31.5: A New Way to Interpret a over b

Lesson

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

Exercise \(\PageIndex{1}\): Recalling Ways of Solving

Solve each equation. Be prepared to explain your reasoning.

\(0.07=10m\qquad 10.1=t+7.2\)

Exercise \(\PageIndex{2}\): Interpreting \(\frac{a}{b}\)

Solve each equation.

1. \(35=7x\)
2. \(35=11x\)
3. \(7x=7.7\)
4. \(0.3x=2.1\)
5. \(\frac{2}{5}=\frac{1}{2}x\)

Are you ready for more?

Solve the equation. Try to find some shortcuts.

\(\frac{1}{6}\cdot\frac{3}{20}\cdot\frac{5}{42}\cdot\frac{7}{72}\cdot x=\frac{1}{384}\)

Exercise \(\PageIndex{3}\): 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.

\(0.7+x=12\qquad \frac{1}{4}x=\frac{3}{2}\)

Summary

In the past, you learned that a fraction such as \(\frac{4}{5}\) can be thought of in a few ways.

• \(\frac{4}{5}\) 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.
• \(\frac{4}{5}\) is the share that each person would have if 4 wholes were shared equally among 5 people. This means that \(\frac{4}{5}\) 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 \(\frac{3}{8}\) finished and the length of the completed part is \(\frac{4}{3}\) miles. How long will the road be when completed?

We can write the equation \(\frac{3}{8}x=\frac{4}{3}\) to represent the situation and solve the equation.

The completed road will be \(3\frac{5}{9}\) or about 3.6 miles long.

\(\begin{aligned} \frac{3}{8}x&=\frac{4}{3} \\ x&=\frac{\frac{4}{3}}{\frac{3}{8}} \\ x&=\frac{4}{3}\cdot\frac{8}{3} \\ x&=\frac{32}{9}=3\frac{5}{9}\end{aligned}\)

Glossary Entries

Definition: 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 \(y=1\cdot y\).

Definition: 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 \(9+1\neq 8\).

Definition: 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 \(10-6=4\).

Practice

Exercise \(\PageIndex{4}\)

Select all the expressions that equal \(\frac{3.15}{0.45}\).

1. \((3.15)\cdot (0.45)\)
2. \((3.15)\div (0.45)\)
3. \((3.15)\cdot\frac{1}{0.45}\)
4. \((3.15)\div\frac{45}{100}\)
5. \((3.15)\cdot\frac{100}{45}\)
6. \((\frac{0.45}{3.15}\)

Exercise \(\PageIndex{5}\)

Which expressions are solutions to the equation \(\frac{3}{4}x=15\)? Select all that apply.

1. \(\frac{15}{\frac{3}{4}}\)
2. \(\frac{15}{\frac{4}{3}}\)
3. \(\frac{4}{3}\cdot 15\)
4. \(\frac{3}{4}\cdot 15\)
5. \(15\div\frac{3}{4}\)

Exercise \(\PageIndex{6}\)

Solve each equation.

\(4a=32\qquad 4=32b\qquad 10c=26\qquad 26=100d\)

Exercise \(\PageIndex{7}\)

For each equation, write a story problem represented by the equation. For each equation, state what quantity \(x\) represents. If you get stuck, consider drawing a diagram.

1. \(\frac{3}{4}+x=2\)
2. \(1.5x=6\)

Exercise \(\PageIndex{8}\)

Write as many mathematical expressions or equations as you can about the image. Include a fraction, a decimal number, or a percentage in each.

Figure \(\PageIndex{1}\): A fundraiser thermometer labeled Fundraiser, our goal, 2 hundred 50 thousand dollars. The numbers 50 thousand through 2 hundred 50 thousand, in increments of 50 thousand dollars, are indicated. There are 4 evenly spaced tick marks between each indicated dollar value. Starting from the bottom, the thermometer is shaded to the first tick mark above 1 hundred thousand dollars.

(From Unit 3.4.4)

Exercise \(\PageIndex{9}\)

In a lilac paint mixture, 40% of the mixture is white paint, 20% is blue, and the rest is red. There are 4 cups of blue paint used in a batch of lilac paint.

1. How many cups of white paint are used?
2. How many cups of red paint are used?
3. How many cups of lilac paint will this batch yield?

If you get stuck, consider using a tape diagram.

(From Unit 3.4.3)

Exercise \(\PageIndex{10}\)

Triangle P has a base of 12 inches and a corresponding height of 8 inches. Triangle Q has a base of 15 inches and a corresponding height of 6.5 inches. Which triangle has a greater area? Show your reasoning.

(From Unit 1.3.3)