How to Solve Equations with Fractions and Decimals in 3 Simple Steps

Fractions and decimals make equations harder than they need to be. Many students stumble on finding the LCD, forget to multiply every term, or feel unsure when converting decimals. By Grade 6-7 at Think Academy (Grade 8 in school), mastering these steps is key to moving forward with algebra. This guide shows the exact process so mistakes don’t hold students back.

Solving Equations with Fractions in 3 Steps

Example:

\[\frac{x}{3} + \frac{1}{6} = \frac{5}{12}\]

Step 1: Multiply by the Least Common Denominator (LCD)

The denominators are 3, 6, and 12. To find their Least Common Denominator (LCD), let’s list out the multiples of them and spot the match:

• The denominators are 3, 6, and 12.
• Multiples of 3: 3, 6, 9, 12, 15…
• Multiples of 6: 6, 12, 18, 24…
• Multiples of 12: 12, 24…

The first number they have in common is 12. 12 is called the Least Common Denominator (LCD) of 3, 6, and 12, which is the smallest number all denominators can evenly divide into.

Multiply both sides of the equation by 12 to remove the fractions:

\[\frac{x}{3} \cdot 12 + \frac{1}{6}\cdot 12 = \frac{5}{12}\cdot 12\]

This simplifies to:

4𝑥 + 2 = 5

Step 2: Move Variables to the Left Side, Constants to the Right

Move the variable terms to the left

In this example, the variable terms are already to the left.

Move the constants to the right

Subtract 2 from both sides:

Therefore:

4𝑥 = 3

Step 3: Solve for the Variable

Divide both sides of the equation by the coefficient 4 to get the final answer:

Therefore:

\[x = \frac{3}{4}\]

Solving Equations with Decimals in 3 Steps

Example:

0.4𝑦 − 1.2 = 2.8

Step 1: Turn Decimals into Whole Numbers

Since the largest decimal place is one decimal digit, multiply both sides of the equation by 10:

0.4𝑦·10 − 1.2·10 = 2.8·10

This simplifies to:

4𝑦 − 12 = 28

Step 2: Move Variables to the Left Side, Constants to the Right

Move the variable terms to the left

In this example, the variable terms are already to the left.

Move the constants to the right

Add 12 from both sides:

Therefore:

4𝑦 = 40

Step 3: Solve for the Variable

Divide both sides of the equation by the coefficient 4 to get the final answer:

Therefore:

𝑦 = 10

Example Problems: How to Solve Equations with Fractions and Decimals

Example 1

Find the solution to the equation.

\[\frac{x}{5} – \frac{2}{15} = \frac{4}{3}\]

Solution:

Step 1: LCD of 5, 15, 3is 15. Multiply both sides of the equation by 15.

\[15 \cdot \left(\frac{x}{5} – \frac{2}{15}\right)= 15 \cdot \frac{4}{3}\]

\[15 \cdot \frac{x}{5} – 15 \cdot \frac{2}{15}= 15 \cdot \frac{4}{3}\]

3𝑥 − 2 = 20

Step 2: Move variables left, constants right

3𝑥 − 2 + 2 = 20 + 2

3𝑥 = 22

Step 3: Solve for the variable

3𝑥 ÷ 3 = 22 ÷ 3

\[x = \frac{22}{3}\]

Example 2

Find the solution to the equation.

5.1𝑥 − 2.9 = 3.4𝑥 + 5.6

Solution:

Step 1: Turn Decimals into Whole Numbers

10·(5.1𝑥 − 2.9) = 10·(3.4𝑥 + 5.6)

10 · 5.1𝑥 − 10 · 2.9 = 10 · 3.4𝑥 + 10 · 5.6

51𝑥 − 29 = 34𝑥 + 56

Step 2: Move variables left, constants right

51𝑥 – 29 – 34 𝑥 = 34𝑥 + 56 – 34𝑥

17𝑥 – 29 = 56

17𝑥 – 29 + 29 = 56 + 29

17𝑥 = 85

Step 3: Solve for the variable

17𝑥 ÷ 17 = 85 ÷ 17

𝑥 = 5

Summary: How to Solve Equations with Fractions and Decimals

• Fractions: Multiply through by the LCD to make the equation fraction-free.
• Decimals: Multiply by 10, 100…(depending on decimal places) to make the equation whole-numbered.
• Always move variables to one side and constants to the other before solving.
• The process is the same for both: clear the fractions/decimals → isolate the variable → solve.
• For extra safety, always substitute our solution back into the original equation to make sure it works perfectly.

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More related articles

• How to Solve Equations with Variables on Both Sides in 3 Simple Steps
• How to Solve Equations with Parentheses in 4 Simple Steps
• Area of a Circle: Definition, Formula and Example

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