Why Do Equations Sometimes Contain Fractions and Decimals?

Many real-world quantities are not whole numbers.

Examples include:

• money
• measurements
• percentages
• rates
• probability
• portions of a whole

So equations often contain coefficients such as:

0.4x

or:

3/5 x

The good news is that these equations are solved using the same basic principle as other equations:

whatever operation you perform on one side of the equation, you must perform on the other side.

An equation behaves like a balanced scale. If both sides are equal, performing the same valid operation on both sides keeps them equal.

---

Review: Coefficients

A coefficient is a number multiplying a variable.

For example:

4x

has coefficient:

4

In:

0.7x

the coefficient is:

0.7

In:

3/5 x

the coefficient is:

3/5

A coefficient does not need to be a whole number.

---

Solving Equations with Decimal Coefficients

Consider:

0.5x = 6

One approach is simply to divide both sides by 0.5:

x = 6 ÷ 0.5

x = 12

That is perfectly valid.

However, equations containing several decimals can often be easier to solve if we first eliminate the decimals.

---

Clearing Decimals

Decimals are another way of representing fractions.

For example:

0.3 = 3/10

0.25 = 25/100

0.007 = 7/1000

Therefore, we can eliminate decimals by multiplying every term by a suitable power of 10.

Use:

• 10 if the greatest number of decimal places is 1
• 100 if the greatest number of decimal places is 2
• 1000 if the greatest number of decimal places is 3

---

Example 1: One Decimal Place

Solve:

0.4x + 1.2 = 5.6

The greatest number of decimal places is:

1

Multiply every term by:

10

10(0.4x + 1.2) = 10(5.6)

Simplify:

4x + 12 = 56

Subtract 12:

4x = 44

Divide by 4:

x = 11

---

Why Must We Multiply Every Term?

A very common error is to multiply only the decimal coefficient.

For example, starting with:

0.4x + 1.2 = 5.6

it would be wrong to write:

4x + 1.2 = 5.6

You must multiply the entire equation by 10:

10(0.4x + 1.2) = 10(5.6)

which gives:

4x + 12 = 56

Every term must be affected.

---

Example 2: Two Decimal Places

Solve:

0.06x + 0.02 = 0.25x − 1.5

The greatest number of decimal places is:

2

So multiply the entire equation by:

100

100(0.06x + 0.02) = 100(0.25x − 1.5)

Simplify:

6x + 2 = 25x − 150

Now the equation contains no decimals.

This method is especially useful when several decimal coefficients appear in the same equation.

---

Continue Solving Example 2

From:

6x + 2 = 25x − 150

Subtract 6x from both sides:

2 = 19x − 150

Add 150:

152 = 19x

Divide by 19:

x = 8

---

Another Decimal Example

Solve:

1.4x − 1.8 + 2.35x = 0.21

Because the greatest number of decimal places is 2, multiply every term by 100:

140x − 180 + 235x = 21

Combine like terms:

375x − 180 = 21

Add 180:

375x = 201

Divide:

x = 201/375

x = 0.536

This example shows how removing decimals can make the algebra easier to manage.

---

Do You Always Have to Eliminate Decimals?

No.

Consider:

0.8x − 5 = 7

You could solve directly:

Add 5:

0.8x = 12

Divide by 0.8:

x = 15

Or multiply the entire original equation by 10:

8x − 50 = 70

Then:

8x = 120

x = 15

Both methods are correct.

Choose the method that makes the arithmetic simplest.

---

Solving Equations with Fractional Coefficients

Now consider:

1/3 x = 5

The variable is multiplied by:

1/3

We can undo this by multiplying both sides by:

3

3(1/3 x) = 3(5)

Therefore:

x = 15

---

Example: Fractional Coefficient

Solve:

3/4 x = 12

Multiply both sides by the reciprocal of 3/4:

4/3

So:

4/3 × 3/4 x = 12 × 4/3

The fractions cancel:

x = 16

This works well when there is only one fractional coefficient.

