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Difference between proportions and ratios

proportions ratios

In this lesson we’ll be learning about proportions, ratios and their use in real situations.

First, let’s define a ratio.

A ratio is a comparison between numbers of two different things.
For example, if you look at group of twelve ducks. 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}.

ratios

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 who have the same value.

The equality between two ratios, a:b and c:d, is called the proportion (\frac{a}{b} = \frac{c}{d}).

simple proportions equation

The easiest way is to examine fractions. When we shorten both fractions, they should be the same.

Example 1:

determination of proportionality

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, than they are not proportional.

Example 2: Let’s say that you are starting your own business selling muffins and chocolates. First day you sell 4 muffins and 3 chocolates, 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 tell?

Simply put all in ratios, and then look at proportions.

rations with muffins and chocolates

You shorten the fractions:

rations with muffins and chocolates simplyfied

Since all fractions are not the same, you didn’t sell proportionally.

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:

means extremes property

Example 3:

\frac{5}{6} = \frac{10}{12} => 5 * 12 = 10 * 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 * 6 = 8 * 3 => 24 = 24
These ratios are proportional.

b) 7 : 2 and 14 : 3 => \frac{7}{2} = \frac{14}{3}(means-extremes property) 7 * 3 = 14 * 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 * 9 = 3 * 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 * x = 5 * 6 => 10x = 30 => x = 3.

 

Proportions and ratios worksheets

  Proportions - Integers (176.5 KiB, 525 hits)

  Proportions - Decimals (116.8 KiB, 428 hits)

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