Equations with Fractions & Decimals

TipLearning Objectives
  • Solve linear equations that contain fractions.
  • Use the LCD to clear fractions.
  • Solve linear equations with decimals by scaling.
  • Avoid errors with negative fractions and decimals.

Key Ideas

For equations with fractions:

  • Find the least common denominator (LCD).
  • Multiply both sides of the equation by the LCD to clear fractions.

For equations with decimals:

  • Multiply both sides by a power of 10 (10, 100, 1000, …) to remove decimals.

After clearing, solve like a normal linear equation.

Common Problem Types

1. Fractions with Same Denominator

Solve:
\[ \frac{x}{5} + \frac{2}{5} = 3 \]

Multiply both sides by 5:
\[ x + 2 = 15 \Rightarrow x = 13 \]

2. Fractions with Different Denominators

Solve:
\[ \frac{x}{3} - \frac{1}{2} = 1 \]

LCD of 3 and 2 is 6.
Multiply both sides by 6:
\[ 2x - 3 = 6 \]

Add 3: \(2x = 9\)
Divide: \(x = \dfrac{9}{2}\)

3. Decimals

Solve:
\[ 0.4x + 1.2 = 3 \]

Multiply both sides by 10:
\[ 4x + 12 = 30 \]

Subtract 12: \(4x = 18\)
Divide: \(x = 4.5\)

Strategies

  • For fractions, always clear denominators first if possible.
  • Choose the smallest LCD to keep numbers manageable.
  • For decimals, match the largest number of decimal places and scale.
  • Rewrite negative fractions carefully (e.g., \(-\dfrac{3}{4}x\)).

Worked Examples

Example 1

Solve:
\[ \frac{2x}{7} + \frac{1}{7} = 3 \]

Multiply both sides by 7:
\[ 2x + 1 = 21 \]

Subtract 1: \(2x = 20\)
Divide: \(x = 10\)

Example 2

Solve:
\[ 0.25x - 0.5 = 1.75 \]

Multiply both sides by 100:
\[ 25x - 50 = 175 \]

Add 50: \(25x = 225\)
Divide: \(x = 9\)

Common Mistakes

WarningCommon Mistakes
  • Multiplying some terms by the LCD but not all.
  • Forgetting to distribute the LCD across parentheses.
  • Handling negative fractions incorrectly.
  • Scaling one side for decimals but not the other.

Practice Problems

  1. \(\dfrac{x}{4} + \dfrac{1}{4} = 3\)
  2. \(\dfrac{x}{2} - \dfrac{3}{4} = 1\)
  3. \(0.3x = 4.5\)
  4. \(0.6x + 1.8 = 4.2\)
  5. \(\dfrac{2x}{5} + \dfrac{1}{10} = \dfrac{3}{2}\)

1. Multiply both sides by 4:
\(x + 1 = 12\) → \(x = 11\).

2. LCD is 4. Multiply by 4:
\(2x - 3 = 4\) → \(2x = 7\) → \(x = \dfrac{7}{2}\).

3. \(0.3x = 4.5\)
Multiply both sides by 10: \(3x = 45\) → \(x = 15\).

4. \(0.6x + 1.8 = 4.2\)
Multiply both sides by 10: \(6x + 18 = 42\)
Subtract 18: \(6x = 24\) → \(x = 4\).

5. LCD for 5, 10, 2 is 10:
\(\dfrac{2x}{5} + \dfrac{1}{10} = \dfrac{3}{2}\)

Multiply both sides by 10:
\(4x + 1 = 15\)
Subtract 1: \(4x = 14\) → \(x = \dfrac{7}{2}\).