---

Another Method: Divide by the Fraction

We could also write:

x = 12 ÷ 3/4

Dividing by a fraction means multiplying by its reciprocal:

x = 12 × 4/3

x = 16

Both approaches are equivalent.

---

Equations with Several Fractions

Consider:

x/2 + x/3 = 10

Solving while keeping the fractions is possible, but it is usually easier to eliminate them first.

The denominators are:

2 and 3

The least common denominator is:

6

Multiply every term by 6:

6(x/2) + 6(x/3) = 6(10)

Simplify:

3x + 2x = 60

Combine:

5x = 60

Therefore:

x = 12

---

Why Use the Least Common Denominator?

The least common denominator, or LCD, is the smallest number divisible by all denominators in the equation.

For denominators:

2 and 3

LCD = 6

For:

3 and 4

LCD = 12

For:

4 and 6

LCD = 12

For:

5 and 10

LCD = 10

Multiplying by the LCD usually clears all the fractions in one step.

---

Example: Different Denominators

Solve:

x/4 + 1/3 = 5/6

The denominators are:

4, 3, 6

The LCD is:

12

Multiply the entire equation by 12:

12(x/4) + 12(1/3) = 12(5/6)

Simplify:

3x + 4 = 10

Subtract 4:

3x = 6

Divide by 3:

x = 2

---

Why Does the LCD Work?

Look at:

12 × x/4

The 12 and 4 simplify:

12/4 = 3

so:

12 × x/4 = 3x

Similarly:

12 × 1/3 = 4

and:

12 × 5/6 = 10

The denominators disappear because the LCD contains each denominator as a factor.

---

Example with Brackets

Solve:

1/2(x − 4) = 1/3(x + 2)

The denominators are:

2 and 3

LCD:

6

Multiply both sides by 6:

6 × 1/2(x − 4) = 6 × 1/3(x + 2)

Simplify:

3(x − 4) = 2(x + 2)

Expand:

3x − 12 = 2x + 4

Subtract 2x:

x − 12 = 4

Add 12:

x = 16

---

Simplify Before Solving

Sometimes the best first step is not immediately clearing decimals or fractions.

Look first for:

• like terms
• brackets
• common factors
• fractions that can simplify
• decimals that are easy to combine

For example:

0.5x + 0.2x = 4.9

Combine like terms first:

0.7x = 4.9

Then divide:

x = 7

This is simpler than multiplying the original equation by 10 first.

---

Another Simplification Example

Solve:

3/5x + 2/5x = 10

Because the fractions already have the same denominator:

5/5x = 10

Therefore:

x = 10

Clearing fractions first would also work, but combining like terms is quicker.

---

Fractions Containing Expressions

Be careful when the numerator contains several terms.

For example:

(x + 3)/4 = 5

The entire expression:

x + 3

is being divided by 4.

Multiply both sides by 4:

x + 3 = 20

Subtract 3:

x = 17

---

Example with Fractions on Both Sides

Solve:

(2x + 1)/3 = (x + 5)/2

LCD:

6

Multiply both sides by 6:

2(2x + 1) = 3(x + 5)

Expand:

4x + 2 = 3x + 15

Subtract 3x:

x + 2 = 15

Therefore:

x = 13

---

Checking the Solution

A solution should make the original equation true.

Always substitute into the original equation, not only into a simplified version.

Suppose we solved:

x/4 + 1/3 = 5/6

and found:

x = 2

Check:

2/4 + 1/3

Simplify:

1/2 + 1/3

Common denominator 6:

3/6 + 2/6

= 5/6

The right-hand side is also:

5/6

Therefore:

x = 2 is correct.

---

Checking a Decimal Solution

Suppose:

0.4x + 1.2 = 5.6

and we found:

x = 11

Substitute:

0.4(11) + 1.2

= 4.4 + 1.2

= 5.6

The right side is:

5.6

Therefore the solution is correct.

