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

Multiplying and Dividing Positive and Negative Fractions: Formulas and Linear Relations

B3.4 A1 A2About 75 minutes
Learning goals
  • I can multiply and divide positive and negative fractions and mixed numbers, simplifying before I multiply.
  • I can predict the sign of a product or quotient of fractions.
  • I can substitute fractions and mixed numbers into formulas, such as A = 1/2 bh and F = 9/5 C + 32.
  • I can evaluate a linear relation with fraction coefficients and use technology to check my work.

Words to know

reciprocal
The number you multiply by to get 1. The reciprocal of −3/4 is −4/3 (the sign stays the same).
simplify before multiplying
Dividing a numerator and a denominator by a common factor before you multiply, to keep numbers small.
formula
An equation that shows how quantities are related, such as A = 1/2 bh.
substitute
Replace a variable with a number.
linear relation
A relationship whose graph is a straight line, such as y = −2/3 x + 4.

Remember from Grade 8

In Grade 8 you multiplied and divided fractions and mixed numbers, and you multiplied and divided integers. Now you will combine them with negative fractions and use them in formulas and linear relations.

Multiplying fractions

Multiply the numerators and multiply the denominators, after changing mixed numbers to improper fractions. Decide the sign first (same signs → positive, different signs → negative), then work with the sizes.

Why does it work? 2/3 × 3/4 means "2/3 of 3/4". The area model below shows it.

3/4 of the width2/3 of the heightoverlap: 6 of 12 parts2/3 × 3/4 = 6/12 = 1/2in both parts
An area model for 2/3 × 3/4. The whole is split into 4 columns and 3 rows (12 parts). The green region is 2/3 of 3/4: 6 of the 12 parts, so 2/3 × 3/4 = 6/12 = 1/2.
Worked example

Example 1: Multiply negative mixed numbers

Calculate (−2 1/2)(1 3/5).

  1. Sign: one negative, so the answer is negative.
  2. Improper fractions: 2 1/2 = 5/2 and 1 3/5 = 8/5.
  3. Simplify before multiplying: 5/2 × 8/5. The 5s cancel; 8 and 2 share 2, leaving 4/1.
  4. So the size is 4, and the answer is −4.

−4

Dividing fractions

Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of a negative number is negative: the reciprocal of −9/8 is −8/9.

Why? Ask "how many 9/8s fit in 3/4?" The answer is 3/4 × 8/9, because multiplying by 8/9 undoes multiplying by 9/8.

Worked example

Example 2: Divide two negative fractions

Calculate (−3/4) ÷ (−9/8).

  1. Sign: two negatives, so positive.
  2. Multiply by the reciprocal: 3/4 × 8/9.
  3. Simplify: 3 and 9 share 3 (→ 1 and 3); 8 and 4 share 4 (→ 2 and 1). So 1/1 × 2/3 = 2/3.

2/3

Worked example

Example 3: A mixed number divided by a fraction

Calculate −3 1/3 ÷ 2/9.

  1. Sign: negative.
  2. −10/3 × 9/2. Simplify: 10 and 2 → 5 and 1; 9 and 3 → 3 and 1.
  3. 5 × 3 = 15.

−15. Check: −15 × 2/9 = −30/9 = −10/3 = −3 1/3 ✓

Worked example

Example 4: The temperature formula

Use F = 9/5 C + 32 to change −12 1/2 °C to degrees Fahrenheit.

  1. Substitute: F = 9/5 × (−25/2) + 32.
  2. 9/5 × (−25/2) = −225/10 = −45/2 = −22 1/2. (Simplify first: 25 ÷ 5 = 5, so it is −9 × 5/2 = −45/2.)
  3. F = −22 1/2 + 32 = 9 1/2.

9 1/2 °F (9.5 °F)

Worked example

Example 5: Area of a triangle

A triangular sail has base 5 1/4 m and height 3 1/3 m. Find its area using A = 1/2 bh.

  1. A = 1/2 × 21/4 × 10/3.
  2. Simplify: 10 and 2 → 5 and 1; 21 and 3 → 7 and 1.
  3. A = 1/1 × 7/4 × 5/1 = 35/4 = 8 3/4.

8 3/4 m²

Fractions in linear relations

A linear relation like y = −2/3 x + 4 has a fraction as its rate of change (the number multiplying x). For each 3 units right, y changes by −2 (goes down 2). Choosing x-values that are multiples of 3 makes the table easy, but you can substitute any rational number.

−3326xy(−3, 6)(0, 4)(3, 2)(6, 0)y = −2/3 x + 4
The graph of y = −2/3 x + 4. Every step of 3 to the right takes the line 2 down, which is the rate −2/3. The points (−3, 6), (0, 4), (3, 2) and (6, 0) are all on the line.
Worked example

Example 6: Evaluate and solve a linear relation

For y = −2/3 x + 4: a) find y when x = −4 1/2; b) find x when y = 0.

