Simple equations with fractions and decimals

You have already solved equa­tions like $x + 2 = 8$ and $x \times 5 = 250$ with inte­gers. Now the num­bers grow up. The same four rules work when the num­bers are frac­tions, dec­i­mals or even roots. In this les­son you will use those rules to solve one-step equa­tions with ratio­nal num­bers, and check every answer.

Words to know#

An equa­tion is a state­ment that two things are equal, such as $\displaystyle x + \frac{2}{3} = \frac{5}{6}$. The let­ter $x$ stands for a num­ber we do not know yet. It is called the vari­able.

To solve an equa­tion means to find the value of $x$ that makes it true. That value is the solu­tion.

The sign $\Rightarrow$ means "implies" or "so". We write it at the start of each new step.

The four rules#

When a num­ber moves from one side of the equals sign to the other, it does the oppo­site job.

1. If $x + a = b$, then $x = b - a$. A num­ber added on one side is taken away on the other.
2. If $x - a = b$, then $x = b + a$. A num­ber taken away on one side is added on the other.
3. If $a \times x = b$, then $\displaystyle x = \frac{b}{a}$. A num­ber mul­ti­ply­ing on one side divides on the other ($a \neq 0$).
4. If $\displaystyle \frac{x}{a} = b$, then $x = b \times a$. A num­ber divid­ing on one side mul­ti­plies on the other ($a \neq 0$).

Mov­ing a num­ber across the equals sign like this is called trans­pos­ing. It works because we are really doing the same thing to both sides, so they stay equal.

Remem­ber. Plus becomes minus, minus becomes plus, times becomes divide, divide becomes times.

Equa­tions with frac­tions#

Exam­ple 1#

Solve $\displaystyle x + \frac{2}{3} = \frac{5}{6}$.

$$\displaystyle \begin{aligned} x + \frac{2}{3} &= \frac{5}{6} \\ \Rightarrow x &= \frac{5}{6} - \frac{2}{3} \\ \Rightarrow x &= \frac{5}{6} - \frac{4}{6} \\ \Rightarrow x &= \frac{1}{6} \end{aligned}$$

Check: $\displaystyle \frac{1}{6} + \frac{2}{3} = \frac{1}{6} + \frac{4}{6} = \frac{5}{6}$. It works.

Exam­ple 2#

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

$$\displaystyle \begin{aligned} x - \frac{3}{4} &= -\frac{1}{2} \\ \Rightarrow x &= -\frac{1}{2} + \frac{3}{4} \\ \Rightarrow x &= -\frac{2}{4} + \frac{3}{4} \\ \Rightarrow x &= \frac{1}{4} \end{aligned}$$

Exam­ple 3#

Solve $\displaystyle \frac{3}{4}x = \frac{9}{8}$.

Here $\displaystyle \frac{3}{4}$ mul­ti­plies $x$. So it divides on the other side. Divid­ing by a frac­tion is mul­ti­ply­ing by it turned upside down.

$$\displaystyle \begin{aligned} \frac{3}{4}x &= \frac{9}{8} \\ \Rightarrow x &= \frac{9}{8} \div \frac{3}{4} \\ \Rightarrow x &= \frac{9}{8} \times \frac{4}{3} \\ \Rightarrow x &= \frac{36}{24} = \frac{3}{2} \end{aligned}$$

Check: $\displaystyle \frac{3}{4} \times \frac{3}{2} = \frac{9}{8}$. It works.

Exam­ple 4#

Solve $\displaystyle \frac{x}{4} = -\frac{5}{6}$.

$$\displaystyle \begin{aligned} \frac{x}{4} &= -\frac{5}{6} \\ \Rightarrow x &= -\frac{5}{6} \times 4 \\ \Rightarrow x &= -\frac{20}{6} = -\frac{10}{3} \end{aligned}$$

Exam­ple 5#

Solve $\displaystyle -\frac{2}{5}x = \frac{4}{15}$.

$$\displaystyle \begin{aligned} -\frac{2}{5}x &= \frac{4}{15} \\ \Rightarrow x &= \frac{4}{15} \times \left(-\frac{5}{2}\right) \\ \Rightarrow x &= -\frac{20}{30} = -\frac{2}{3} \end{aligned}$$

A plus times a minus gives a minus, so the answer is neg­a­tive.

Equa­tions with dec­i­mals#

Exam­ple 6#

Solve $x + 2.75 = 1.5$.

$$\begin{aligned} x + 2.75 &= 1.5 \\ \Rightarrow x &= 1.5 - 2.75 \\ \Rightarrow x &= -1.25 \end{aligned}$$

$2.75$ is big­ger than $1.5$, so the answer is neg­a­tive. Work out $2.75 - 1.50 = 1.25$, then put the minus sign in front.

Exam­ple 7#

Solve $x - 0.6 = 3.25$.

$$\begin{aligned} x - 0.6 &= 3.25 \\ \Rightarrow x &= 3.25 + 0.6 \\ \Rightarrow x &= 3.85 \end{aligned}$$

Exam­ple 8#

Solve $0.5x = 7.5$.

$$\displaystyle \begin{aligned} 0.5x &= 7.5 \\ \Rightarrow x &= \frac{7.5}{0.5} = \frac{75}{5} \\ \Rightarrow x &= 15 \end{aligned}$$

To divide by $0.5$, mul­ti­ply the top and the bot­tom by $10$.

Exam­ple 9#

Solve $\displaystyle \frac{x}{0.2} = 3.4$.

$$\displaystyle \begin{aligned} \frac{x}{0.2} &= 3.4 \\ \Rightarrow x &= 3.4 \times 0.2 \\ \Rightarrow x &= 0.68 \end{aligned}$$

Check: $\displaystyle \frac{0.68}{0.2} = \frac{6.8}{2} = 3.4$. It works.

Always check#

Put your answer back into the orig­i­nal equa­tion. If the two sides come out equal, the answer is right. Check­ing takes a few sec­onds and catches most sign slips.

Com­mon mis­takes#

• Mov­ing a num­ber across with­out chang­ing its sign. In $\displaystyle x + \frac{2}{3} = \frac{5}{6}$, the $\displaystyle \frac{2}{3}$ must become $\displaystyle -\frac{2}{3}$.
• Divid­ing by a frac­tion with­out turn­ing it upside down.
• For­get­ting the minus sign when the num­ber taken away is big­ger, as in $1.5 - 2.75 = -1.25$.
• Leav­ing a frac­tion answer unsim­pli­fied. $\displaystyle \frac{36}{24}$ should be $\displaystyle \frac{3}{2}$.

Prac­tice#

1. Solve $\displaystyle x + \frac{1}{4} = \frac{3}{4}$.
2. Solve $\displaystyle x - \frac{2}{5} = \frac{1}{10}$.
3. Solve $\displaystyle x + \frac{5}{6} = -\frac{1}{3}$.
4. Solve $x - 1.8 = -0.45$.
5. Solve $x + 3.6 = 10$.
6. Solve $\displaystyle \frac{2}{3}x = \frac{8}{9}$.
7. Solve $1.2x = 4.8$.
8. Solve $\displaystyle \frac{x}{3} = \frac{7}{12}$.
9. Solve $\displaystyle \frac{x}{0.4} = 2.5$.
10. Solve $\displaystyle -\frac{3}{7}x = \frac{9}{14}$.

Answers#