Equations with Fractions and Decimals

Fractions and decimals in equations make many students nervous, but there is a powerful shortcut: clear them out entirely before you start solving. By multiplying every term by the right number, you can transform a messy equation into a clean one with whole-number coefficients. This page teaches you when and how to do that, plus how to handle situations where working directly with fractions is the smarter move.

Strategy 1: Clearing Fractions with the LCD

The Least Common Denominator (LCD) is the smallest number that all denominators divide into evenly. When you multiply every term in the equation by the LCD, every fraction becomes a whole number.

Steps

1. Identify every denominator in the equation.
2. Find the LCD of those denominators.
3. Multiply every term on both sides by the LCD.
4. Simplify — the fractions vanish.
5. Solve the resulting equation using standard methods.

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

Step 1 — Identify denominators: 3 and 6.

Step 2 — Find the LCD: The LCD of 3 and 6 is 6.

Step 3 — Multiply every term by 6:

$6 \cdot \frac{x}{3} + 6 \cdot \frac{x}{6} = 6 \cdot 5$

$2x + x = 30$

Step 4 — Combine like terms:

$3x = 30$

Step 5 — Divide by 3:

$x = 10$

Answer: $x = 10$

Check: $\frac{10}{3} + \frac{10}{6} = \frac{20}{6} + \frac{10}{6} = \frac{30}{6} = 5$ . Correct.

Example 2: $\frac{2x + 1}{4} = \frac{x - 3}{2}$

Denominators: 4 and 2. LCD = 4.

Multiply every term by 4:

$4 \cdot \frac{2x + 1}{4} = 4 \cdot \frac{x - 3}{2}$

$2x + 1 = 2(x - 3)$

$2x + 1 = 2x - 6$

Subtract $2x$: $1 = -6$

This is false, so no solution exists.

Example 3: $\frac{3}{4}x - \frac{1}{2} = \frac{5}{8}x + 1$

Denominators: 4, 2, and 8. LCD = 8.

Multiply every term by 8:

$8 \cdot \frac{3}{4}x - 8 \cdot \frac{1}{2} = 8 \cdot \frac{5}{8}x + 8 \cdot 1$

$6x - 4 = 5x + 8$

Subtract $5x$: $x - 4 = 8$

Add 4: $x = 12$

Answer: $x = 12$

Check: Left: $\frac{3}{4}(12) - \frac{1}{2} = 9 - 0.5 = 8.5$. Right: $\frac{5}{8}(12) + 1 = 7.5 + 1 = 8.5$ . Correct.

Strategy 2: Clearing Decimals by Multiplying by Powers of 10

Decimals are just fractions in disguise — tenths, hundredths, and so on. Multiplying by the right power of 10 clears them all at once.

Steps

1. Find the term with the most decimal places.
2. Multiply every term by $10^n$ where $n$ is that number of decimal places.
3. Solve the resulting whole-number equation.

Example 4: $0.3x + 1.5 = 4.2$

Most decimal places: 1 (tenths). Multiply everything by 10.

$3x + 15 = 42$

Subtract 15: $3x = 27$

Divide by 3: $x = 9$

Answer: $x = 9$

Example 5: $0.25x - 0.10 = 0.15x + 0.40$

Most decimal places: 2 (hundredths). Multiply everything by 100.

$25x - 10 = 15x + 40$

Subtract $15x$: $10x - 10 = 40$

Add 10: $10x = 50$

Divide by 10: $x = 5$

Answer: $x = 5$

Check: Left: $0.25(5) - 0.10 = 1.25 - 0.10 = 1.15$. Right: $0.15(5) + 0.40 = 0.75 + 0.40 = 1.15$ . Correct.

Example 6: $1.2(x + 5) = 0.6x + 9$

Multiply everything by 10 first:

$12(x + 5) = 6x + 90$

Distribute: $12x + 60 = 6x + 90$

Subtract $6x$: $6x + 60 = 90$

Subtract 60: $6x = 30$

Divide by 6: $x = 5$

Answer: $x = 5$

When to Clear vs. When to Keep Fractions

Clearing fractions is usually the best approach, but there are situations where working with fractions directly is simpler.

