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.
- \(35=7x\)
- \(35=11x\)
- \(7x=7.7\)
- \(0.3x=2.1\)
- \(\frac{2}{5}=\frac{1}{2}x\)
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 \(\frac{3.15}{0.45}\).
- \((3.15)\cdot (0.45)\)
- \((3.15)\div (0.45)\)
- \((3.15)\cdot\frac{1}{0.45}\)
- \((3.15)\div\frac{45}{100}\)
- \((3.15)\cdot\frac{100}{45}\)
- \((\frac{0.45}{3.15}\)
Exercise \(\PageIndex{5}\)
Which expressions are solutions to the equation \(\frac{3}{4}x=15\)? Select all that apply.
- \(\frac{15}{\frac{3}{4}}\)
- \(\frac{15}{\frac{4}{3}}\)
- \(\frac{4}{3}\cdot 15\)
- \(\frac{3}{4}\cdot 15\)
- \(15\div\frac{3}{4}\)
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.
- \(\frac{3}{4}+x=2\)
- \(1.5x=6\)
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.
(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.
- How many cups of white paint are used?
- How many cups of red paint are used?
- 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)