2.5 Solve Equations with Fractions or Decimals

Learning objectives

Learning objectives

• Solve equations with fraction coefficients

• Solve equations with decimal coefficients

Solve Equations with Fraction Coefficients

Let’s use the general strategy for solving linear equations introduced earlier to solve the equation, $\frac{1}{8}x+\frac{1}{2}=\frac{1}{4}$.

This method worked fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates 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.

Let’s solve a similar equation, but this time use the method that eliminates the fractions.

Example 2.48

Notice in Example 2.48, once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

Example 2.49

Solution

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

In the next example, we again have variables on both sides of the equation.

Example 2.50

Solution

In the next example, we start by using the Distributive Property. This step clears the fractions right away.

Example 2.51

Solution

In the next example, even after distributing, we still have fractions to clear.

Example 2.52

Solution

Example 2.53

Solution

Example 2.54

Solution

Example 2.55

Solution

Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also 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.

Example 2.56

Solution

Look at the decimals and think of the equivalent fractions.

Notice, the LCD is 100.

By multiplying by the LCD, we will clear the decimals from the equation.

The next example uses an equation that is typical of the money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.

Example 2.57

Solution

Section 2.5 Exercises

Section 2.5 Exercises

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

1. Solve: $\frac{1}{4}x+\frac{1}{2}=\frac{5}{8}$.

2. Solve: $\frac{1}{8}x+\frac{1}{2}=\frac{1}{4}$.

3. Solve: $7=\frac{1}{2}x+\frac{3}{4}x-\frac{2}{3}x$.

4. Solve: $-1=\frac{1}{2}u+\frac{1}{4}u-\frac{2}{3}u$.

1. Solve: $x+\frac{1}{3}=\frac{1}{6}x-\frac{1}{2}$.

2. Solve: $c+\frac{3}{4}=\frac{1}{2}c-\frac{1}{4}$.

3. Solve: $-11=\frac{1}{2}(6p+2)$.

4. Solve: $8=\frac{1}{3}(9q+6)$.

1. Solve: $\frac{1}{5}(n+3)=\frac{1}{4}(n+2)$.

2. Solve: $\frac{1}{2}(m-3)=\frac{1}{4}(m-7)$.

3. Solve: $\frac{4y-7}{3}=\frac{y}{6}$.

4. Solve: $\frac{-2z-5}{4}=\frac{z}{8}$.

1. Solve: $\frac{b}{10}+2=\frac{b}{4}+5$.

2. Solve: $\frac{c}{6}+3=\frac{c}{3}+4$.

3. Solve: $\frac{3r+5}{6}+1=\frac{4r+3}{3}$.

4. Solve: $\frac{2s+3}{2}+1=\frac{3s+2}{4}$.

1. Solve: $0.14h+0.12=0.35h-2.4$.

2. Solve: $0.65k-0.1=0.4k-0.35$.

3. Solve: $0.25n+0.05(n+5)=2.95$.

4. Solve: $0.10d+0.05(d-5)=2.15$.

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

2. $\frac{5}{6}y-\frac{1}{3}=-\frac{7}{6}$

3. $\frac{5}{8}b+\frac{1}{2}=-\frac{3}{4}$

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

1. $\frac{5}{6}n-\frac{1}{4}n-\frac{1}{2}n=-2$

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

3. $\frac{3}{2}z+\frac{1}{3}=z-\frac{2}{3}$

4. $\frac{1}{2}a-\frac{1}{4}=\frac{1}{6}a+\frac{1}{12}$

1. $\frac{1}{3}x+\frac{2}{5}=\frac{1}{5}x-\frac{2}{5}$

2. $1=\frac{1}{5}(15x-10)$

3. $\frac{1}{5}(q+3)=\frac{1}{2}(q-3)$

4. $\frac{1}{2}(x+4)=\frac{3}{4}$

1. $\frac{1}{3}(x+5)=\frac{5}{6}$

2. $\frac{4m+2}{6}=\frac{m}{3}$

3. $\frac{3p+6}{3}=\frac{p}{2}$

4. $\frac{v}{10}+1=\frac{v}{4}-2$

1. $\frac{d}{6}+3=\frac{d}{8}+2$

2. $\frac{10y-2}{3}+3=\frac{10y+1}{9}$

3. $\frac{3v-6}{2}+5=\frac{11v-4}{5}$

4. $0.4y-4=2$

1. $2.1k+3=7.2$

2. $0.7x+0.4=0.6x+2.4$

3. $0.48x+1.56=0.58x-0.64$

4. $1.2x-0.91=0.8x+2.29$

1. $0.05n+0.10(n+7)=3.55$

2. $0.10d+0.25(d+7)=5.25$

3. $0.05(q-8)+0.25q=4.10$

4. Stamps Paula bought $22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 8 less than the number of 49-cent stamps. Solve the equation$0.49s+0.21(s-8)=22.82$ for s, to find the number of 49-cent stamps Paula bought.

1. If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?

2. In the equation $0.35x+2.1=3.85$ what is the LCD? How do you know?