Lesson 2.5: Solve Equations with Fractions or Decimals

New Concept

This lesson introduces a powerful technique for equations with fractions or decimals. By multiplying the entire equation by the least common denominator (LCD), you can "clear" these terms, converting the problem into a simpler integer equation you already know how to solve.

What’s next

Soon, you'll tackle practice cards that guide you through clearing fractions and decimals, followed by challenge problems to test your new skills.

Property

An alternate method to solve equations with fractions is to eliminate the fractions. We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called “clearing” the equation of fractions.

Strategy to solve equations with fraction coefficients.
Step 1. Find the least common denominator of all the fractions in the equation.
Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.
Step 3. Solve using the General Strategy for Solving Linear Equations.

Examples

• Solve $\frac{1}{5}y - \frac{1}{2} = \frac{3}{10}$. The LCD is 10. Multiplying by 10 gives $10(\frac{1}{5}y) - 10(\frac{1}{2}) = 10(\frac{3}{10})$, which simplifies to $2y - 5 = 3$. Solving gives $y = 4$.

Property

When a fraction contains a variable expression in the numerator, such as $\frac{5x-3}{4}$, treat the numerator as a group. When clearing fractions, the LCD multiplies the entire numerator. For $\frac{5x-3}{4} = \frac{x}{2}$, multiplying by the LCD of 4 gives $4(\frac{5x-3}{4}) = 4(\frac{x}{2})$, simplifying to $5x-3 = 2x$.

Examples

• Solve $\frac{4q+3}{2} = 5$. The LCD is 2. Multiplying by 2 gives $2(\frac{4q+3}{2}) = 2(5)$ which becomes $4q+3 = 10$. So, $4q=7$ and $q = \frac{7}{4}$.

• Solve $\frac{5x-8}{5} = \frac{2x}{10}$. The LCD is 10. Multiplying by 10 gives $10(\frac{5x-8}{5}) = 10(\frac{2x}{10})$ which becomes $2(5x-8) = 2x$. This is $10x-16=2x$, so $8x=16$ and $x=2$.

Property

When an equation involves a fraction multiplied by a quantity in parentheses, you can either distribute the fraction first and then clear all resulting fractions with the LCD, or you can clear the initial fraction by multiplying by its denominator first. For example, in $\frac{1}{2}(y-5) = \frac{1}{4}(y-1)$, you can distribute to get $\frac{1}{2}y - \frac{5}{2} = \frac{1}{4}y - \frac{1}{4}$ and then multiply by the LCD, 4.

Examples

• Solve $-5 = \frac{1}{4}(8x+4)$. Distribute first: $-5 = \frac{1}{4}(8x) + \frac{1}{4}(4)$, which simplifies to $-5 = 2x+1$. Then solve for $x$: $-6 = 2x$, so $x=-3$.

• Solve $\frac{1}{5}(q+3) = \frac{1}{2}(q-3)$. The LCD is 10. Multiply by 10: $10 \cdot \frac{1}{5}(q+3) = 10 \cdot \frac{1}{2}(q-3)$ becomes $2(q+3) = 5(q-3)$. Then $2q+6 = 5q-15$, so $21 = 3q$ and $q=7$.

Property

Some equations have decimals in them. Decimals can be expressed as fractions. For example, $0.3 = \frac{3}{10}$ and $0.17 = \frac{17}{100}$. So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator. For $0.06x + 0.02 = 0.25x - 1.5$, the LCD of the equivalent fractions is 100. Multiplying by 100 will clear the decimals.

Examples

• Solve $0.05a + 0.03 = 0.2a - 1.2$. The place with the most decimals is hundredths, so multiply by 100. This gives $5a + 3 = 20a - 120$. Solving gives $123 = 15a$, so $a = 8.2$.

• Solve $0.14y + 0.07 = 0.3y - 1.53$. Multiply by 100: $14y + 7 = 30y - 153$. Solving gives $160 = 16y$, so $y=10$.

Property

In equations with decimals and parentheses, it is often best to use the Distributive Property first. After distributing, combine any like terms. Then, clear the decimals from the simplified equation by multiplying by the appropriate power of 10. For $0.25x + 0.05(x+3) = 2.85$, distribute to get $0.25x + 0.05x + 0.15 = 2.85$. Combine like terms to get $0.30x + 0.15 = 2.85$.

Examples

• Solve $0.25x + 0.05(x+3) = 2.85$. Distribute: $0.25x + 0.05x + 0.15 = 2.85$. Combine terms: $0.30x + 0.15 = 2.85$. Multiply by 100: $30x + 15 = 285$. Solve: $30x=270$, so $x=9$.

• Solve $0.4(y-2) + 0.1 = 3.3$. Distribute: $0.4y - 0.8 + 0.1 = 3.3$. Combine terms: $0.4y - 0.7 = 3.3$. Multiply by 10: $4y - 7 = 33$. Solve: $4y=40$, so $y=10$.