = 1 10/12.
  2. 3/12 is less than 10/12, so regroup one whole: 3 3/12 = 2 + 12/12 + 3/12 = 2 15/12.
  3. Subtract the wholes: 2 − 1 = 1. Subtract the fractions: 15/12 − 10/12 = 5/12.
  4. Check by adding: 1 5/12 + 1 10/12 = 2 15/12 = 3 3/12 = 3 1/4. ✓

3 1/4 − 1 5/6 = 1 5/12

123− 1− 10/123 1/42 1/41 5/12small ticks are twelfths
Another way to see 3 1/4 − 1 5/6: jump back 1 whole to 2 1/4, then back 5/6 = 10/12 more. You land on 1 5/12.
Worked example

Example 3: Three fractions and simplifying

A muffin recipe uses 1/2 cup of oats, 2/3 cup of flour and 3/4 cup of milk. How many cups is that altogether?

  1. LCD of 2, 3 and 4 is 12.
  2. 1/2 = 6/12, 2/3 = 8/12, 3/4 = 9/12.
  3. 6/12 + 8/12 + 9/12 = 23/12 = 1 11/12.

1 11/12 cups altogether (a little less than 2 cups).

Think like a mathematician

Estimate first with benchmarks 0, 1/2 and 1. 5/6 + 3/8 is about 1 + 1/2 = 1 1/2, so 1 5/24 is reasonable. If you got 8/14, you would know right away something went wrong: it's less than 5/6, but you added to 5/6.

Hands-on activity

Measuring-cup fractions

You need: a set of measuring cups (1 cup, 1/2, 1/3 and 1/4 cup), water, a clear 1-cup measure or a glass, and a towel. Do this at the sink.

  1. Pour 1/2 cup and then 1/4 cup of water into the 1-cup measure. Predict the total first. (3/4 cup.)
  2. Empty it. Pour 1/2 cup and 1/3 cup. Predict: 3/6 + 2/6 = 5/6. Is it a bit less than full?
  3. Empty it. Fill the 1-cup measure, then pour out 1/3 cup. What is left? (2/3 cup.)
  4. Try 1/3 + 1/4. Predict using twelfths: 4/12 + 3/12 = 7/12 (a bit more than half).
  5. Write each one as an addition or subtraction sentence.

Math talk: "Which LCD did you use for each one, and why?"

Just for fun

Why couldn't 1/3 and 1/4 play on the same team right away?

They had to find a common denominator first. (The LCM was the coach.)

Your turn

1

2/3 + 1/4 = ?

2

5/6 − 1/4 = ?

3

3/10 + 2/5 = ?

4

7/8 − 5/12 = ?

5

1/2 + 1/3 + 1/6 = ?

6

2 3/4 + 1 2/3 = ?

7

4 1/3 − 2 3/4 = ?

More practice: warm-up

1

1/4 + 1/2 = ?

2

3/5 − 1/5 = ?

3

1/3 + 1/6 = ?

4

7/10 − 1/2 = ?

5

2/9 + 4/9 = ?

More practice: on your own

1

5/6 + 4/9 = ?

2

11/12 − 3/8 = ?

3

A recipe needs 3/4 cup of brown sugar and 2/3 cup of white sugar. How much sugar is that?

4

5 − 1 3/8 = ?

5

Jaya skated for 1 1/2 hours on Saturday and 2 1/4 hours on Sunday. How long did she skate altogether?

Challenge

1

A board is 2 1/2 m long. You cut off 3/4 m and then 1 1/3 m. How much is left?

2

Find the missing fraction: 3/8 + ? = 5/6.

3

On a school day, Theo sleeps for 3/8 of the day and is at school for 1/4 of the day. What fraction of the day is left? How many hours is that?

Check your understanding

Quiz

1.

What is 1/4 + 2/3?

2.

What is the lowest common denominator for 5/6 and 3/8?

3.

What is 5/6 − 1/2?

4.

What is 2 1/3 + 1 3/4?

5.

What is 3 − 1 1/4?

6.

Why can't you add 1/2 + 1/3 by adding the tops and the bottoms to get 2/5?

For parents and teachers

Big idea: The Ontario curriculum (B2.5) asks Grade 7 students to add and subtract fractions, including by creating equivalent fractions, in various contexts. Students already met unlike denominators in Grade 6; Grade 7 adds efficiency (the LCM as the lowest common denominator and the GCF to simplify) and more complex situations.

Common difficulties:

  • Adding or subtracting the denominators.
  • Forgetting to multiply the numerator when making an equivalent fraction (3/4 → 3/12).
  • Regrouping errors with mixed numbers: borrowing "10" instead of one whole (12/12).
  • Leaving answers like 17/12 when the situation calls for a mixed number.

How to help: Use measuring cups while baking and ask your child to predict totals before pouring. Ask for an estimate first. This lesson builds A1 skills: representing with models, reasoning and checking that answers make sense.