First, we need to define a ratio, the relationship between two measurements or numbers.

The first form of a unit rate is made by simply taking one number and placing it over the other, as in division.



 


For example, if you want to express the statement, “There are 24 eggs in 2 dozens of eggs.” in fraction form it would be 24/2.


The second form is to use the phrase, “is to”, between the things you are comparing, to denote that they are a ratio.






If you take the example using this form, the answer would be, “24 is to 2”.


The third form is to use a colon to between the numbers that we are comparing; this is the standard ratio form.






The ratio form of the example is 24:2.


A rate is a special ratio that is composed of two different units or things; think of it as a ratio with words.
It's like describing what things are equal to in the simplest and most convenient form.






For example, there are 7.57 liters (2.0 US gal) in two gallons. We can write the statement as:


Unit rates are rates whose bottom number, or denominator, is equal to 1.






In other words, unit rates are basic rates defined for a single set – per hour, per minute, or per one meter. For example:;
, The unknown is what is being asked in the problem.


In order to determine the unknown, you need to read the problem carefully and sort out the data.
The unknown is usually preceded by the question part of the problem, indicated by: how, what, when, where, or who.
To determine the unit of the unknown, look for indications of measurements.
For example, if the question asks for distance, look for units like: meters, feet, inches, or centimeters; for mass: kilogram, gram, pound, or ton; and for time: seconds, hours, or minutes.
Try this sample problem: "Find the unit rate of a train if it can travel 236 miles (380 km) in 4 hours. And find how far will the train travel in six hours."






Since the problem asks for how far the train will travel, you can be sure that it is asking for the distance.






Out of the two units given: miles and hours, we know that miles is the unit for distance or displacement.
 After finding the unknown, sort out the data by dividing it into pairs with its appropriate units.


Read the problem and think about which number goes with what.
For easier calculations, you can place the unknown on the left hand side, and the other given data on the other side.
The unknown goes at the top of the division bar, or in the numerator’s place, and its partner goes at the bottom, or in the denominator's place.
Since we have another pair of given numbers, place the number with the same unit at the same place where you put the unknown.
The last piece of given data will go in the denominator's place.
The most important part is that the numerator should always be the same unit as the unknown you were looking for.
Returning to the example, since the unit of the unknown is miles, the given that should be adjacent to it should also be in miles.
Also, since the problem is asking for the unit rate, the denominator has to be 1 of whatever unit it is. So put 1 in the denominator:

, There are a number of ways in which you can solve these “double division” problems.

First, you can multiply the denominator of the left side by the numerator of the right side and divide that number by the denominator of the right hand side.






Alternatively, you can divide the right side first, and then multiply the denominator by the answer obtained.






The resulting unit rate will be a composite unit – two units that are combined by elementary operations, multiplication or division, to form one unified unit.
So, using the second method: dividing 236 by 4 and multiplying it by one:






Therefore, the train has a unit rate of 59 miles (95 km)/hour.
Answering the second question (And find how far will the train travel in six hours.), you can use another variable, "Y", to avoid confusion with the answers.






Since you know that the question is asking for the distance, you once again place the number that has units of distance in the numerator, and the remaining data in the denominator.






Another method is to just multiply the unit rate with the hours asked, since the unit is in a per hour basis.






Taking the unit rate of 59 miles (95 km)/hour and multiplying it by 6 hours yields the same answer.






So the train can travel 354 miles (570 km) in 6 hours.
Understand what unit rates are.
Determine the unknown.
List the given numbers in a fraction form.
Do the required operations to find the unit rate.
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