In this lesson we’ll be learning about proportions, ratios and their use in real life situations.
Firstly, let’s define a ratio.
A ratio is a comparison between numbers of two different things.
For example, imagine that we look at a group of twelve ducks and that there are 7 adult ducks and 5 ducklings. The ratio of adult ducks to ducklings is 7:5.
You can write ratios using the “:” (7 : 5), you can write 7 to 5, or as a fraction $\frac{7}{5}$.
Equivalent ratios
Let’s get back to our ducks. If there are 12 adult ducks and 6 little ducklings, the ratio of adult ducks to ducklings is now 12 : 6. This gives us the information that there are twice as more ducks than there are ducklings. But the ratio 2 : 1 also gives that information and also 6 : 3, 10 : 5 and so on. Those ratios are called equivalent ratios.
So, basically, equivalent ratios are just like equivalent fractions-those ratios that have the same value.
The equality between two ratios, a:b and c:d, is called the proportion ($\frac{a}{b} = \frac{c}{d}$).
The easiest way is to examine fractions. When we shorten both fractions, they should be the same.
Example 1:
Also, if you have more than two ratios and you want to see if they are proportional, you do it the same way. If any shorten fraction is different from the other it means they are not proportional.
Example 2: Let’s say you are starting your own business selling muffins and chocolates. On first day you sell 4 muffins and 3 chocolates, on second day 8 muffins and 6 chocolates and on your third day you sell 9 muffins and 6 chocolates. Suddenly you start wondering about proportions and ask yourself is your selling proportional. How can you examine that?
Simply put all in ratios, and then look at proportions.
After that, you need to shorten the fractions:
Since all the fractions are not the same, your sale wasn’t proportional.
There is another way of determining whether something is proportional or not. You can use the Means-Extremes property of proportions. The means-extremes property of proportions allows you to cross multiply:
Example 3:
$\frac{5}{6} = \frac{10}{12} => 5 \cdot 12 = 10 \cdot 6 => 60 = 60$
Example 4: Are these ratios proportional?
a) 4 : 3 and 8 : 6 => $\frac{4}{3} = \frac{8}{6}$ (means-extremes property) $ 4 \cdot 6 = 8 \cdot 3$ => $ 24 = 24$
These ratios are proportional.
b) $ 7 : 2$ and $ 14 : 3$ => $ \frac{7}{2}$ = $\frac{14}{3}$(means-extremes property) $ 7 \cdot 3 = 14 \cdot 2$ => $21 \neq 28$
These ratios are not proportional.
c) $ 8 : 3$ and $ 24 : 9$ => $\frac{8}{3}$ = $\frac{24}{9}$ (means – extremes property) $ 8 \cdot 9 = 3 \cdot 24$ => $ 72 = 72$
These ratios are proportional.
In proportions can also appear unknowns, and are solved using means-extremes property.
Example 5: Find $ x : \frac{x}{5} = \frac{6}{10}$
Using means-extremes property we come to:
$ 10 \cdot x = 5 \cdot 6$ => $ 10x = 30$ => $ x = 3$.
Proportions and ratios worksheets
Proportions - Integers (176.5 KiB, 1,594 hits)
Proportions - Decimals (116.8 KiB, 1,099 hits)
