## 5.1: Recalling Ways of Solving

Solve each equation. Be prepared to explain your reasoning.

- $0.07 = 10m$

- $10.1 = t + 7.2$

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

Solve each equation. Be prepared to explain your reasoning.

- $0.07 = 10m$

- $10.1 = t + 7.2$

Solve each equation.

- $35=7x$

- $35=11x$

- $7x=7.7$

- $0.3x=2.1$

- $\frac25=\frac12 x$

Solve the equation. Try to find some shortcuts.

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

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, draw a diagram.

- $0.7 + x = 12$

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

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 division* 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: $\frac{4.5}{1.5} = 4.5\div{1.5} = 3$.

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.

$$\begin {align}

\frac38x&=\frac43\\[5pt]

x&=\frac{\frac43}{\frac38}\\[5pt]

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