1. Rates Rule...Ratios Drool

Rates can come in many different forms, and can be quite confusing.  A Rate is a fraction describing how many of one thing compared to another thing, such as miles per gallon.

A "Ratio" is a rather simplified form of Rates where the fraction is rather generic, such as 3 to 1, or 3:1.  We'll be discussing ratios later on.

English phrases used to describe a rate or ratio in a word problem can be: 

"per" -  as in miles per gallon

"for every" -  as in one cup of sugar for every two cups of flour

"compared to" - as in I am only 4 feet tall compared to Billy who is 5 feet tall

"in"or "in about" - as in I can eat 4 of those in about 3 minutes

There is no magic as to how describe a rate: one thing compared to another.  Miles per gallon and gallon per miles are both valid, and can be used depending on what you are trying to solve.  

So the key is, whatever you are solving for, put THAT number on top in your rate.  For example:

Your car gets 30 miles per gallon, and you have 12 gallons in your tank.  How far (i.e. how many miles) will you be able to drive until the tank is empty?  The problem wants to solve for distance, or miles.  Therefore, describe the rate with miles on top:

However, what if the question was how many gallons would you need to drive 500 miles?  In this case, you are solving for gallons not miles…so describe the rate with gallons on top.

Cancelation of Units
An important thing to remember when setting up equations from word problems is to pay attention to the units "above the line" and "below the line", and set up the equation so that the units cancel out yielding the answer you want.  For example, in the sample problem above, we set the rate with gallons on top and miles below so when we multiply by the number of miles, the two "miles" (one "above the line", and the one "below the line" cancel out...and only gallons are left.

Let’s now try a few rate-type word problems (you'll find the answers in the next blog post below):

Rate Q1: The 90 minute math test has 25 word problems.   You know it takes you about 15 minutes to solve 3 word problems.  At that rate, how many word problems can you solve in the 90 minutes?  Are you going to get through all of the problems?  At what rate are you solving word problems?

Rate Q2:  The teacher has to grade all of these math tests.  By watching the clock, she noticed that she graded 5 word problems in about 12 minutes.   Since there are 25 word problems per test, how long will it take her to grade one test?  What is the rate of which she is grading the word problems?

If there are 30 students taking the tests, how long will it take her to grade all of the tests?  At what rate is she grading each test?

Rate Q3:  It is 1:20 pm on a Friday afternoon.  The school bell will ring exactly at 2 o’clock, so you have 40 minutes remaining until the weekend starts.  However, the teacher is planning to hand out a pop math quiz before the end of the school day.  You and your friends decide that if you ask enough questions to keep the teacher busy, maybe there won’t be enough time for the test!   Yesterday, you noticed that 3 math questions took about 8 minutes to ask and answer.  How many questions do you and your friends need to come up with to take up the 40 minutes left in the day?  What is the rate of questions to be asked?

Rates as Percentages

Percentages are simply rates with "100" as the denominator, for example 33% is equal to 33 out of 100, or 30/100.  If you have a different "total" amount, such as 500 and you want to find 33%, then you set up the rates to be equal.

In this case above, the rate of 33 to 100 is the same as ? will be to 500...both are 33%.  Hence the "?" will be 33% of the 500.  So, if you solve for "?", 

OK, got it?  Let's try a few percentage type problems.  Again, the answers are presented in the next blog post below.

Rate Q4:  The Math test has 30 word problems.  30% are hard, and 40% are medium.  How many are going to be easy?

Rate Q5:  There are 30 students in the class.  60% are boys.  How many are girls?

Rate Q6:  The girls in the class tend to get 80% of the math answers correct.  However, the boys only get 60% of the answers correct.  Using your answers from Rate Q5 above, and assuming there are 30 word problems in each test, how many total word problems will the girls get correct?  How many total word problems will the boys get correct?   Which group wins with the most total correct answers?