Subjects
Grades
Identify the quantities (values) to compare.
Determine if it is a ratio or a rate.
- If the quantities are the same in nature, it is a ratio
- If the quantities are not the same, it is a rate.
If it is a ratio, ensure the units are the same and convert them, if necessary.
Express the ratio or rate in its appropriate form.
- For ratios, simplify the units and reduce them as needed.
- For rates, calculate the unit rate or an equivalent rate.
How to choose the position of the terms in a ratio or a rate:
Unless otherwise instructed, the smaller quantity is written in the numerator and the larger in the denominator.
In a word problem, prepositions such as “for”, “by”, or “in” are usually used to express a rate. The quantity in front of the keyword is the numerator’s value and the quantity after the keyword is the denominator’s value.
Frank is |1.35| metres tall. During afternoon recess, he notices his shadow measures |35| centimetres. What ratio or rate can be established from this situation?
Identify the quantities (values) to compare.
The 2 quantities to compare are Frank’s height |(1.35\ \text{m})| and the length of his shadow |(35\ \text{cm}).|
Determine if it is a ratio or a rate.
Both quantities are lengths, so it is a ratio.
If it is a ratio, ensure the units are the same and convert them, if necessary.
One of the quantities is expressed in metres and the other is expressed in centimetres. Therefore, unit conversion must be performed.
||1.35\ \text{m}\ \stackrel{\times 100\ }{\Longrightarrow}\ 135\ \text{cm}||
Express the ratio in its appropriate form.
The ratio between the length of the shadow and Frank’s height is the following:
||\dfrac{35\ \cancel{\text{cm}}}{135\ \cancel{\text{cm}}}=\dfrac{35}{135}||
We can reduce the ratio, as follows:
||\dfrac{35\color{#3a9a38}{\div 5}}{135\color{#3a9a38}{\div 5}}=\dfrac{7}{27}||
Therefore, Frank’s shadow is |\dfrac{7}{27}| of his height.
Yusef is a swimmer in the Sport-Études program at his school. At his last competition, he completed a |50| metre butterfly swim in |32| seconds. What ratio or rate can be determined from this situation?
Identify the quantities (values) to compare.
The 2 quantities to compare are the distance traveled |(50\ \text{m})| and the time, |(32\ \text{s}).|
Determine if it is a ratio or a rate.
The 2 quantities are not of the same nature, so it is a rate.
If it is a ratio, ensure the units are the same and convert them, if necessary.
Since it is a rate, no conversion is required.
Express the ratio or rate in its appropriate form.
In this situation, the rate is worded as follows: “a |50| metre butterfly swim in |32| seconds.” So, |50\ \text{m}| goes in the numerator and |32\ \text{s}| in the denominator.
||\dfrac{50\ \text{m}}{32\ \text{s}}||
The unit rate is calculated by dividing the numerator by the denominator.
||50\ \text{m}\div 32\ \text{s}\approx 1.56\ \text{m/s}||
This unit rate represents Yusef's average speed.
It is often best to express a rate as a unit rate, because it is more convenient to compare different rates if they all have a denominator of |1.|
Also, most of the rates used in daily life are given as a unit rate. We use |\text{km}/\text{h}| or |\text{m}/\text{s}| for speeds, |\$/\text{h}| for salaries, |\$/\text{L}| for the price of gas, etc.