# Lesson 5

A New Way to Interpret $a$ over $b$

Let's investigate what a fraction means when the numerator and denominator are not whole numbers.

### 5.1: Recalling Ways of Solving

Solve each equation. Be prepared to explain your reasoning.

$$0.07 = 10m$$

$$10.1 = t + 7.2$$

### 5.2: Interpreting $\frac{a}{b}$

Solve each equation.

1. $$35=7x$$

2. $$35=11x$$

3. $$7x=7.7$$

4. $$0.3x=2.1$$

5. $$\frac25=\frac12 x$$

Solve the equation. Try to find some shortcuts.

$$\displaystyle \frac{1}{6} \boldcdot \frac{3}{20} \boldcdot \frac{5}{42} \boldcdot \frac{7}{72} \boldcdot x = \frac{1}{384}$$

### 5.3: Storytime Again

Take turns with your partner telling a story that might be represented by each equation. Then, for each equation, choose one story, state what quantity $$x$$ describes, and solve the equation. If you get stuck, consider drawing a diagram.

$$0.7 + x = 12$$

$$\frac{1}{4}x = \frac32$$

### Summary

In the past, you learned that a fraction such as $$\frac45$$ can be thought of in a few ways.

• $$\frac45$$ is a number you can locate on the number line by dividing the section between 0 and 1 into 5 equal parts and then counting 4 of those parts to the right of 0.
• $$\frac45$$ is the share that each person would have if 4 wholes were shared equally among 5 people. This means that $$\frac45$$ is the result of dividing 4 by 5.

We can extend this meaning of a fraction as a quotient to fractions whose numerators and denominators are not whole numbers. For example, we can represent 4.5 pounds of rice divided into portions that each weigh 1.5 pounds as: $$\frac{4.5}{1.5} = 4.5\div{1.5} = 3$$. In other words, $$\frac{4.5}{1.5}=3$$ because the quotient of 4.5 and 1.5 is 3.

Fractions that involve non-whole numbers can also be used when we solve equations.

Suppose a road under construction is $$\frac38$$ finished and the length of the completed part is $$\frac43$$ miles. How long will the road be when completed?

We can write the equation $$\frac38x=\frac43$$ to represent the situation and solve the equation.

The completed road will be $$3\frac59$$ or about 3.6 miles long.

\displaystyle \begin {align} \frac38x&=\frac43\\[5pt] x&=\frac{\frac43}{\frac38}\\[5pt] x&=\frac43\boldcdot \frac83\\[5pt] x&=\frac{32}{9}=3\frac59\\ \end {align}

### Glossary Entries

• coefficient

A coefficient is a number that is multiplied by a variable.

For example, in the expression $$3x+5$$, the coefficient of $$x$$ is 3. In the expression $$y+5$$, the coefficient of $$y$$ is 1, because $$y=1 \boldcdot y$$.

• solution to an equation

A solution to an equation is a number that can be used in place of the variable to make the equation true.

For example, 7 is the solution to the equation $$m+1=8$$, because it is true that $$7+1=8$$. The solution to $$m+1=8$$ is not 9, because $$9+1 \ne 8$$

• variable

A variable is a letter that represents a number. You can choose different numbers for the value of the variable.

For example, in the expression $$10-x$$, the variable is $$x$$. If the value of $$x$$ is 3, then $$10-x=7$$, because $$10-3=7$$. If the value of $$x$$ is 6, then $$10-x=4$$, because $$10-6=4$$.