Ratios

A ratio is a comparison by division between two quantities or two measurements with the same measurement unit.

A ratio involves division and can be written as |\dfrac{a}{b}| or |a:b.|

Typically, a ratio does not include the units of measurement. Given that the  units are the same for the two compared quantities, they cancel each other out.

Martin ate three clementines as a snack and his little sister ate two.

The ratio between the number of clementines Martin ate and the ones his sister ate is |3:2|.

When expressing the ratio in the form of a fraction, the units of measurement for each quantity compared are simplified – they cancel each other out because they are the same.
||\displaystyle \frac{3 \color{red}{\text{ clementines}}}{2 \color{red}{\text{ clementines}}}=\frac{3}{2}||

A theater troupe includes |7| girls and |9| boys.

The ratio between the number of girls and the number of boys is |7| to |9|.

The ratio can be represented as |\displaystyle\frac{7}{9}| or |7:9|.

It may seem as though this example is not a ratio because the units of measurement are not the same. However, the two quantities being compared are the same - human beings. In other words, even if girls and boys are being compared, the person is the unit of measurement – not the sex.

A model airplane’s wings measure |4\ \text{cm},| while the original airplane’s wings are |5{.}4\ \text{m}.|

The ratio between the length of the scale model’s wings and that of the original airplane is |4\ \text{cm}| to |540\ \text{cm}.|

The ratio can be represented as |\dfrac{4}{540}| or |4:540.|

As shown below, it is possible to simplify this ratio.

Before comparing the two lengths, make sure they are expressed using the same unit of measurement. In this example, one of the lengths is in |\text{cm}| and the other is in |\text{m}|.

Convert the original airplane’s wing measurement to centimetres to obtain:
||5{.}4\ \text{m}\stackrel{\times 100}{\Rightarrow}540\ \text{cm}||

A simplified ratio is a ratio where the terms are relatively prime.
In other words, a simplified ratio is represented by an simplified fraction in the form |\displaystyle \frac{a}{b}|.

1. Express the ratio in the form |\displaystyle \frac{a}{b}| if it has been written in the form |a:b|.

2. Simplify the ratio to obtain a simplified fraction.

To simplify a ratio, simply divide the terms of the ratio by their GCF.

Simplify the ratio |24:40|.

1. Express the ratio in the form |\displaystyle \frac{a}{b}| if it has been written in the form |a:b|.
   ||\displaystyle 24:40=\frac{24}{40}||

2. Simplify the ratio to obtain a simplified fraction.
   ||\begin{align} \frac{24}{40}&\Rightarrow \frac{24\color{green}{\div 2}}{40\color{green}{\div 2}}=\frac{12}{20}\\
   \\
   &\Rightarrow \frac{12\color{green}{\div 4}}{20\color{green}{\div 4}}=\frac{3}{5}\end{align}||
   Therefore, the simplified ratio of |24:40| is |3:5|.
   We can also say that the ratios are equivalent ratios.

Equivalent ratios are ratios that have the same quotient. Equivalent ratios form a proportion.

1. Obtain the quotient of each ratio by dividing the numerator by the denominator.

2. Compare the quotients obtained. If the quotients are equal, the ratios are equivalent.

It is also possible to determine whether two ratios are equivalent by expressing them as simplified ratios. The ratios are equivalent if the simplified ratios are the same.

Are the ratios |\displaystyle \frac{3}{12}| and |\displaystyle \frac{2}{8}| equivalent?

1. Obtain the quotient of each ratio by dividing the numerator by the denominator.
   ||\begin{align} \frac{3}{12}&=3\div12=0{.}25 & \frac{2}{8}&=2\div8=0{.}25\end{align}||

2. Compare the quotients obtained. The ratios are equivalent if the quotients are equal.
   Note that the quotients are equal: |0{.}25=0{.}25|
   Therefore, the ratios are equivalent.

Each ratio can also be simplified to determine if they are equivalent.

||\begin{align}\frac{3\color{green}{\div 3}}{12\color{green}{\div 3}}&=\frac{1}{4} &
\frac{2\color{green}{\div 2}}{8\color{green}{\div 2}}&=\frac{1}{4}\end{align}||
Note that once simplified, the ratios are identical. Therefore, we can determine that they are equivalent.

Are the ratios |15:8| and |16:10| equivalent?