Clear fractions when:

• Multiple different denominators appear
• The equation is long and you want cleaner arithmetic
• You tend to make errors with fraction operations

Keep fractions when:

• There is only one fraction and it is easy to handle
• The LCD would be very large, making the multiplied terms unwieldy
• The fraction reduces to a simple expression quickly

Example — keeping the fraction is easier: Solve $\frac{x}{2} = 7$.

Just multiply both sides by 2: $x = 14$. No need for a multi-step LCD process.

Mixed Fractions and Decimals in One Equation

Occasionally you encounter both fractions and decimals in the same equation. Convert everything to one form first, then clear.

Example 7: $\frac{x}{4} + 0.5 = 2.75$

Convert the decimal to a fraction (or the fraction to a decimal — either works):

Converting $0.5 = \frac{1}{2}$ and $2.75 = \frac{11}{4}$:

$\frac{x}{4} + \frac{1}{2} = \frac{11}{4}$

LCD = 4. Multiply every term by 4:

$x + 2 = 11$

$x = 9$

Answer: $x = 9$

Alternatively, convert $\frac{x}{4}$ to $0.25x$ and solve with decimals:

$0.25x + 0.5 = 2.75$

Subtract 0.5: $0.25x = 2.25$

Divide by 0.25: $x = 9$

Same answer either way.

Real-World Application: Nursing — Medication Dosage Calculation

A nurse needs to administer two liquid medications into the same IV line. The first medication requires $\frac{3}{4}$ mL and the second requires $\frac{x}{6}$ mL. The total volume must equal $\frac{13}{12}$ mL. How much of the second medication is needed?

$\frac{3}{4} + \frac{x}{6} = \frac{13}{12}$

LCD of 4, 6, and 12 is 12. Multiply every term by 12:

$12 \cdot \frac{3}{4} + 12 \cdot \frac{x}{6} = 12 \cdot \frac{13}{12}$

$9 + 2x = 13$

$2x = 4$

$x = 2$

Answer: The second medication requires $\frac{2}{6} = \frac{1}{3}$ mL. Check: $\frac{3}{4} + \frac{1}{3} = \frac{9}{12} + \frac{4}{12} = \frac{13}{12}$ mL. Confirmed.

Real-World Application: Carpentry — Cutting Lumber to Size

A carpenter needs to cut a board so that one piece is $\frac{2}{3}$ the length of the other piece. The total board is 7.5 feet long. What is the length of each piece?

Let $x$ = the longer piece. The shorter piece is $\frac{2}{3}x$.

$x + \frac{2}{3}x = 7.5$

Convert 7.5 to a fraction: $\frac{15}{2}$

$x + \frac{2}{3}x = \frac{15}{2}$

LCD of 1, 3, and 2 is 6. Multiply every term by 6:

$6x + 4x = 45$

$10x = 45$

$x = 4.5$

The shorter piece: $\frac{2}{3}(4.5) = 3$

Answer: The longer piece is 4.5 feet and the shorter piece is 3 feet. Check: $4.5 + 3 = 7.5$ . Correct.

Common Mistakes to Avoid

1. Multiplying only some terms by the LCD. Every single term on both sides must be multiplied — including stand-alone constants like the 5 in $\frac{x}{3} + 2 = 5$.
2. Using a common multiple that is not the least. A larger multiple works but creates bigger numbers and more room for arithmetic errors. Always find the LCD.
3. Forgetting to distribute when a fraction multiplies a grouped expression. In $\frac{2x + 1}{4}$, multiplying by 4 clears the denominator from the entire numerator: you get $2x + 1$, not $2x + \frac{1}{4}$.
4. Losing the sign on a negative fraction. $-\frac{3}{5}$ times the LCD 5 gives $-3$, not $3$.
5. Rounding decimals prematurely. When clearing decimals, multiply first, then do arithmetic with whole numbers. Rounding before clearing introduces errors.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Clearing fractions by multiplying every term by the LCD converts a fractional equation into a simpler whole-number equation
• Clearing decimals works the same way — multiply by $10$, $100$, or $1000$ depending on the most decimal places present
• Always multiply every term on both sides, including stand-alone constants
• When fractions and decimals appear together, convert to one form first, then clear
• After clearing, solve using the standard multi-step equation strategy
• Keeping fractions is sometimes simpler for very basic equations with a single fraction

Return to Algebra for more topics in this section.