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.

\(0.7 + x = 12\)

\(\frac{1}{4}x = \frac32\)



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 \(10-6=4\).