Grade by Grade
B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Adding and Subtracting Positive and Negative Fractions and Mixed Numbers

B3.4 A1About 70 minutes
Learning goals
  • I can add and subtract positive and negative fractions using a lowest common denominator.
  • I can add and subtract positive and negative mixed numbers, choosing between improper fractions and working with wholes and parts.
  • I can solve problems with fractions in measurement, formulas and changes, and check that my answer is reasonable.
  • I can use technology, such as a fraction calculator, to check my work when appropriate.

Words to know

lowest common denominator (LCD)
The least common multiple of the denominators. For 6 and 4, the LCD is 12.
improper fraction
A fraction whose numerator is greater than or equal to its denominator, like 11/4.
mixed number
A whole number and a fraction, like 2 3/4. A negative mixed number like −2 3/4 means −(2 + 3/4).
additive inverse (opposite)
The number that adds to give 0: the additive inverse of −5/6 is 5/6.
estimate
A quick approximate answer, used to check that an exact answer makes sense.

Remember from Grade 8

In Grade 8 you added and subtracted fractions and mixed numbers, including a few negative fractions, and you used integer rules. Grade 9 puts it all together: any mix of positive and negative fractions and mixed numbers, used in measurements, formulas and real changes.

Two rules do almost all the work

  1. You can only add pieces of the same size. Rewrite the fractions with a common denominator (the LCD keeps the numbers small).
  2. Subtracting is adding the opposite. a − (−b) = a + b, and a − b = a + (−b). Then use integer rules on the numerators.
2/3= 8/123/4= 9/12Twelfths fit both: 2/3 = 8/12 and 3/4 = 9/12, so 3/4 − 2/3 = 1/12
Thirds and quarters are different-sized pieces, so you cannot subtract 2/3 from 3/4 directly. Cut both into twelfths (the LCD of 3 and 4), and the pieces match.
Worked example

Example 1: Different signs

Calculate −5/6 + 3/4.

  1. LCD of 6 and 4 is 12. −5/6 = −10/12 and 3/4 = 9/12.
  2. Add the numerators with integer rules: −10 + 9 = −1.
  3. Estimate: −5/6 is a bit less than −3/4, so the sum should be a small negative number. ✓

−1/12

Worked example

Example 2: Subtracting a negative

Calculate 2/3 − (−1/4).

  1. Subtracting a negative is adding the opposite: 2/3 + 1/4.
  2. LCD 12: 8/12 + 3/12 = 11/12.

11/12

Mixed numbers with signs

A negative mixed number like −2 1/3 means −(2 + 1/3), so it is −2 and another −1/3, not −2 + 1/3. That is why the safest method is to change mixed numbers to improper fractions first: −2 1/3 = −7/3.

When both numbers are positive, you can also add the wholes and the parts separately, regrouping if needed.

−2−1 1/2−103/41+2 1/4
−1 1/2 + 2 1/4 on a number line marked in quarters: start at −1 1/2 and jump 2 1/4 to the right. You land at 3/4.
Worked example

Example 3: Positive and negative mixed numbers

Calculate −2 1/3 + 4 3/4.

  1. Improper fractions: −2 1/3 = −7/3 and 4 3/4 = 19/4.
  2. LCD 12: −7/3 = −28/12 and 19/4 = 57/12.
  3. −28 + 57 = 29, so the sum is 29/12 = 2 5/12.
  4. Estimate: about −2 + 5 = 3, a bit less. 2 5/12 ✓

2 5/12

Worked example

Example 4: Subtracting to get further below zero

Calculate −1 5/8 − 2 3/4.

  1. Improper: −13/8 − 11/4. LCD 8: −13/8 − 22/8.
  2. Add the opposite: −13/8 + (−22/8) = −35/8.
  3. −35/8 = −4 3/8.

−4 3/8. Two negatives going the same way: the size grows.

Worked example

Example 5: Measurement with a formula

A triangular garden bed has sides 2 1/2 ft, 3 3/4 ft and 4 1/8 ft. Use P = a + b + c to find the perimeter.

  1. LCD 8: 2 1/2 = 2 4/8, 3 3/4 = 3 6/8, 4 1/8.
  2. Wholes: 2 + 3 + 4 = 9. Parts: 4/8 + 6/8 + 1/8 = 11/8 = 1 3/8.
  3. Total: 9 + 1 3/8 = 10 3/8.

