Difference between Ratio and Rate
A ratio is a comparison of two quantities having the same unit. We use ratios to compare dollars to dollars, miles to miles, pounds to pounds, and so on. Just like ratios, rates allow us to compare two quantities.
However, unlike ratios, the two quantities compared in rates are different, that is, they have different units. The general form of a ratio is a:b, and the general form of a rate is a units:b units. A rate can be further modified and expressed as a comparison of one quantity to one unit of another quantity. This type of comparison is known as unit rate. If we know the rate of two quantities, we can calculate the unit rate using a simple formula.
List of Formulas
Just like ratios, rates are generally expressed as a:b. However, in the case of rates, a and b have different units. So, the rate is expressed as a units : b units. We can use a simple formula to convert this rate into a unit rate.
- Unit rate \(=\frac{a}{b}\) units : 1 unit
How to use the Unit Rate Formula?
A ratio is generally expressed as a : b, and a rate is generally expressed as a units:b units. In either case, we can calculate the unit rate if we know the values of a and b.
Suppose a person runs 15 miles in 3 hours. We know the values of a and b in this case.
a = 15 miles
b = 3 hours
We know that a : b is the rate at which the person runs.
Rate = a miles : b hours = 15 miles : 3 hours
Unit rate \(=\frac{a}{b}\) miles:1 hour \( =\frac{15}{3}\) miles:1 hour
= 5 miles : 1 hour
So, the person runs 5 miles per hour.
We can use the same logic to find the unit rate for different cases.
Solved Examples
Example 1: A car travels 150 miles in 5 hours. What is the average distance traveled by the car in an hour?
Solution:
The average distance traveled by the car in an hour is the unit rate of distance, also known as speed.
Rate of distance traveled = a miles : b hours = 150 miles : 5 hours
The unit rate of distance covered \(=\frac{a}{b}\) miles : 1 hour \(=\frac{150}{5}\) miles : 1 hour
= 30 miles : 1 hour
Hence, the average distance covered by the car is 30 miles per hour.
Example 2: If a sculptor completes one-fifth of a statue in three weeks, what portion of the statue does he complete per week?
Solution:
The portion of the statue completed per week is the unit rate of completing the statue.
The portion of the statue completed = a \(=\frac{1}{5}\)
The time taken to complete \(\frac{1}{5}\) of the statue = b = 3 weeks
Rate of completion = a of statue:b weeks \(=\frac{1}{5}\) of statue : 3 weeks
Unit rate \(=\frac{a}{b}\) of statue : 1 week\(=\frac{1}{5\times 3}\) of statue : 1 week
\(=\frac{1}{15}\) of statue : 1 week
Therefore, the sculptor completes \(\frac{1}{15}\) or one-fifteenth of the statue per week.
Example 3: A worker painted four-fifths of a building in 8 days. At what rate does he paint per day?
Solution:
The rate at which the worker paints per day is the unit rate of painting the building.
The portion of the building he painted = a \(=\frac{4}{5}\)
The time taken to paint 45 of the building = b = 8 days
Rate of completion = a of building : b days\(=\frac{4}{5}\) of building : 8 days
Unit rate \(=\frac{a}{b}\) of building:1 day \(=\frac{4}{5\times 8}\) of building : 1 day
\(=\frac{4}{40}\) of building : 1 day
\(=\frac{1}{10}\) of building : 1 day
Hence, the worker paints \(\frac{1}{10}\) or one-tenth of the building per day.
Example 4: A person buys 9 pounds of apples from store A for $12 and 15 pounds of apples from store B for $21. Which shop sells apples at a lower rate?
Solution: To find which shop sells apples at a lower rate we will determine the unit rate of apples at each shop.
Unit Rate \(=\frac{a}{b}\) units : 1 unit
For store A, a = $12 and b = 9 pounds
For store B, a = $21 and b = 15 pounds
The unit rate of apples at store A \(=\frac{a}{b}\) units : 1 unit
\(=\frac{12}{9}\) dollars : 1 pound
= $1.33 : 1 pound or $1.33 per pound
The unit rate of apples at store B = \(=\frac{a}{b}\) units : 1 unit
\(=\frac{21}{15}\) dollars : 1 pound
= $1.4 : 1 pound or $1.4 per pound
Therefore, store A sells apples at a lower rate.
A rate compares the values of two different quantities. However, in a unit rate, one quantity is compared to a unit of the other quantity.
Equivalent ratios are the fractions that give us the same fraction when simplified. Equivalent ratios can be obtained by multiplying or dividing both the numerator and the denominator of the ratio by the same number.
A ratio or a rate becomes a unit rate when the denominator is 1. Hence, to convert to a unit rate, we need to multiply or divide both the numerator and the denominator by a number such that the denominator is 1. The ratio thus obtained represents the unit rate.
The mixed number is first converted to an improper fraction. Then the numerator and the whole number is multiplied, the product becomes the numerator of the required fraction and the denominator is the same as that of the improper fraction. Thus, the required fraction is obtained.
We use unit rates to express speed, prices of goods and services, statistics related to demographics and the economy, and so on.