Ratios and proportions and how to solve them
Let's talk about ratios and proportions. When we talk about the speed of a car or an airplane we measure it in miles per hour. This is called a rate and is a type of ratio. A ratio is a way to compare two quantities by using division as in miles per hour where we compare miles and hours.
A ratio can be written in three different ways and all are read as "the ratio of x to y"
$$x\: to\: y$$
$$x:y$$
$$\frac{x}{y}$$
A proportion on the other hand is an equation that says that two ratios are equivalent. For instance if one package of cookie mix results in 20 cookies than that would be the same as to say that two packages will result in 40 cookies.
$$\frac{20}{1}=\frac{40}{2}$$
A proportion is read as "x is to y as z is to w"
$$\frac{x}{y}=\frac{z}{w} \: where\: y,w\neq 0$$
If one number in a proportion is unknown you can find that number by solving the proportion.
Example
You know that to make 20 pancakes you have to use 2 eggs. How many eggs are needed to make 100 pancakes?
| Eggs | pancakes | |
| Small amount | 2 | 20 |
| Large amount | x | 100 |
$$\frac{eggs}{pancakes}=\frac{eggs}{pancakes}\: \: or\: \: \frac{pancakes}{eggs}=\frac{pancakes}{eggs}$$
If we write the unknown number in the nominator then we can solve this as any other equation
$$\frac{x}{100}=\frac{2}{20}$$
Multiply both sides with 100
$${\color{green} {100\, \cdot }}\, \frac{x}{100}={\color{green} {100\, \cdot }}\, \frac{2}{20}$$
$$x=\frac{200}{20}$$
$$x=10$$
If the unknown number is in the denominator we can use another method that involves the cross product. The cross product is the product of the numerator of one of the ratios and the denominator of the second ratio. The cross products of a proportion is always equal
If we again use the example with the cookie mix used above
$$\frac{{\color{green} {20}}}{{\color{blue} {1}}}=\frac{{\color{blue} {40}}}{{\color{green} {2}}}$$
$${\color{blue} {1}}\cdot {\color{blue} {40}}={\color{green} {2}}\cdot {\color{green} {20}}=40$$
It is said that in a proportion if
$$\frac{x}{y}=\frac{z}{w} \: where\: y,w\neq 0$$
$$xw=yz$$
If you look at a map it always tells you in one of the corners that 1 inch of the map correspond to a much bigger distance in reality. This is called a scaling. We often use scaling in order to depict various objects. Scaling involves recreating a model of the object and sharing its proportions, but where the size differs. One may scale up (enlarge) or scale down (reduce). For example, the scale of 1:4 represents a fourth. Thus any measurement we see in the model would be 1/4 of the real measurement. If we wish to calculate the inverse, where we have a 20ft high wall and wish to reproduce it in the scale of 1:4, we simply calculate:
$$20\cdot 1:4=20\cdot \frac{1}{4}=5$$
In a scale model of 1:X where X is a constant, all measurements become 1/X - of the real measurement. The same mathematics applies when we wish to enlarge. Depicting something in the scale of 2:1 all measurements then become twice as large as in reality. We divide by 2 when we wish to find the actual measurement.
Video lesson
Find x
$$\frac{x}{x + 20} = \frac{24}{54}$$