Grade by Grade
C. Algebra · Algebra: Expressions, Equations and Coding

Equations with Fractions and Decimals, and Rearranging Formulas

C1.5 C1.4 A1About 75 minutes
Learning goals
  • I can solve equations with fractions by multiplying both sides by a common denominator.
  • I can solve equations with decimals, by working with them directly or by multiplying by a power of 10.
  • I can rearrange a formula to isolate a different variable.
  • I can use a rearranged formula to solve a problem and check the answer.

Words to know

lowest common denominator (LCD)
The smallest number that all the denominators divide into evenly. The LCD of 3 and 4 is 12.
clear the fractions
Multiply every term on both sides by the LCD so no fractions are left.
formula
An equation that shows how quantities are related, such as P = 2l + 2w.
rearrange (isolate)
Use inverse operations to get one chosen variable by itself on one side of a formula.
subject of a formula
The variable on its own on one side. In d = st, d is the subject.

Let's start with a temperature puzzle

Canada uses degrees Celsius, but many recipes, ovens and websites from the United States use degrees Fahrenheit. The formula is

F = 1.8C + 32

It's easy to go from Celsius to Fahrenheit. But what if a recipe says to bake at 350°F and your oven shows °C? You need the formula the other way round. In this lesson you will learn to rearrange formulas, and to solve equations that contain fractions and decimals.

Equations with fractions: clear them first

Fractions make equations messy. Get rid of them in one step: multiply every term on both sides by the lowest common denominator (LCD). The equation stays balanced, and the fractions disappear.

For x/3 + x/4 = 7, the LCD of 3 and 4 is 12:

12 × x/3 + 12 × x/4 = 12 × 7 → 4x + 3x = 84 → 7x = 84 → x = 12.

The bar model shows why it works.

xx/3 = 4 partsx/4 = 3 partsx/3 + x/4 = 7 parts = 7, so 1 part = 1 and x = 12 parts = 12
Split x into 12 equal parts. A third of x is 4 parts, and a quarter of x is 3 parts. Together they are 7 parts, which equals 7. So each part is 1, and x = 12.
Worked example

Example 1: A fraction on each side

Solve (x + 2)/5 = (x − 1)/2 and verify.

  1. The LCD of 5 and 2 is 10. Multiply both sides by 10: 10 × (x + 2)/5 = 10 × (x − 1)/2.
  2. Simplify: 2(x + 2) = 5(x − 1). Keep the brackets; each numerator is a whole quantity.
  3. Expand: 2x + 4 = 5x − 5.
  4. Subtract 2x and add 5: 9 = 3x, so x = 3.
  5. Verify: LS = (3 + 2)/5 = 1. RS = (3 − 1)/2 = 1 ✓.

x = 3

Worked example

Example 2: Decimals

Solve 0.2x + 1.5 = 0.5x − 0.3.

  1. Multiply every term by 10 to make whole numbers: 2x + 15 = 5x − 3.
  2. Subtract 2x: 15 = 3x − 3. Add 3: 18 = 3x.
  3. x = 6.
  4. Verify: LS = 0.2(6) + 1.5 = 1.2 + 1.5 = 2.7. RS = 0.5(6) − 0.3 = 3 − 0.3 = 2.7 ✓.

x = 6. (You could also solve with the decimals as they are: 1.5 + 0.3 = 0.5x − 0.2x, so 1.8 = 0.3x and x = 6.)

Rearranging formulas

A formula is an equation with more than one variable. To make a different variable the subject, use the same inverse operations you use to solve equations. Think of the formula as a machine: to run it backwards, undo the last operation first.

Formula: C to FC× 1.8+ 32FUndo it, in reverse order: F to CF− 32÷ 1.8CC = (F − 32) ÷ 1.8
To find F, you multiply C by 1.8 and then add 32. To go backwards, undo the steps in reverse order: subtract 32, then divide by 1.8.
Worked example

Example 3: Rearrange the temperature formula

Rearrange F = 1.8C + 32 to make C the subject. Then convert 350°F to Celsius.

  1. Subtract 32 from both sides: F − 32 = 1.8C.
  2. Divide both sides by 1.8: C = (F − 32) ÷ 1.8.
  3. For F = 350: C = (350 − 32) ÷ 1.8 = 318 ÷ 1.8 ≈ 176.7.
  4. Check: 1.8(176.7) + 32 ≈ 318 + 32 = 350 ✓.

C = (F − 32) ÷ 1.8; 350°F is about 177°C (oven dials often round this to 175°C or 180°C).

