Ratios and Rates

Ratios

A ratio compares multiple quanitites measured in the same units. We write ratios with a colon (:) inbetween the quanties. For example, if you had \(10\) marbles and \(6\) were red, the ratio of red : blue is \( 6 : 4 \). We read this as "the ratio of red to blue marbles is \(6\) to \(4\)".

You can write a ratio comparing more than one quantity. Consider a classroom of \(30\) students picking their favourite subject at school. We could write a ratio like:

math : english : art

\( 8 : 10 : 7\)


This means that there are \(8\) students whose favourite subject is math, \(10\) whose favourite is english and \(7\) whose favourite is art. Since we know there are \(30\) students, we can assign the remaining \(5\) to an other category:

math : english : art : other

\( 8 : 10 : 7 : 5\)


We can also include the total in the ratio:

math : english : art : other : total

\( 8 : 10 : 7 : 5 : 30\)


There are a lot of numbers in this ratio! You can pick the numbers you are interested in, such as the ratio of art to total:

art : total

\( 7 : 30\)

Ratios as Fraction

We can also write ratios as a fractions. The ratio of math : english is \(8 : 10\). You can write this as a fraction like:

\( \cfrac{\text{math}}{\text{english}} = \cfrac{8}{10} \)

Often we will reduce the ratio/fraction to simpliest terms. Recall that reduce fractions means to find an equilvant fraction that is in lowest terms. We should divide both the numerator and denominator by the greatest common factor. In the example above, we can divide top and bottom by \(2\):

\( \cfrac{8}{10} \)

\( = \cfrac{8 \div 2}{10 \div 2} \)

\( = \cfrac{4}{5} \)

Solving Problem with Ratios

We can interpret the ratio as for every \(4\) students that like math, \(5\) like english. The ratio is perserved as the number of students increases. For example, if there were \(40\) students that like math, then \(50\) would like english. Basically, the ratio is constant so each fraction has to be equivalent:

\( \cfrac{\text{math}}{\text{english}} = \cfrac{4}{5} \)

\( \cfrac{\text{math}}{\text{english}} = \cfrac{4}{5} = \cfrac{40}{x}\)

\( \cfrac{4 * ?}{5 * ?} = \cfrac{40}{x}\)

\( \cfrac{4 * 10}{5 * 10} = \cfrac{40}{50}\)

Solve for the missing number: \( 1 : 6 : x = 3 : y : 9 \)

Rates

A rate is a ratio that compares two quantities with different units. Rates are used to help us with conversions. For example, \(1\) hour is \(60\) minutes. A unit rate is a special rate where the second quantity is \(1\). For example, \(20 \; [km/hr]\) means \(20 \; [km]\) are travelled in \( 1 \; [hr]\).

Unit rates can be helpful for comparing different options. For example, you are at the grocery store and see two options for juice. One brand sells \(1 \; [L]\) for \($2.99\). Another brand sells \(700 \; [mL]\) for \($1.99\). Which price is better?

The price per litre ($/L)can be represented as a rate! The first option is per \(1 \; [L]\) already. Let's convert the second option:

\( \cfrac{$1.99}{700 \; [mL]} \)

\(= \cfrac{$1.99}{0.700 \; [L]} \)

\(= \cfrac{$2.84}{1 \; [L]} \)

Thus, the second option is cheaper by \(15\) cents!

The cost of electricity is \( \cfrac{20 \; [c]}{2 \; \text{kWh} } \). How much does \( 10 \; \text{kWh} \) cost?

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