4.9: Ratios and Proportions
Learning Objectives
- Write and simplify ratios.
- Formulate proportions.
- Use ratios and proportions in problem solving.
Ratios
A ratio is a fraction (and can be simplified just like a fraction). Usually a ratio is a fraction that compares two parts. “The ratio of \begin{align*}x\end{align*} to \begin{align*}y\end{align*}” can be written in several ways:
\begin{align*}\frac{x}{y} && \text{or} && x : y && \text{or} && x && \text{to}\ y\end{align*}
A ratio is another word for ________________________________.
Proportions
A proportion is an equation. The two sides of the equation are ratios that are equal to each other. A proportion looks like this:
\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}
A proportion is an ___________________________________ with equal fractions.
Cross Multiplication Theorem
Let \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*} be real numbers, with \begin{align*}b \neq 0\end{align*} and \begin{align*}d \neq 0\end{align*}.
If \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*} then \begin{align*}ad = bc\end{align*}.
When you have a proportion, you can solve the equation by cross multiplying. This means multiplying the parts of the fractions across the equal sign (top left times bottom right and bottom left times top right) like the arrows show:
Solve a proportion by cross ___________________________________________.
Reading Check:
1. True/False: Another name for a fraction is a ratio.
2. True/False: Every proportion always has an equal sign in between the fractions.
Properties of Proportions
Here are a few more ways of looking at the Cross Multiplication Theorem:
If \begin{align*}a \neq 0, b \neq 0, c \neq 0,\end{align*} and \begin{align*}d \neq 0\end{align*}, and \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then all of the following are true:
- \begin{align*}\frac{a}{c} = \frac{b}{d}\end{align*} Notice that \begin{align*}b\end{align*} and \begin{align*}c\end{align*} have changed positions.
- \begin{align*}\frac{d}{b} = \frac{c}{a}\end{align*} Notice that \begin{align*}a\end{align*} and \begin{align*}d\end{align*} have changed positions.
- \begin{align*}\frac{b}{a} = \frac{d}{c}\end{align*} Notice that we have the reciprocals of the original ratios.
A reciprocal is a fraction whose numerator and denominator are flipped from the original fraction.
In other words, when you flip the top and bottom of a fraction, your result will be the reciprocal.
An easy way to remember the word reciprocal is that it sounds like “re-flip-rocal” which reminds you to flip the fraction!
When the numerator and denominator are flipped, it is called the _________________________.
Reading Check:
What is the reciprocal of the fraction \begin{align*}\frac{5}{6}\end{align*}?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
What does it mean to be proportional?
As you will learn in the next lesson, sometimes parts of triangles are proportional to one another. When two values are proportional, you can set up a proportion to describe their relationship.
When two things are proportional, you can set up a __________________________________ between them to describe how they are related.
You may remember from Unit 3 that proportional describes a relationship between two values where you can multiply one of the values by some number to get the second value.
In the example on the following page, we will create a proportion between the lengths of the sides of the two triangles. Until the next lesson, you don’t need to worry about why this is. The more important thing is that you understand how to solve the proportion.
Example 1
Look at the diagram below:
Suppose we’re given that \begin{align*}\frac{10}{6} = \frac{15}{9} = \frac{x}{y}\end{align*}.
We know \begin{align*}\frac{10}{6} = \frac{15}{9}\end{align*}, since \begin{align*}10 \cdot 9 = 6 \cdot 15 = 90\end{align*} (from the Cross Multiplication Theorem)
Here are some other proportions that must also be true based on our given information:
\begin{align*}&&\frac{15}{x} = \frac{9}{y}&&&&\frac{y}{6} = \frac{x}{10}\\ \frac{15}{9} = \frac{x}{y}&&&&\frac{15}{10} = \frac{9}{6}&&&&\frac{x}{15} = \frac{y}{9}\end{align*}
There are two more true statements related to the Cross Multiplication Theorem below:
The “if” part of these is the same as above.
- If \begin{align*}a \neq 0, b \neq 0, c \neq 0,\end{align*} and \begin{align*}d \neq 0\end{align*}, and \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}\frac{a + b}{b} = \frac{c + d}{d}\end{align*}. (This is the equivalent of adding 1 to each ratio.)
And another, nearly the same as the previous,
- If \begin{align*}a \neq 0, b \neq 0, c \neq 0,\end{align*} and \begin{align*}d \neq 0\end{align*}, and \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}\frac{a - b}{b} = \frac{c - d}{d}\end{align*}. (This is the equivalent of subtracting 1 from each ratio.)
Reading Check:
1. True/False: To solve a proportion, use cross multiplication.
2. True/False: The reciprocal of a fraction is when both the numerator and the denominator are multiplied by each other.
3. There are many true mathematical statements that you can make about a proportion. If you have the proportion \begin{align*}\frac{5}{15} = \frac{x}{3}\end{align*}, what is an example of another true statement you can make?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
4. Complete the sentence:
When two values are proportional,...
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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