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# Unit Rate

## Introduction

A unit rate, often known as a single-unit rate, compares one unit of one quantity with a different unit of another quantity. Unit rates are frequently encountered in daily life. Some typical example of unit rates is the distance covered per time, like kilometers per hour and meters per second. In the supermarket, when we get the cost per item is an example of the unit rate.

A unit rate is a ratio between 2 distinct units with one as the denominator. To put it simply, it tells the rate in the lowest terms or the amount for one.

In this article, we will have the definition of rate, unit rate, and unit price and some solved examples in calculating unit rates.

## What is the rate?

### Definition

Two quantities in different units are compared using a rate. It can be expressed as either a fraction or a ratio. A rate is a ratio that usually involves a unit of time.

For example, an automobile travels 40 kilometers in 2 hours. It means that in 2 hours, the automobile covers forty kilometers. The way we compare the two different units of distance and time as a single ratio is a rate.

## What is a unit rate?

### Definition

A unit rate, often known as a single-unit rate, compares one unit of one quantity with a different unit of another quantity. A unit rate is a ratio between 2 distinct units with one as the denominator. To put it simply, it tells the rate in the lowest terms or the amount for one.

Some examples of unit rates are kilometer per hour (kph), miles per hour (mph), words per minute (wpm), etc. In problems, the symbol “/” may substitute the word “per.”

The following are examples of unit rates:

( a ) The car travelled 80 kilometers in 2 hours. The unit rate is 40 kilometers per hour or 40 kph.
( b ) There are 200 students and 5 teachers. The unit rate is 40 students per teacher or 40 students/teacher.
( c ) A gardener earned 520 dollars in 40 hours of work. The unit rate is 13 dollars per hour or 13 dollars/hour
( d ) There are 420 apples sold for 6 days. The unit rate is 70 apples sold per day or 70 apples/day.
( e ) The patient has 148 heartbeats in 2 minutes. The unit rate is 74 heartbeats per minute or 74 heartbeats/minute.
( f ) A sprinter runs 120 meters in 20 seconds. The unit rate is 6 meters per second or 6 meters/second.

## Calculating the Unit Rate

A unit rate compares one unit of one quantity with a different unit of another quantity. To calculate the unit rate, we must divide the numerator by the denominator of the given rate.

Let us calculate the unit rate of each of the following scenarios below:

( a ) $45 for 15 pencils ( b ) 71 miles in 2 gallons ( c ) 320 words in 8 minutes ( d ) 18 miles in 150 minutes ( e ) 96 dollars in 8 hours ( f ) 18 dollars for 3 gallons of milk Solution ( a )$45 for 15 pencils

To calculate the unit rate, let us divide 45 dollars by 15 pencils.

Unit Rate = $\frac{45\: dollars}{15\: pencils}$

Unit Rate = 3 dollars/pencil

Therefore, the unit rate of 45 dollars per 15 pencils is 3 dollars per pencil.

( b ) 71 miles in 2 gallons

To calculate the unit rate, let us divide 71 miles by 2 gallons.

Unit Rate = $\frac{71\: miles}{2\: gallons}$

Unit Rate = 35.5 miles/gallon

Therefore, the unit rate for 71 miles per 2 gallons is 35.5 miles per gallon.

( c ) 320 words in 8 minutes

To calculate the unit rate, let us divide 320 words by 8 minutes.

Unit Rate = $\frac{320\: words}{8\: minutes}$

Unit Rate = 40 words/minute

Therefore, the unit rate for 320 words per 8 minutes is 40 words per minute.

( d ) 18 miles in 150 minutes

To calculate the unit rate, let us divide 18 miles by 150 minutes.

Unit Rate = $\frac{18\: miles}{150\: minutes}$

Unit Rate = 0.12 miles/minute

Therefore, the unit rate for 18 miles per 150 minutes is 0.12 miles per minute.

( e ) 96 dollars in 8 hours

To calculate the unit rate, let us divide 96 dollars by 8 hours.

Unit Rate = $\frac{96\: dollars}{8\: hours}$

Unit Rate = 12 dollars/hour

Therefore, the unit rate of 96 dollars per 8 hours is 12 dollars per hour.

( f ) 18  dollars for 3 gallons of milk

To calculate the unit rate, let us divide 18 dollars by 3 gallons of milk.

Unit Rate = $\frac{18\: dollars}{3\: gallons\: of\: milk}$

Unit Rate = 6 dollars/gallon of milk

Therefore, the unit rate of 18 dollars per 3 gallons of milk is 6 dollars per gallon or 6 dollars/gallon.

