2.5 Solve Equations with Fractions or Decimals

Solving Linear Equations with Fractions or Decimals

Fractions and decimals in equations look intimidating, but there's a simple trick: you can clear them out entirely before you start solving. Multiply strategically, and you're left with a straightforward equation using whole numbers.

Clearing Fractions in Equations

The core idea is to multiply every term in the equation by the least common denominator (LCD) of all the fractions. The LCD is the smallest number that all denominators divide into evenly. This eliminates every fraction in one step.

How to do it:

1. Find the LCD of all denominators in the equation.
2. Multiply every term on both sides by the LCD.
3. Simplify. You should now have an equation with only integers.
4. Solve using the usual steps: combine like terms, isolate the variable, divide.

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

• The denominators are 3 and 4, so the LCD is 12.
• Multiply every term by 12:

$12 \cdot \frac{2}{3}x + 12 \cdot \frac{1}{4} = 12 \cdot 5$

$8x + 3 = 60$

• Subtract 3 from both sides: $8x = 57$
• Divide both sides by 8: $x = \frac{57}{8}$

Solving Equations with Decimals

Decimals are just fractions in disguise (0.25 is $\frac{25}{100}$), so you could convert them to fractions and use the LCD method above. But there's a faster approach: multiply both sides by a power of 10 to turn all decimals into whole numbers.

How to do it:

1. Count the most decimal places any number in the equation has.
2. Multiply every term on both sides by the matching power of 10 (1 decimal place → multiply by 10; 2 decimal places → multiply by 100; and so on).
3. Solve the resulting whole-number equation normally.

Example: Solve $0.5x + 0.25 = 1$

• The most decimal places here is 2 (in 0.25), so multiply everything by 100:

$100 \cdot 0.5x + 100 \cdot 0.25 = 100 \cdot 1$

$50x + 25 = 100$

• Subtract 25 from both sides: $50x = 75$
• Divide both sides by 50: $x = 1.5$

You could also multiply by 10 if you first simplify 0.25 to $\frac{1}{4}$, but multiplying by 100 is usually quicker and less error-prone.

Strategies for Fractional Word Problems

Word problems with fractions follow the same translation process as any other word problem. The fractions just add one extra solving step.

1. Identify the unknown. Read the problem and assign a variable (like $x$) to whatever you're looking for.
2. Translate into an equation. Turn the relationships described in words into math. Phrases like "half of" become $\frac{1}{2}x$, and "one-third of" becomes $\frac{1}{3}x$.
3. Clear fractions and solve. Multiply both sides by the LCD, then solve the simpler equation.
4. Check your answer against the original problem. Make sure it actually makes sense in context.

Example: "Half of the apples plus one-third of the apples equals 10. How many apples are there?"

• Let $x$ = total number of apples.
• Equation: $\frac{1}{2}x + \frac{1}{3}x = 10$
• The LCD of 2 and 3 is 6. Multiply every term by 6:

$3x + 2x = 60$

$5x = 60$

$x = 12$

• Check: half of 12 is 6, one-third of 12 is 4, and $6 + 4 = 10$. It works.

Key Concepts to Remember

• Clearing fractions and decimals is optional but highly recommended. You can solve without clearing them, but whole-number equations are much easier to work with and less prone to arithmetic mistakes.
• The LCD must work for every denominator in the equation. If you miss one, you'll still have fractions left over.
• Algebraic properties still apply after clearing. You're using the same distributive property, combining like terms, and properties of equality as always. The clearing step just makes the numbers friendlier.