Rational Numbers
6.4 Proportional Reasoning
Ratios and Proportions
Ratios are as ubiquitous as they are useful. For example, driving $5$ mph means that for every hour you drive, you will have gone $5$ miles. The miles per hours describes the ratio as a relationship between miles and hours. This relationship between two numbers is described as a rational number.
$$\frac{5 \text{ miles}}{1 \text{ hour}}$$
Definition: Ratio
A ratio compares two quantities and may be written as
$$a:b,\quad \frac ab,\quad a/b.$$
MyOpenMath: Find the ratio
Whenever we have another ratio that is equivalent to the first, we have a proportion. For example, there are $5280$ feet in a $1$ mile. We could describe our $5$ mph as an equivalent ratio of some number of inches per hour.
$$\frac{5 \textcolor{blue}{\text{ miles}}}{1 \text{ hour}}=\frac{5 \cdot\textcolor{blue}{5280\text{ feet}}}{1 \text{ hour}}$$ $$\frac{5 \text{ miles}}{1 \text{ hour}}=\frac{26400 \text{ feet}}{1 \text{ hour}}$$
We now have an equivalence between two ratios, or a proportion. These ratios may look different, but they are the same in value.
Definition: Proportion
A proportion is an equation stating two ratios are equal.
$$\frac ab=\frac cd$$
When we have a proprotion, we can use what we know about rational numbers to figure out missing information. Namely, we can use what it means for rational numbers to be equal.
Fact: Proportion Equivalence
For $b,d\ne0$,
$$\frac ab=\frac cd \iff ad=bc.$$
Example: Solve a typing proportion
Allie and Bente type at fixed rates. When Allie types 8 pages, Bente types 4. When Bente types 10, how many has Allie typed?
Show Solution
This is a prime type of problem for proprortions. If the rates at which the two people are typing is fixed, then that fist bit of information is constant and makes a valid ratio.
$$\frac{\text{Ally's 8 pages}}{\text{Bene's 4 pages}}$$
So if more time passes and the number of pages change, the ratio between them will not change. So we can make a porportion.
$$\frac{\text{Ally's 8 pages}}{\text{Bene's 4 pages}}=\frac{\text{Ally's ? pages}}{\text{Bene's 10 pages}}$$
To simplify this a little, we can use a placeholer number for what we're trying to find and we get to an expression like this.
$$\frac{8}{4}=\frac{x}{10}.$$
Based on our definition of the equality of rational numbers, this will only be true if $8\cdot 10 = 4\cdot x$, commonly referred to as cross-multiplying.
$$8\cdot10=4x\Rightarrow x=20.$$
Allie has typed 20 pages in the time it took Bene to type 10.
Example: Cars-to-students proportion
If there are 3 cars per 8 students, how many cars for 1200 students?
Show Solution
$$\frac{3}{8}=\frac{x}{1200}\Rightarrow 8x=3600\Rightarrow x=450.$$
So about 450 cars.
MyOpenMath: Ratios and proportions
Example: Map scale
A map has a scale of $1:300$. If map distance is 17 cm, find actual distance.
Show Solution
Actual length is 300 times map length:
$$\frac{1\text{ map cm}}{300 \text{ real cm}}=\frac{17 \text{map cm}}{? \text{ real cm}}$$
$$17\cdot300=?\cdot 1$$ $$? = 5100\text{ cm}$$
The real distance is $5100$cm or $51$ meters.
MyOpenMath: Ratio and Proportions
Bar Model for Proportions
A bar model is a visual representation of a proportion. For example, if we're told that the ratio of 8th graders to 7th graders is 4:7, we can create a bar which is made up of 11 equal parts, and then color in 4 of those parts to represent the 8th graders and 7 of those parts to represent the 7th graders.
Using that bar, we can reason through more complicated expressions. For example, if we know that $\frac{1}{4}$ of the 8th graders are boys and $\frac{2}{7}$ of the 7th graders are boys, then we can deduce that of the whole (which is now $11$ parts), $3$ pieces of the bar are boys. So $\frac{3}{11}$.
Example: Using a bar model
Suppose the number of blue cars in the parking lot to the the red cars in the parking lot are in a $3:5$ ratio. Draw the bar model to represent this.
If half the blue cars are SUVs and and $\frac{2}{5}$ of the red cars are SUVs, what is the ratio of SUV to total cars?
