What is a “rate”?
A rate is similar to a ratio. It is a comparison of two quantities. But in the case of ratios, the comparison is between two quantities of the same unit. On the other hand, a rate compares two quantities of different units. In mathematical terms,
Rate: a units : b units
For example, we can express the price of fuel using the rate: $1.2 per ¼ gallons.
What is a unit rate?
When a quantity is compared to one unit of another quantity, the rate is known as the unit rate. In mathematical terms,
Unit rate is: \( \frac{a}{b}\) units : 1 unit
In the previous example, the rate of fuel was expressed per ¼ gallon. We can also express the rate of one gallon of fuel using unit rates. Rate of fuel: $4.8 per gallon.
How do we represent rates and unit rates?
Rates and unit rates can be expressed in two ways:
1) Using the symbol ‘/’ (a forward slash): 60 miles/hour
2) Using the word ‘per’ : 60 miles per hour
Relation between rate and unit rate
A “rate” is a unit rate only when one of the quantities has a unit value. For example, think of how we measure speed. If you need 3 hours to travel 90 miles, we can express the rate as 90 miles per 3 hours. But if you want to express it in unit rate, one of the quantities should have a unit value. While expressing speed, the unit value of time, one hour, is usually taken into consideration. Hence, we need to divide the distance by 3.
So, the unit rate of speed is 30 miles per hour.
Application of rates
Rates have several applications. Here are a few cases in which rates can be used in real life:
1) To compare the value of goods (to compare the weight of chocolate per dollar among different varieties of chocolates)
2) Used to express quantities like speed, density, cost per area or volume, and so on.
3) Used in statistics to measure literacy rate, fertility rate, and so on.
Finding unit rates
Unit rate can be calculated by performing basic operations on a given rate. We just have to rearrange the rate expression in such a way that one of the quantities has a unit value.
Examples
Example 1: Suppose a bar of chocolate “A” weighs 2.5 oz. and costs $1.5, and another bar of chocolate “B” weighs 3.5 oz. and costs $2. Determine which chocolate has a better return value for money in terms of its quantity.
Solution:
Chocolate bar A: 2.5 oz. per $\( 1.5=\frac{2.5}{1.5}~oz\) per $
= 1.67 oz. per $
Chocolate bar B: 3.5 oz. per $\( 2=\frac{3.5}{2}~oz\) . per $ = 1.75 oz. per $
Hence, chocolate bar B is of better value in terms of quantity.
Example 2: If a train travels 60 miles every thirty minutes, what is the distance it will cover in a day if it moves continuously at the same speed?
Solution: Let’s make a ratio table and find the missing values.
Since a day has 24 hours, it is clear that the train travels 2800 miles per day.
Example 3: A 40-gram serving of breakfast cereal has 12 grams of sugar in it. If the entire container of cereal contains eight servings, find the amount of sugar per container.
Solution:
We know that one serving of cereal has 12 grams of sugar and the box contains eight servings. So, we can multiply the amount of sugar in each serving by eight to get the amount of sugar per container.
Hence, the amount of sugar per container is 96 grams.
Example 4: If a person can walk half a mile in 10 minutes, find the distance that the person can walk in six hours, assuming that the person walks at the same pace.
Solution:
We know the pace at which the person walks for 10 minutes. We just need to find the distance covered in six hours at the same pace.
First, we can find the distance covered by the person in an hour by multiplying the given values by 6. To find the distance covered in 6 hours, we can multiply the result by 6 again.
Therefore, the distance that the person can walk in six hours is 18 miles.
In ratios, the quantities being compared have the same unit. In rates, the quantities being compared may have different units.
A rate is a comparison of two quantities of different units. A unit rate is a special case of rate wherein we compare a quantity to one unit of another quantity. For example, “30 miles per three hours” is a rate, and “10 miles per hour” is a unit rate.
Ratios and fractions are mathematically the same: 1 per 2 is the same as ½. But in most cases, they carry a different meaning. For example, 200 miles per 3 hours in a rate and but 200/3 as such has a different meaning.