---

Why Checking Is Especially Useful

Fractions and decimals increase the chance of arithmetic mistakes.

Common problems include:

• multiplying only some terms
• using the wrong LCD
• moving a decimal point incorrectly
• forgetting negative signs
• incorrectly dividing fractions

A quick substitution can often reveal the error immediately.

---

Worked Example: Fraction Equation

Solve:

3/2x + 1/2(x − 4) = 6

Both fractional terms have denominator 2.

Multiply every term by 2:

3x + (x − 4) = 12

Simplify:

3x + x − 4 = 12

4x − 4 = 12

Add 4:

4x = 16

Therefore:

x = 4

---

Worked Example: Decimal Equation

Solve:

0.03y − 0.2 = 1.15

The greatest number of decimal places is 2.

Multiply by 100:

3y − 20 = 115

Add 20:

3y = 135

Divide by 3:

y = 45

Check:

0.03(45) − 0.2

= 1.35 − 0.2

= 1.15

Correct.

---

Worked Example: Mixed Fractions and Decimals

Solve:

0.5x + x/4 = 6

You could convert everything into fractions:

1/2x + 1/4x = 6

LCD = 4.

Multiply by 4:

2x + x = 24

3x = 24

Therefore:

x = 8

Check:

0.5(8) + 8/4

= 4 + 2

= 6

Correct.

---

Fractions and Decimals Are Closely Related

Remember:

0.5 = 1/2

0.25 = 1/4

0.75 = 3/4

0.2 = 1/5

0.125 = 1/8

Sometimes converting between fractions and decimals makes an equation much easier.

For example:

0.25x + 1/2x = 9

Convert 0.25:

1/4x + 1/2x = 9

Multiply by 4:

x + 2x = 36

3x = 36

x = 12

---

Deciding Which Method to Use

There is no single required method.

You might:

Keep the decimals

Useful when they are easy to work with.

Multiply by 10, 100 or 1000

Useful when many decimals appear.

Multiply by the LCD

Useful when fractions have several denominators.

Convert decimals to fractions

Useful when the decimal has a simple fraction equivalent.

Simplify first

Useful when like terms can be combined easily.

The best method is usually the one that reduces arithmetic complexity.

---

Common Mistake: Forgetting a Term

Solve:

x/2 + 3 = x/4 + 8

LCD = 4.

Correct:

4(x/2) + 4(3) = 4(x/4) + 4(8)

which becomes:

2x + 12 = x + 32

Incorrect:

2x + 3 = x + 8

The whole equation must be multiplied.

---

Common Mistake: Using the Wrong LCD

For denominators:

4 and 6

a student may choose:

24

This works, but it is not the least common denominator.

The LCD is:

12

Using the smallest useful multiplier usually keeps the numbers easier to manage.

---

Common Mistake: Moving Decimal Points

Students sometimes write:

0.06x + 0.2 = 1.5

then "move the decimal point" without treating every term consistently.

A safer mathematical description is:

multiply the entire equation by 100

This preserves equality and gives a clear reason why the decimals disappear.

---

Common Mistake: Dividing by a Fraction

Remember:

divide by a fraction = multiply by its reciprocal

For example:

x = 10 ÷ 2/5

becomes:

x = 10 × 5/2

x = 25

Do not divide numerator by numerator and denominator by denominator.

---

Common Mistake: Checking Only Part of the Equation

Suppose you find:

x = 6

Do not simply check whether one intermediate step works.

Substitute x = 6 into the original equation.

The original equation is the one your answer must satisfy.

---

Real-World Example: Money

Suppose a service charges:

$4.50 plus $0.75 per kilometre

and the total cost is:

$15.75

Equation:

0.75x + 4.50 = 15.75

Subtract 4.50:

0.75x = 11.25

Divide by 0.75:

x = 15

So the trip was:

15 km

Decimal equations often arise naturally in money problems.