Show Solution
First, we can set up a bar to show the $3:4: ratio.
But we need to take half of the $3$ blue cars and we can't because there are only $3$ blocks which doesn't cleanly divide into halves. We have to get that common denominator! Lets duplicate this bar so that we'll have $2\cdot 3=6$ units of area for the blue cars and we can then get half of them.
Then we need $\frac{2}{5}$ of the red cars. Because we've doubled our bar, we'll need $4$ of those units to get the appropriate amount.
All together, thats $3$ (of the $6$ blue to make half) and $4$ (of the $10$ red to make $\frac{2}{5}$) which makes $3+4=7$ of our now $16$ total units of area. The result is
$$\frac{3+4}{16}=\frac{7}{16}.$$
MyOpenMath: Bar Model Proportions
Compound Ratios
We can also talk about relating more than just two things at a time. For example, if we have three types of computers in a room, say tablets, laptops, and pcs, and they're in a $$2:3:5$$ ratio, then our model would look something like this.
Really, this is shorthand for showing several ratios. We have tablets to laptops as $2:3$, but tablets to pcs are $2:5$, while laptops to pcs are $3:5$.
Definition: Compound Ratio
A compound proportion uses more than one linked ratio, often written
$$a:b:c$$
or interpreted through systems like
$$\frac ab=\frac bc$$
when context supports that relationship.
Example: Shared pay by hours
Kai, Paul, and Judy all worked at the same rate of pay. Together they made \$2520. But in hours worked the proportion K:P:J is to $30:50:60$ hours. How much did each person make?
Show Solution
$30:50:60$ can be reduced to a $3:5:6$ ratio. If were were to use a bar model, the total length of the bar would be $3+5+6=14$.
First up, Kai. Kai would represent $3$ of those $14$ parts, so they made $\frac{3}{14}$ of the total amount of money.
$$\frac{3}{14}=\frac{k}{2520}$$ $$14k=3\cdot 2520$$ $$k=540$$
Kai made \$540.
Paul represents $5$ of the $14$ parts.
$$\frac{5}{14}=\frac{p}{2520}$$ $$14p=5\cdot 2520$$ $$p=900$$
Paul earned \$900.
Finally, Judy represents $6$ of the $14$ parts.
$$\frac{6}{14}=\frac{j}{2520}$$ $$14j=6\cdot 2520$$ $$p=1080$$
Checking to make sure this is a reasonable answer: $$540+900+1080=2520.$$
MyOpenMath: Compound Ratios
Constant of Proportionality
We need our ratios to be fixed in value to solve proportions. If we know that some classroom has girls to boys in a ratio of $4$ to $3$, the representation of the ratio ratio can change, but the value will remain the same $\frac{4}{3}$. It could be the case that there are $4$ girls and $3$ boys. However, it could also describe a classroom that has $8$ girls and $6$ boys.
$$\frac{4}{3}=\frac{8}{6}=\frac{12}{9}=\ldots$$
This fixed, reduced, value is the ratio, and we can describe the relationship as a constant (meaning fixed or unchanging) of proportionality.
Definition: Constant of Proportionality
If variables satisfy $y=kx$ (direct variation) or $y=\frac{k}{x}$ (inverse variation), then $k$ is the constant of proportionality.
Lets take another example to solidify that idea.
Example: Direct Variation
The number of girls in a classroom is directly proportional to the number of boys in the classroom. If the number of girls in a classroom is $2:1$ with the boys, find the constant of proportionality.
Show Solution
We have a direct variation. $2$ girls for every $1$ boy means that if we know how many boys there are, we can multiply that by $2$ to find the total number of girls. $$\frac{2 \text{ girls}}{1 \text{ boy}}=2=k$$
So our relationship is $$1 \text{girl }= 2\text{ boys}$$
and following your definition,
$$y=2x$$
MyOpenMath: Direct Variation
Example: Inverse Variation
The length of a string on a guitar is inversely proportional to the frequency it makes. For example, a string of $24$ inches on a particular guitar emits a frequency of $440$hz. Find the constant of proportionality.
Show Solution
Becuase our relationship is inverse, we know that the form is $$\text{length }=\frac{k}{\text{frequency}}$$
We also know that if the string is $24$ inches, then the frequency would be $440$hz.
$$24=\frac{k}{440}$$ $$k=10560$$
So our complete realtionship, with y being string length and x being string frequency, is
$$y=\frac{10560}{x}$$