8.4 Solve Equations with Fraction or Decimal Coefficients

Solving Equations with Fractions and Decimals

Solving equations with fractions and decimals can feel messy, but there's a simple trick: you can clear them out entirely before solving. By multiplying strategically, you turn a fraction or decimal equation into a regular equation with whole numbers.

This section covers how to clear denominators, handle decimal coefficients, and convert between fractions and decimals so you can pick whichever approach makes a problem easier.

Solving Equations with Fractions and Decimals

Clearing Denominators in Equations

The core idea here is simple: if you multiply every term in an equation by the least common denominator (LCD), all the fractions disappear. Then you just solve a normal equation.

How to clear denominators, step by step:

1. Identify all the denominators in the equation.
2. Find the LCD of those denominators. The LCD is the least common multiple (LCM) of all the denominators. For example, if your denominators are 3, 4, and 6, the LCD is 12.
3. Multiply every term on both sides by the LCD. This cancels out every denominator.
4. Simplify and solve the resulting equation using standard methods.

Example: Solve $\frac{2x}{3} + \frac{1}{4} = \frac{5}{6}$

• The denominators are 3, 4, and 6. The LCD is 12.
• Multiply every term by 12:
  • $12 \cdot \frac{2x}{3} = 8x$
  • $12 \cdot \frac{1}{4} = 3$
  • $12 \cdot \frac{5}{6} = 10$
• The equation becomes $8x + 3 = 10$
• Subtract 3 from both sides: $8x = 7$
• Divide both sides by 8: $x = \frac{7}{8}$

The key step is making sure you multiply every single term by the LCD. If you miss one, the equation won't balance.

Strategies for Decimal Coefficients

Decimals work the same way as fractions, just with a different clearing strategy. Instead of the LCD, you multiply every term by a power of 10 large enough to turn all decimals into whole numbers.

How to clear decimals, step by step:

1. Look at the decimal with the most decimal places. If the smallest value goes to the hundredths place, you'll multiply by 100. If it goes to the tenths place, multiply by 10.
2. Multiply every term on both sides by that power of 10.
3. Simplify and solve the resulting whole-number equation.

Example: Solve $0.3x - 0.07 = 1.2$

• The most decimal places here is two (in 0.07), so multiply everything by 100:
  • $100 \cdot 0.3x = 30x$
  • $100 \cdot 0.07 = 7$
  • $100 \cdot 1.2 = 120$
• The equation becomes $30x - 7 = 120$
• Add 7 to both sides: $30x = 127$
• Divide both sides by 30: $x = \frac{127}{30}$

You don't have to clear decimals. For a simple equation like $0.2x = 1.4$, you can just divide both sides by 0.2 directly. Clearing is most helpful when the decimals make the arithmetic annoying.

Fraction-Decimal Conversions for Equations

Sometimes an equation mixes fractions and decimals together. When that happens, convert everything to one form so you can work with it consistently.

Converting fractions to decimals: Divide the numerator by the denominator.

• $\frac{3}{4} = 3 \div 4 = 0.75$
• $\frac{1}{3} = 1 \div 3 = 0.333...$ (repeating decimal, so fractions might be the better choice here)

Converting decimals to fractions: Write the decimal over the appropriate power of 10, then simplify.

• $0.6 = \frac{6}{10} = \frac{3}{5}$
• $0.75 = \frac{75}{100} = \frac{3}{4}$

Which form should you choose? Pick whichever makes the math cleaner:

• If all the fractions convert to nice terminating decimals (like $\frac{1}{4} = 0.25$), decimals might be easier.
• If you'd get repeating decimals (like $\frac{1}{3}$), stick with fractions and clear the denominators instead.

Example: Solve $\frac{2x}{5} + 0.3 = \frac{3}{4}$

Converting everything to decimals: $0.4x + 0.3 = 0.75$. Now multiply by 100: $40x + 30 = 75$. Subtract 30: $40x = 45$. Divide by 40: $x = \frac{45}{40} = \frac{9}{8}$.

Working with Algebraic Fractions

Algebraic fractions have variables in the numerator, the denominator, or both (like $\frac{x+1}{3}$ or $\frac{5}{x-2}$).

You solve these using the same clearing strategy: multiply both sides by the LCD. But there's one extra thing to watch for. If a variable appears in a denominator, certain values of that variable would make the denominator zero, which is undefined. Always check that your solution doesn't create a zero in any denominator.

For Pre-Algebra, most of your algebraic fractions will have variables only in the numerator (like $\frac{2x}{3}$), so this concern won't come up often. But it's a good habit to build now for when equations get more complex later.