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.

1. $0.07 = 10m$
1. $10.1 = t + 7.2$

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

Solve each equation.

1. $35=7x$
1. $35=11x$
1. $7x=7.7$
1. $0.3x=2.1$
1. $\frac25=\frac12 x$

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, draw a diagram.

1. $0.7 + x = 12$
1. $\frac{1}{4}x = \frac32$

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 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}