## Unit Price

The cost of a single item is expressed as a unit price. Unit Rate and Unit Price are highly comparable. To determine the unit price of a given good, divide the total cost by the number of items.

The formula calculates the unit price as the total price over total units.

Unit Price=$\frac{Total\: Price}{Total\: Units}$

Let us say, for example, you spent 24 dollars for 6 dozen eggs. To calculate the unit price, we must divide the total cost, 24 dollars, by the total units, 6 dozen. Hence, we have,

Unit Price = $\frac{Total\: Price\: of\: Eggs}{Number\: of\: Dozen\: of\: Eggs}$

Unit Price = $\frac{24\: dollars}{6\: dozen}$

Unit Price  = 4 dollars/dozen

Therefore, the unit price of 24 dollars for 6 dozen is 4 dollars per dozen.

For another example, suppose you paid 120 dollars for a car you rented for 3 days. To calculate the unit cost, let us divide the cost of 320 dollars by the total units of 3 days.

Thus, we have,

Unit Price = $\frac{Total\: amount\: paid}{number\: of\: days}$

Unit Price = $\frac{120\: dollars}{3\: days}$

Unit Price = 40 dollars/day

Therefore, the unit price for the car rental is 40 dollars per day.

## Solved Examples on Unit Rate

Example 1

The tourists travelled 30 kilometers in 2 hours. Calculate the unit rate.

Solution

Since the tourists travelled 30 kilometers in 2 hours, we must divide 30 kilometers by 2 hours to calculate the unit rate. Hence, we have,

Unit Rate = $\frac{Total\: distance\: covered}{Number\: in\: hours}$

Unit Rate = $\frac{30\: kilometers}{2\: hours}$

Unit Rate = 15 kilometers/hour

Therefore, the unit rate for 30 km per 2 hours is 15 kilometers per hour or 15 kph.

Example 2

A printer prints 60 pages in 20 seconds. Determine the unit rate of pages printed each second.

Solution

The number of pages a printer makes in 20 seconds is 60 pages. To calculate the unit rate, we must divide the total number of pages printed, 60 pages, by the total number of seconds, 20 seconds. Hence, we have,

Unit Rate = $\frac{Total\: pages}{Number\: of\: seconds}$

Unit Rate = $\frac{60\: pages}{20\: seconds}$

Unit Rate = 3 pages/second

Therefore, the unit rate for 60 pages in 20 seconds is 3 pages per second.

Example 3

Find the unit rate if a worker earns $800 in 40 hours. Solution Since the man earns$800 in 40 hours, we must divide 800 by 40 to calculate the unit rate.

Thus, we have,

Unit Rate = $\frac{Total\: earnings}{Number\: of\: hours}$

Unit Rate = $\frac{800\: dollars}{40\: hours}$

Unit Rate = 20 dollars/hour

Therefore, the unit rate of $800 for 40 hours is$20 per hour.

Example 4

Maria bought 4 chocolate bars for her sisters. How much did each chocolate bar cost if she purchased them for 8 dollars?

Solution

Since we are asked to find the price of each chocolate bar, we must divide the total cost by the total units. Hence, we have the solution,

Unit Price = $\frac{Total\: Price\: of\: chocolate\: bars}{Number\: of\: chocolate\: bars}$

Unit price = $\frac{8\: dollars}{4\: chocolate\: bars}$

Unit Price = 2 dollars/chocolate bar

Therefore, each chocolate bar costs 2 dollars.

Example 5

Martin took 6 hours to run 9 kilometers. Find the unit rate.

Solution

To find the unit rate, divide 9 kilometers by 6 hours. So, we have,

Unit Rate = $\frac{Total\: distance\: covered}{Number\: in\: hours}$

Unit Rate = $\frac{9\: kilometers}{6\: hours}$

Unit Rate = 1.5 kilometers/hour

Therefore, the unit rate of 9 kilometers in 6 hours is 1.5 kilometers per hour or 1.5 kph.

Example 6

A businessman bought 5 television for $3150. Calculate the unit cost. Solution To calculate the unit cost, let us divide$3150 by the total number of television the businessman bought. So, we have,

Unit Cost = $\frac{Total\: Price\: of\: television}{Number\: of\: televisions}$

Unit Cost = $\frac{3150\: dollars}{5\: televisions}$

Unit Cost = 630 dollars/television

Therefore, the unit cost is $630 per television. Example 7 Calculate the unit rate of each of the following: ( a )$200 for 8 bags
( b ) 50 kilometers in 4 gallons
( c ) 450 words in 9 minutes
( d ) 250 meters in 8 minutes
( e ) 490 dollars in 35 hours
( f ) 60 dollars for 12 dozen eggs

Solution