Rates

A rate is a relationship between 2 quantities of a different nature expressed as a fraction and denoted as |a/b| or |\dfrac{a}{b}.|

When stating a rate, it is important to write the units. If they are not indicated, it is implied that they are the same and that it is, instead, a ratio.

A unit rate is a rate with a denominator of |1.|

At |20\ ^\circ\text{C},| |10| litres of water weighs |9.98| kilograms. Which unit rate reflects this situation?

The unit rate is found by dividing the numerator by the denominator.||9.98\ \text{kg}\div10\ \text{L}=0.998\ \text{kg}/\text{L}||For example, |0.998\ \text{kg}/\text{L}| is the unit rate that is equivalent to |\dfrac{9.98\ \text{kg}}{10\ \text{L}}.|

Division is not the only way of calculating a unit rate. In fact, it can also be found by simplifying the fraction.

At the grocery store, Caroline paid |\$4.32| for |6| avocados. What was the price of one avocado?

To find the unit rate, divide the numerator and denominator by |6.| ||\dfrac{ \$4.32\color{#3a9a38}{\div6}}{6\ \text{avocados}\color{#3a9a38}{\div6}}=\dfrac{ \$0.72}{1\ \text{avocado}}||Caroline paid |\$0.72| for 1 avocado. So, |\$0.72/\text{avocado}| is equivalent to |\dfrac{\$4.32}{6\ \text{avocados}}.|

An hourly rate is a unit rate that expresses a quantity of money in relation to an hourly basis (of |1| hour).

Charlotte earns |\$138| in an |8\ \text{h}| work day. What is her hourly rate (her hourly wage)?||\$138\div{8\ \text{h}}=\$17.25/\text{h}||Charlotte’s hourly wage (wage for |1\ \text{h}| of work) is |\$17.25/\text{h}.|

To get to Montreal, Gaston travelled |240\ \text{km}| in |3\ \text{hours}.| What was Gaston’s average speed?||240\ \text{km}\div{3\ \text{h}}=80\ \text{km}/\text{h}||Gaston’s average speed was |80\ \text{km}/\text{h}.|

Equivalent rates are rates that have the same units of measurement and can be reduced to the same unit rate.

Xavier earns |\$80| for |5\ \text{h}| of work, while Julie earns |\$112| for |420\ \text{min}.| Are their wages equivalent?

Notice that the units of time are not the same, which means the units must be converted. Changing minutes to hours gives the following:||420\ \text{min}\div 60\ \text{min/h}=7\ \text{h}||

Now that the rates have the same units, they can be simplified to determine the unit rates.

Since the unit rates are equal, we can see that Xavier’s salary and Julie’s salary are equivalent.||\dfrac{\$80}{5\ \text{h}}=\dfrac{\$112}{420\ \text{min}}||

1. Ensure the rates have the same units of measure. Convert as needed.

2. Express the rates as unit rates.

3. Compare the unit rates.

Car A burns |11\ \text{L}| of gas every |100\ \text{km},| while car B burns |18\ \text{L}| of gas every |150\ \text{km}.| Which of the 2 cars burns less gas?

1. Ensure the rates have the same units of measure. Convert as needed.
   In this example, the rates being compared have the same units of measure.

2. Express the rates as unit rates.

3. Compare the unit rates.
   Since |0.11\ \text{L}/\text{km}<0.12\ \text{L}/\text{km}|, car A has the lowest fuel consumption.

Stephanie looks at local grocery store flyers to figure out which one offers the best price on ground beef. Dufour’s sells its ground beef at |\$8.50| for |2\ \text{kg},| while Buy-in-Bulk sells it at |\$12.24| for |3000\ \text{g}.| Which grocery store offers Stephanie the best value for her money?

1. Ensure the rates have the same units of measure. Convert as needed.
   The weights are not given in the same unit. Convert the |\textbf{g}| to |\textbf{kg}.|||3000\ \text{g}\ \stackrel{\!\!\div\,1000}{\Large\longrightarrow}\ 3\ \text{kg}||

2. Express the rates as unit rates

3. Compare the unit rates.
   Find the grocery shop that sells the cheapest ground beef. |\$4.25/\text{kg}>\$4.08/\text{kg},| so Stephanie should buy from Vrac-à-Vrac.

The previous problem can also be solved using equivalent rates, rather than unit rates. To compare both prices, find equivalent rates with the same denominator. For example, the prices at both grocery stores for the same quantity of ground beef, namely |6\ \text{kg}| can be found.

For the same quantity of ground beef, Vrac-à-Vrac has the better deal.

Let’s say the rate is |\dfrac{a}{b}.|

Pierre is currently earning |\$525| for |35| hours of work.

The rate representing this situation is |\dfrac{\$525}{35\ \text{h}}.|

a) There are 2 ways Pierre’s employer can increase the value of his hourly rate. What are they?

• 1st way: Increase the amount of money for the same number of hours worked.

  If Pierre earns |\color{#3a9a38}{\$70}| more, it gives:||\begin{align}\dfrac{\$525 \color{#3a9a38}{+\$70}}{35\ \text{h}}&=\dfrac{\$595}{35\ \text{h}}\\\\ \dfrac{\$595}{35\ \text{h}}&>\dfrac{\$525}{35\ \text{h}}\end{align}||

• 2nd way: Decrease the number of hours worked while keeping the same salary.

  If Pierre is asked to work |\color{#3a9a38}{5\ \text{h}}| less, it gives:||\begin{align}\dfrac{\$525}{35\ \text{h}\color{#3a9a38}{-5\ \text{h}}}&=\dfrac{\$525}{30\ \text{h}}\\\\ \dfrac{\$525}{30\ \text{h}}&>\dfrac{\$525}{35\ \text{h}}\\\\\dfrac{\$17.50}{1\ \text{h}}&>\dfrac{\$15}{1\ \text{h}}\end{align}||

b) There are 2 ways Pierre’s employer can decrease the value of his hourly rate. What are they?

• 1st way: Decrease the amount of money for the same number of hours worked.

  If Pierre earns |\color{#3a9a38}{\$35}| less, it gives:||\begin{align}\dfrac{\$525\color{#3a9a38}{-\$35}}{35\ \text{h}}&=\dfrac{\$490}{35\ \text{h}}\\\\ \dfrac{\$490}{35\ \text{h}}&<\dfrac{\$525}{35\ \text{h}}\end{align}||

• 2nd way: Increase the number of hours worked while keeping the same salary.

  If Pierre is asked to work |\color{#3a9a38}{3\ \text{h}}| more, it gives:||\begin{align}\dfrac{\$525}{35\ \text{h}\color{#3a9a38}{+3\ \text{h}}} &= \dfrac{\$525}{38\ \text{h}}\\\\ \dfrac{\$525}{38\ \text{h}}&<\dfrac{\$525}{35\ \text{h}}\\\\\dfrac{\$13.82}{1\ \text{h}}&<\dfrac{15}{1\ \text{h}}\end{align}||