---

Real-World Example: Fraction of a Quantity

Suppose:

3/5 of a number plus 4 equals 19

Equation:

3/5x + 4 = 19

Subtract 4:

3/5x = 15

Multiply by 5/3:

x = 25

Check:

3/5(25) + 4

= 15 + 4

= 19

---

Real-World Example: Measurement

A piece of material has:

0.25x + 1.5 = 5.0

where x represents an unknown length.

Subtract 1.5:

0.25x = 3.5

Divide by 0.25:

x = 14

Or multiply the original equation by 100:

25x + 150 = 500

which gives the same answer.

---

A Reliable Strategy for Decimal Equations

When solving an equation containing decimals:

1. Simplify like terms if useful.
2. Identify the greatest number of decimal places.
3. Multiply every term by 10, 100, 1000, etc., if clearing decimals will help.
4. Simplify.
5. Isolate the variable.
6. Solve.
7. Substitute the answer into the original equation.

---

A Reliable Strategy for Fraction Equations

When solving an equation containing fractions:

1. Simplify if possible.
2. Identify every denominator.
3. Find the least common denominator.
4. Multiply every term by the LCD.
5. Cancel the denominators.
6. Simplify the resulting equation.
7. Solve for the variable.
8. Check in the original equation.

---

Challenge Example

Solve:

(2x − 1)/3 + (x + 2)/4 = 5

Denominators:

3 and 4

LCD:

12

Multiply every term by 12:

4(2x − 1) + 3(x + 2) = 60

Expand:

8x − 4 + 3x + 6 = 60

Combine like terms:

11x + 2 = 60

Subtract 2:

11x = 58

Therefore:

x = 58/11

This is already an exact answer.

As a decimal:

x ≈ 5.27

Unless instructed otherwise, an exact fractional answer is often preferable.

---

Challenge Example with Decimals

Solve:

0.15x + 0.35 = 0.05x + 1.75

Multiply every term by 100:

15x + 35 = 5x + 175

Subtract 5x:

10x + 35 = 175

Subtract 35:

10x = 140

Therefore:

x = 14

Check:

Left:

0.15(14) + 0.35 = 2.45

Right:

0.05(14) + 1.75 = 2.45

Correct.

---

Did You Know?

Clearing fractions and decimals does not change the solution of an equation.

Why?

Because we multiply both sides by the same nonzero number.

For example:

0.5x = 3

and:

5x = 30

have exactly the same solution:

x = 6

The second equation is simply a more convenient equivalent form.

---

Key Terms

Equation – A mathematical statement showing that two expressions are equal.

Coefficient – A number multiplying a variable.

Decimal coefficient – A coefficient written as a decimal, such as 0.6.

Fractional coefficient – A coefficient written as a fraction, such as 3/4.

Denominator – The bottom number of a fraction.

Least common denominator (LCD) – The smallest common multiple of all denominators involved.

Reciprocal – A fraction turned upside down; for example, the reciprocal of 3/4 is 4/3.

Equivalent equation – An equation with the same solution set as another equation.

Substitution – Replacing a variable with a value.

Verify – Check that a proposed solution makes the original equation true.

---

Key Takeaways

• Equations can contain both fractional and decimal coefficients.
• The same balance principle used for ordinary equations still applies.
• Decimal equations can often be simplified by multiplying the entire equation by 10, 100, 1000, or another suitable power of 10.
• Fraction equations can often be simplified by multiplying every term by the least common denominator.
• Every term on both sides must be multiplied.
• Simplifying before solving can sometimes make the equation much easier.
• Dividing by a fraction is equivalent to multiplying by its reciprocal.
• Fractions and decimals can often be converted into one another.
• Exact fractional answers are often preferable to rounded decimals.
• Always verify your solution by substituting it into the original equation.
• A strong strategy is:

simplify → clear fractions/decimals if helpful → solve → verify.