Worked example

Example 4: Rearrange a perimeter formula

The perimeter of a rectangle is P = 2l + 2w. A rink's boards measure 50 m around and the rink is 15 m long. Rearrange for w and find the width.

  1. Subtract 2l from both sides: P − 2l = 2w.
  2. Divide both sides by 2: w = (P − 2l)/2.
  3. Substitute: w = (50 − 2(15))/2 = (50 − 30)/2 = 10.
  4. Check: 2(15) + 2(10) = 30 + 20 = 50 ✓.

w = (P − 2l)/2; the width is 10 m.

Watch out!

  • Multiply every term by the LCD, including terms with no fraction: in x/2 + 3 = 5, multiplying by 2 gives x + 6 = 10, not x + 3 = 10.
  • When a numerator has two terms, like (x + 2)/5, keep it in brackets after clearing the fraction.
  • To rearrange a formula, list the operations done to your variable, then undo them in reverse order.
Hands-on activity

Formulas at home

You need: a tape measure or ruler, a calculator, a weather app or newspaper forecast.

  1. Temperature: find tomorrow's high for 4 Canadian cities in °C. Convert each to °F with F = 1.8C + 32. Then use your rearranged formula C = (F − 32) ÷ 1.8 to convert back and check that you get the starting number.
  2. Perimeter: choose a rectangle at home (a table, rug or door). Measure its length and its perimeter (run the tape all the way around, or add all 4 sides). Use w = (P − 2l)/2 to calculate the width. Then measure the width to check.
  3. Speed: use d = st. Rearranged, t = d/s. If a car drives from Ottawa to Toronto, about 450 km, at an average of 90 km/h, how long does it take? (450 ÷ 90 = 5 hours.)

What to look for: the formula is rearranged before numbers are substituted, and every answer is checked.

Just for fun

There is one temperature where Celsius and Fahrenheit agree. Set C = F in the formula: C = 1.8C + 32, so −0.8C = 32 and C = −40.

At −40 degrees, you don't need to convert. You just need a very warm parka.

Your turn

1

Solve x/4 − 2 = 3.

2

Solve x/2 + x/3 = 10.

3

Solve (x − 4)/3 = 5.

4

Solve (x + 1)/2 = (x + 4)/3 and verify.

5

Solve 0.3x − 1.2 = 0.1x + 0.4.

6

Rearrange d = st for t. How long does a 240 km trip take at 80 km/h?

7

Rearrange P = 2l + 2w for l.

More practice: warm-up

1

Solve x/6 = 4.

2

Solve 0.5x = 3.5.

3

Solve 2x/3 = 8.

4

Rearrange A = lw for w.

5

Rearrange y = mx + b for b.

More practice: on your own

1

Solve 3x/4 − 2 = x/2 + 1 and verify.

2

Solve (2x + 1)/5 = (x + 5)/4.

3

Solve 1.25x + 2 = 0.75x + 6.

4

Use C = (F − 32) ÷ 1.8 to convert 14°F to Celsius.

5

A triangle has area A = ½bh. Rearrange for h, and find h when A = 36 cm² and b = 8 cm.

Challenge

1

Solve (x + 3)/2 − (x − 1)/3 = 4 and verify.

2

A phone plan costs T = 25 + 0.15m dollars for m minutes of long-distance calls. Rearrange for m. How many minutes were used on a $37 bill?

3

The surface area of a cylinder is S = 2πr² + 2πrh. Rearrange for h.

Check your understanding

Quiz

1.

Solve x/3 + 2 = 6.

2.

To clear the fractions in x/4 + x/6 = 5, multiply every term by:

3.

Solve (x − 2)/3 = 4.

4.

Solve 0.4x = 2.

5.

Which is P = 2l + 2w rearranged for w?

6.

Rearrange d = st for s.

For parents and teachers

Big idea: C1.5 asks students to create and solve equations for various contexts; this lesson adds fractions, decimals and formulas. Rearranging a formula uses the same inverse operations as solving an equation, which also connects to simplifying expressions (C1.4). Formulas like these appear constantly in Grade 10 math and science.

Common difficulties:

  • Not multiplying every term by the LCD (especially whole-number terms).
  • Dropping brackets around a numerator like (x − 1) after clearing fractions, which causes sign errors.
  • Undoing operations in the wrong order when rearranging (dividing by 1.8 before subtracting 32).

How to help: Convert a few temperatures or recipe oven settings together. Ask your teen to rearrange the formula first, then substitute, then check by going back the other way.