What is “a rate”?
When two quantities of different units are compared and expressed as a ratio, it is known as a rate. For example, the ratio of the distance covered by a car in miles to the time taken in hours. Miles and hours are two different units here.
What is the unit rate?
The unit rate compares a certain number of units of one quantity to units of another quantity. In other words, the second quantity in the comparison is always 1.
Walking 3 kilometers per day, or reading 20 pages per hour are two examples of unit rates. In both examples, a particular quantity is compared to 1 unit of a different quantity.
What are equivalent rates?
Equivalent rates have the same unit rates. Equivalent rates can be found by dividing or multiplying the numerator and the denominator by the same number, just like equivalent ratios.
Example 1:
A car travels 390 miles every 5 hours. Calculate the unit rate.
Solution:
The miles to hours ratio is 390 :5. To get the unit rate in miles per hour, we divide the ratio by 5.
Hence, the unit rate is 78 miles per hour.
Example 2:
Every 6 hours, a pond fills up by 120 liters from the local freshwater spring. What is the quantity of water that fills in 10 hours?
Solution:
The ratio of the quantity of water to hours is 120 :6. To get the unit rate in liters per hour, we divide by 6.
So, the unit rate is 20 liters per hour.
To find the quantity of water flowing into the pond in 10 hours, we multiply both the quantities of the unit rate by 10.
Hence, 200 liters of water enter the pond in 10 hours.
Solved Unit Rate Examples
Example 1:
A motorcycle can travel 240 miles on 5 gallons of gasoline. Calculate the unit rate in miles per gallon.
Solution:
Simply divide 240 by 5 to solve this problem.
\(\frac{240~miles~\div ~5}{5~gallons~\div~5}=\frac{48~miles}{1~gallons}\)
The unit rate is 48 miles per gallon.
Hence, the motorcycle can cover 48 miles on a gallon of fuel.
Example 2:
Every 6 seconds, a jet plane travels 5 miles. Calculate the unit rate.
Solution:
The miles to seconds ratio is 5 :6. To get the unit rate in miles per second, divide by 6.
Hence, the unit rate is \(\frac{5}{6}\) miles per second.
Example 3:
A local store charges the following prices for jars of jelly. Find the best deal.
Solution:
The container with the lowest cost per unit is the best buy. That is, the jar of jelly that has the lowest unit rate in terms of cost per ounce is the best deal.
Divide the price of the jar by the number of ounces in it to get the cost per ounce for each jar. If necessary, round to the nearest thousandth.
Therefore, the 24-ounce jar has the lowest unit rate, and is hence the best buy because it has the lowest cost per ounce of $0.129.
Example 4:
Juice is available in two forms: cans with concentrated juice cans, and ready-to-serve cartons.
Which of the following options is the best buy?
12 oz can make 48 ounce of jouce for $1.69 60 oz cartoon for $2.59
Solution:
The best buy is the item that has the lowest cost per ounce of juice. That is, we need to buy the carton, either concentrated or ready to serve, that has the lowest unit rate in terms of cost per ounce.
To find that, divide the price by the ounces to determine the best buy.
For the concentrated cans, though the quantity in the can is only 12 ounces, using that we can make 48 ounces of juice. Hence, to find the unit rate, we take 48 ounces and not 12 ounces.
Concentrate \(\frac{$1.69}{48~ounces}=$0.0352\) per ounce
Carton \(\frac{$2.59}{60~ounces}=$0.0432\) per ounce
The concentrated cans are the better buy, even if you must mix it yourself, to take a sip!
Units are the tools that are used to measure and compare various objects. When all of the measurement units are the same, the comparison is simple. That is, we can compare 1 meter to 5 meters and identify which one is larger.
However, if we need to compare 5 meters to 60 centimeters, we will have to convert the quantities to a common unit and then make the comparison.
When two ratios are reduced to their lowest form and they have the same value, they are termed equivalent ratios. Equivalent ratios can be obtained by multiplying or dividing both the numerator and the denominator of the ratio by the same number.
A rate compares the values of two different quantities. In a unit rate, however, one quantity is compared to 1 unit of the other quantity.