10 3/8 ft

Worked example

Example 6: Net change

Over three days, the level of a river changes by +3/8 m, −5/6 m and +1/2 m. What is the net change?

  1. LCD of 8, 6 and 2 is 24: 9/24 − 20/24 + 12/24.
  2. 9 − 20 + 12 = 1.

+1/24 m: the river ended just slightly higher (about 4 cm).

Using technology

Many scientific calculators have a fraction key (often shown as a/b, or a box over a box). A spreadsheet can also help: typing =-7/3+19/4 gives 2.41666…, which matches 2 5/12. Use technology to check your work or for messy numbers, but be able to do the steps by hand so you can tell when an answer does not make sense.

Hands-on activity

Snow-day measuring

You need: a ruler or tape measure with inch marks (or cm), paper, a pencil.

  1. Measure the length and width of 3 rectangular things at home (a cutting board, a book, a window pane) to the nearest 1/8 in.
  2. Use P = 2l + 2w or P = l + w + l + w to find each perimeter with fractions.
  3. Find the difference between the longest and shortest perimeters.
  4. Imagine 2 1/4 in of snow fell on Monday, 3/8 in melted on Tuesday and 1 5/8 in fell on Wednesday. Write the net change as a sum of signed fractions and calculate it. (Model answer: 2 1/4 − 3/8 + 1 5/8 = 18/8 − 3/8 + 13/8 = 28/8 = 3 1/2 in.)
  5. Check one answer with a calculator.
Just for fun

A funny-but-true fact: "a fraction of the cost" can mean 99/100 of the cost. Mathematically, that's still a fraction!

Your turn

1

Calculate −3/5 + 1/2.

2

Calculate −7/8 − 1/6.

3

Calculate 5/9 − (−2/3).

4

Calculate −3 1/2 + 1 2/3.

5

Calculate 2 1/4 − 5 5/6.

6

Calculate −1 3/10 − (−4 1/5).

7

A board is 8 ft long. You cut off 2 5/8 ft and then 3 3/4 ft. How much is left? (Ignore the saw cuts.)

More practice: warm-up

1

Calculate 1/4 + 2/4.

2

Calculate −2/5 + (−1/5).

3

Calculate 1/2 − 3/4.

4

Write −2 3/5 as an improper fraction.

5

What is the LCD of 1/6 and 1/9?

More practice: on your own

1

Calculate −5/12 + 3/8 − 1/6.

2

Calculate 4 2/3 − 6 1/4 + 1 1/2.

3

The temperature was −3 1/2 °C. It rose 5 3/4 °C, then fell 4 1/4 °C. What is the new temperature?

4

Evaluate x − y when x = −2/3 and y = −5/6.

5

A rectangle has length 5 1/3 cm and width 2 3/4 cm. Find its perimeter.

Challenge

1

Find the missing number: −1 3/4 + □ = 2/3.

2

The perimeter of a triangle is 12 1/2 in. Two sides are 3 3/8 in and 4 3/4 in. Find the third side.

3

Jun says −2 1/3 + 1 = −1 1/3 and Ana says it equals −1 2/3. Who is right? Use a number line or improper fractions to explain.

Check your understanding

Quiz

1.

What is −2/3 + 1/6?

2.

What is 3/4 − (−1/2)?

3.

Which is equal to −3 2/5?

4.

What is −1 1/2 − 2 1/3?

5.

What is the LCD for 5/6 + 3/8?

6.

A lake's level changes by −1/4 m and then +2/3 m. What is the net change?

For parents and teachers

Big idea: Expectation B3.4 asks students to solve problems involving operations with positive and negative fractions and mixed numbers, including problems with formulas, measurements and linear relations, using technology when appropriate. This lesson covers addition and subtraction; the next covers multiplication, division and formulas.

Common difficulties:

  • Adding denominators (1/2 + 1/3 = 2/5).
  • Treating −2 1/3 as −2 + 1/3. Changing to an improper fraction first avoids this.
  • Losing signs when subtracting a negative. Rewrite every subtraction as adding the opposite.
  • Not estimating. A quick estimate catches most sign errors.

How to help: Ask your teen to estimate before calculating ("about how much?") and to check one answer per page with a calculator.