1. Obtain the quotient of each ratio by dividing the numerator by the denominator.
   ||\begin{align} \frac{15}{8}&=15\div8=1{.}875\\
   \\
   \frac{16}{9}&=16\div9=1{.}\overline{7}\end{align}||

2. Compare the quotients obtained. If the quotients are equal, the ratios are equivalent.
   Note that the quotients are not equal. ||1{.}875\color{red}{\neq}1{.}\overline{7}||

The ratios are not equivalent.

1. Express the ratio in the form |\displaystyle \frac{a}{b}| if it has been written in the form |a:b|.

2. Convert the fraction |\displaystyle \frac{a}{b}| into a percentage.

Express the ratio |3:50| as a percentage.

1. Express the ratio in the form |\displaystyle \frac{a}{b}| if it has been written in the form |a:b|.
   ||3:50=\displaystyle \frac{3}{50}||

2. Convert the fraction |\displaystyle \frac{a}{b}| to a percentage.
   Several methods can be used. To find out more, click here.
   ||\displaystyle \frac{3\color{green}{\times 2}}{50\color{green}{\times 2}}=\frac{6}{100}=6\%||

Therefore, the ratio |3:50| is equivalent to |6\%|.

1. Express each ratio as a decimal by calculating the quotient.

2. Compare the decimals to choose the better ratio according to the situation.

Ratios can also be compared by expressing them in the form |\displaystyle \frac{a}{b}| using a common denominator. Once they have the same denominator, it is possible to compare the numerators to decide which is the best ratio for the situation.

In a broccoli soup recipe, 250 ml of cream is added for every 1000 ml of soup. For the carrot soup, it is suggested to add 700 ml of cream per every 2500 ml. Which soup is creamier?

1. Express each ratio as a decimal by calculating the quotient.
   There are two ratios.
   |250:1000| and |700:2500|
   The quotients obtained are:
   |\bullet| Broccoli soup: |250\div1000=0.25|
   |\bullet| Carrot soup: |700\div2500=0.28|

2. Compare the decimals to choose the better ratio for the situation.
   We are looking for the creamier soup, or the one with the highest cream/soup ratio.
   Because |0.28>0.25|, the carrot soup is creamier.

*It is also possible to convert the two ratios to fractions with a common denominator and then compare the numerators. ||\begin{align}\frac{250\color{green}{\times 5}}{1000\color{green}{\times 5}}&=\frac{1250}{5000} & &\qquad & \frac{700\color{green}{\times 2}}{2500\color{green}{\times 2}}&=\frac{1400}{5000}\end{align}||Note that |1400>1250|. The end result is the same; the carrot soup is creamier.

Consider a ratio |a:b| or |\displaystyle\frac{a}{b}.|
To increase the ratio, we can:

• increase the value of |a| (numerator).

• decrease the value of |b| (denominator).

To decrease the value of the ratio, we can:

• decrease the value of |a| (numerator).

• increase the value of |b| (denominator).

A farmer owns 45 sheep for every 65 horses.
The ratio representing the situation is |\displaystyle \frac{45}{65}|.

a) Present two ways for the farmer to increase the sheep/horse ratio.

• 1^st way: Buy more sheep.
  If he buys |\color{green}{5}| more sheep, the result is: ||\displaystyle \frac{45\color{green}{+5}}{65}=\frac{50}{65}\Rightarrow \frac{50}{65}\color{red}{>} \frac{45}{65}||

• 2nd way: Sell some horses.
  If he sells |\color{green}{10}| horses, the result is: ||\displaystyle \frac{45}{65\color{green}{-10}}=\frac{45}{55}\Rightarrow \frac{45}{55}\color{red}{>} \frac{45}{65}|| *To prove it, one can calculate the quotient of each ratio.

b) Present two ways for the farmer to decrease the sheep/horse ratio.

• 1st way: Sell some sheep.
  If he sells |\color{green}{2}| sheep, the result is: ||\displaystyle \frac{45\color{green}{-2}}{60}=\frac{43}{60}\Rightarrow \frac{43}{60}\color{red}{<} \frac{45}{60}||

• 2nd way: Buy more horses.
  If he buys |\color{green}{7}| horses, the result is: ||\displaystyle \frac{45}{65\color{green}{+7}}=\frac{45}{72}\Rightarrow \frac{45}{72}\color{red}{<} \frac{45}{65}|| *To prove it, we can calculate the quotient of each ratio.