  1. a) y = −2/3 × (−9/2) + 4. Two negatives: +18/6 = 3. So y = 3 + 4 = 7.
  2. b) 0 = −2/3 x + 4. Subtract 4: −4 = −2/3 x. Multiply both sides by the reciprocal −3/2: x = (−4)(−3/2) = 6.
  3. Check b): −2/3 × 6 + 4 = −4 + 4 = 0 ✓

a) y = 7 b) x = 6

Technology check

Type fractions with brackets: (−2/3)*(−9/2)+4 in a calculator or spreadsheet gives 7. Graphing tools such as Desmos can draw y = −2/3x + 4 so you can check a point by tapping on the line. Use these to check, and keep doing the steps by hand.

Hands-on activity

Recipe and temperature conversions

You need: a recipe, a calculator, an oven dial or a thermometer that shows both °C and °F (if you have one).

  1. Scale a recipe by 1 1/2 and by 2/3. Multiply each amount by the scale factor. (Example: 3/4 cup × 1 1/2 = 3/4 × 3/2 = 9/8 = 1 1/8 cups; 3/4 × 2/3 = 1/2 cup.)
  2. Temperature: Use F = 9/5 C + 32 to convert −20 °C, −5 °C, 0 °C, 37 °C and 180 °C. Then use C = 5/9(F − 32) to convert 350 °F (a common oven setting) to °C.

Model answers: −20 °C = −4 °F; −5 °C = 23 °F; 0 °C = 32 °F; 37 °C = 98 3/5 °F (98.6 °F); 180 °C = 356 °F; 350 °F = 5/9 × 318 = 176 2/3 °C (about 177 °C).

If you use the oven, have a grown-up help.

Just for fun

A funny-but-true fact: −40 °C is exactly −40 °F. Check it: 9/5 × (−40) + 32 = −72 + 32 = −40. So on the coldest Prairie mornings, nobody has to argue about which scale to use.

Your turn

1

Calculate (−4/9)(3/8).

2

Calculate (−1 1/3)(−2 1/4).

3

Calculate 5/6 ÷ (−10/3).

4

Calculate −2 2/5 ÷ (−1 1/5).

5

Find the reciprocal of −1 3/4.

6

Use C = 5/9(F − 32) to convert −4 °F to °C.

7

For y = 3/4 x − 2, find y when x = −2 2/3.

8

Find the area of a triangle with base 4 1/2 cm and height 2 2/3 cm.

More practice: warm-up

1

Calculate 1/2 × 3/5.

2

Calculate (−2/3)(3).

3

Calculate 1/4 ÷ 1/2.

4

What is the sign of (−5/7) ÷ (−2/3)?

5

What is the reciprocal of −3/8?

More practice: on your own

1

Calculate (−3/5)(10/21)(−7/4).

2

Calculate (−7/8 + 1/4) ÷ (−5/16).

3

A rectangular box is 2 1/2 ft long, 1 1/3 ft wide and 3/4 ft high. Find its volume using V = lwh.

4

For y = −5/2 x + 1, find x when y = −9.

5

How many 3/8 in pieces can be cut from a 6 3/4 in strip?

Challenge

1

Find the temperature where the Celsius value is half the Fahrenheit value (C = F/2). Use F = 9/5 C + 32.

2

A triangle has area 7 1/2 cm² and base 3 3/4 cm. Find its height.

3

Without calculating, decide if (−2/3)⁵ ÷ (−3/4)² is positive or negative. Then calculate it.

Check your understanding

Quiz

1.

What is (−3/4)(8/9)?

2.

What is (−2/5) ÷ (−4/15)?

3.

What is the reciprocal of −2 1/2?

4.

Using F = 9/5 C + 32, what is −10 °C in °F?

5.

For y = −2/3 x + 4, what is y when x = 9?

6.

What is the area of a triangle with base 2 2/3 m and height 3 3/4 m?

For parents and teachers

Big idea: This lesson completes expectation B3.4: operations with positive and negative fractions and mixed numbers, in problems with formulas, measurements and linear relations, using technology when appropriate. These are the exact skills used in the Algebra and Linear Relations units of this course.

Common difficulties:

  • Multiplying mixed numbers part by part (2 1/2 × 1 3/5 ≠ 2 × 1 + 1/2 × 3/5). Always change to improper fractions.
  • Flipping the wrong fraction when dividing, or changing the sign of a reciprocal.
  • Deciding the sign at the end and forgetting. Decide it first and write it down.
  • Order of operations in formulas: in F = 9/5 C + 32, multiply before adding.

How to help: Cook together and let your teen scale the recipe. On a cold morning, ask them to convert the temperature to Fahrenheit in their head (double it, take away 10% of that, then add 32; this mental method gives exactly 9/5 C + 32).