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.
Solve each equation.
Solve the equation. Try to find some shortcuts.
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.
In the past, you learned that a fraction such as $\frac45$ can be thought of in a few ways.
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.
$$\begin {align}
\frac38x&=\frac43\\[5pt]
x&=\frac{\frac43}{\frac38}\\[5pt]
x&=\frac43\boldcdot \frac83\\[5pt] x&=\frac{32}{9}=3\frac59\\ \end {align}$$