3. Ratios: 4 Out of 5 Agree

I'll bet the odds are 3:1 that you are just as confused as I am when to use the term "rate" and when to use the term "ratio".  They seem similar, if not the same.  No matter...the common core will be asking both types of questions, so we must be prepared for both.

Ratios can be described in three basic ways:

"for every x of these, there are y of those", for example, for every 3 girls there are 2 boys.
...or... the ratio of girls to boys is 3:2 (this is the most common description of a ratio)
...or... the ratio of girls to boys is 3/2  (but this is also common...so who knows)

Like rates, a ratio can be described as x:y, or as y:x ...and both are valid.  In the girls-to-boys example, the ratio can be described as 3:2 (ratio of girls to boys), or as 2:3 (boys to girls). Both ways are perfectly valid depending one what you are trying to solve.


SAFETY WARNING:    Many word problems will describe a ratio one way, and then ask you to answer for the opposite.  For example,

"for every 5 boys, there are 3 girls.  What is the ratio of girls to boys."

See what they did here?  They offered a ratio with girls first and boys second....then reversed the order for the ratio answer asking for boys first and girls second to see if you are paying attention.  Although the ratio was first described as 5 boys to 3 girls, the question is asking for a girls-to-boys ratio, which is 3:5.


Ratio Tables
Tables seem to prevalent in many common core studies.  They set up a table of numbers and expect you (or the student) to interpret ratios based ion the table...or perhaps completing the table.  For example:

There are three school classrooms of 6th graders.  They each have the same exact ratio of boys to girls in their class.

What is the ratio of boys to girls in these classrooms?

How many girls are in class #3?

Now think....the problems is asking for the quantity of girls.  You can set the ratios up any way you want.  Since we are solving for girls, let's show the ratios with girls on top:

The ratio of Girls to Boys is 2:3.  Hence, the ratio of Boys-to-Girls is the opposite, or 3:2.

The ratio of girls-to-boys in classroom #3 must be the same ratio as classrooms #1 and #2, which is 2:3.  Another way to think about it is:  "? is to 12 as 2 is to 3."  The two relationships must be the same.  Set up an equation which shows that the two ratios are equal to each other:

Now solve for ?, which yields the answer 8 girls.


This seems pretty straight forward, so let's try a few ratio problems (answers to follow in the next post):

Ratio Q1:  Three boys can eat 2 pizzas.  What is the ratio of boys to pizzas?  What is the ratio of pizzas to boys?

Ratio Q2:  The ratio of pieces of sausage to pieces of pepperoni on these pizzas is  3:1.

For every ____ pieces of pepperoni, there are ____ pieces of sausage.


Ratio Q3:  The three grade 6 classrooms have the same ratio of Boys to Girls.

 What is the ratio of Boys-to-Girls?

How many Girls are in Classroom #2?

How many Boys are in Classroom #3?

4. Answers to Ratio Problems

Ratio Q1:  Three boys can eat 2 pizzas.

The ratio of boys to pizzas is 3:2

The ratio of pizzas to boys is 2:3

These seem kinda obvious, so I am not sure what more to add.  If you have any questions, post a comment and I'll think of something clever.


Ratio Q2:  The ratio of pieces of sausage to pieces of pepperoni on these pizzas is 3:1.

For every 1   piece of pepperoni there are  3   pieces of sausage.


Ratio Q3:  Classroom #1 has 15 boys to 20 girls, or 15:20.  Break this down to its simplest form:

For Classroom #2, we need to solve for girls.  So flip the ratio so that girls "?" is on top.  In this case, we also flip the 3/4 ratio to 4/3.  So:  ? is to 12 as 4 is to 3.

 Now solve for ?, which is 16 girls in classroom 2.

For boys in classroom 3, describe the ratio with the boys "?" on top, which is now 3/4. So ? is to 12 as 3 is to 4.  Solve for the ?, which is 9 boys in classroom 3.

The final matrix is:

One important check to make sure you got the matrix correct:  The ratio of Boys on top to Girls on the bottom row is 3:4 (there are more girls than boys).  Hence, the top row must always be smaller than the bottom row.  If they are not, then you forgot to flip the ratio, or something.

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?

2. Answers to Rates Problems

Answer to Rate Q1:  

The question is asking for “how many word problems”, hence we need to show the rate with the number of word problems on top

Knowing the rate, we use it and the total time allowed in the test to determine the quantity of problems:

Answer to Rate Q2:

The rate we are looking for is "5 word problems in about 12 minutes.  This can be described as 5 word problems/12 minutes, or 12 minutes to grade 5 word problems.  The problem is asking for time - how long will it take.  So use the rate with minutes on top.

Use this rate to now solve for the time to do 25 problems.

We know the time for grading one test (which is a rate of 60 minutes/1 test)...now use that time to grade 25 tests.  In this case, I converted 60 minutes to one hour...

30 hours....Yikes!

Answer to Rate Q3

The problem is asking for the total number of questions required.  Hence, specify the rate as so many questions/minute

Knowing the time (minutes) before the end of class, solve for the quantity of questions needed to fill up the remaining time:

Answer to Rate Q4:  

There are two ways to solve this one.  The first method is brute force:  find the number of hard questions, then the number of medium questions, then subtract them both from the total (30) to find the remaining easy questions.  This method is the more traditional approach, and probably the method Common Core is looking for.

The alternative approach (which you can do nearly in your head) is to find the percentage of easy problems.  Percentages always add up to 100.  If we have 30/100 hard and 40/100 medium, then that leaves 30/100 remaining to make up the total 100/100, which are the easy ones.  Since the percentage of easy ones (30%) are the same percentage as the hard ones (30%), then the answers are the same:  9.

Answer to Rate Q5:  

Again, two ways to solve this one.  Find the quantity of boys (60% of 30), then subtract that from the total to find the remaining quantity of girls...or...find the percentage of girls (since the total percentage must add up to be 100), then calculate for the quantity of girls.

Answer to Rate Q6:  

This one is tricky.  The problem is asking for the total quantity of word problems solved correctly by the 18 boys versus the 12 girls. There are more boys than girls, so the total number of correct answers may be higher for the boys...except percentage of correct answers is lower than the girls.

So first, let's find the total quantity of problems being attempted by the girls, then figure out how many of those are correctly solved (in this case 60% for the girls).

Since there are 30 problems in the test, and 12 girls taking the test, then the total quantity of problems being attempted is 30 word problems x 12 girls = 360 girl word problems.

Now, if only 60% of them were solved correctly:  60% x 360 = 216 correct girl answers

Now the boys:  Total boy word problems = 30 word problems x 18 boys = 540

Quantity of Boys correct answers = 40% x 540 = 216 correct boy answers

It’s a Tie!!