How to Change Rates Using Multipliers

Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

Consider miles per hour versus feet per second or pay per hour versus pay per minute. How do you get from one to the other? This lesson provides step by step instructions and examples of changing rates using multipliers.

What is a Rate?

Everywhere you go you will be exposed to rates: rates of speed, rates of pay, rates of currency exchange and plenty more. So what is a rate? A rate is a comparison of how two variables measure against each.

You may be familiar with rates of speed, so we can use this to get a clear idea of the definition of rate. So, your rate of speed (most people just say 'speed') is the distance you travel in a specific amount of time. Do you see the two variables? Distance and time are the variables being compared. Think about how we give speed information: ''I was going 35 miles per hour when they sped past me.'' Miles per hour compares the distance in miles to the amount of time required in hours. In this example, it takes one hour to travel 35 miles.

Rates as Multipliers

Every rate will have two variables compared to each other. Every rate can be described as a ratio or fraction (variable 1 / variable 2). Having two variables in a constant relationship to each other means that rates offer a unique formula for solution:

Variable 1 = rate x Variable 2

Multiply the rate times the second variable to find the first. This is called using a multiplier for solving rate problems; multiply the rate times one variable in the rate to find the value of the other variable. As long as you know at least two parts of the puzzle (variable 1, variable 2, or rate), you can use simple algebra to solve for the third part.

Pay Rate Example

Let's say you just got a new job. You know that you get paid $6.25/hr. You worked 5 hours so far. How much money have you earned?

  • Step 1- Formula: $ (variable 1) = rate x time (variable 2)

You are looking for the total amount of money earned, so the amount of the first variable equals the pay rate of $6.25 times the second variable which in this case is five hours.)

  • Step 2- Substitution: $ = 6.25 x 5
  • Step 3- Solve: $31.25 has been earned so far at your new job.

Should the answer be in dollars? Well anything over itself in a fraction equals 1 (cancels out). So the unit of measure for the second variable is in the denominator of the rate fraction while in the numerator of the value being multiplied, thus the unit of measure actually cancels out leaving only the unit of measure for the first variable.

Pay Rate Per Hour
Rate example 1

Flow Rate Example

Let's say your job is to fill swimming pools. You use a hose with a flow rate of 20 g/hr. Remember, this means 20 gallons of water leaves the hose every hour. How many hours will it take to fill a 600 gallon pool?

  • Formula: gallons (variable 1) = rate x time (variable 2)

The total number of gallons (600) equals the flow rate (20 g/h) times the time (unknown variable).

  • Substitute: 600 = 20 x h
  • Solve: 600/20 = h = 30

So, it will take 30 hours to fill the pool at the current flow rate.

Rate Change Using Multiplier Method

What if you wanted to change the rate to different measurements? How much money do you make in a minute or how many liters of water does the hose put out every second?

To change the rate you are using, multiply a new unit ratio (with the value of one) to the old rate ratio in order to cancel out the old measurement units.

Basically, you are multiplying by 1 in order to cancel the units without changing the value of the rate. Remember, when you multiply any number by 1 you do not change the value (3 x 1 = 3 and Y x 1 = Y).

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