Proportions, Ratios, Rates, Oh My!

Contributor: Mason Smith. Lesson ID: 11129

"I have a map of the United States, actual size...Scale: 1 mile=1 mile. I spent last summer folding it. People ask me where I live...I say, E6." Steven Wright's joke shows why ratios are so important!

categories

High School, Middle School

subject
Math
learning style
Visual
personality style
Lion
Grade Level
Middle School (6-8), High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:

1:3 Odds? 1:15 Faculty-to-Student ratio? What could this possibly mean for statues and maps?

Ratios and proportions surround us and are a part of everyday life.

From maps to recipes to filling up the gas tank, many things can be measured in terms of their parts and how they relate to each other, which is a casual definition for ratio.

The formal definition would be "a comparison of two quantities (or amounts) by division," which means that all ratios can be written as a/b where b ≠ 0 (since we can't divide by zero!) or, in another common form, a:b.

  • But what does that mean?

A ratio says that for every number of part a, there are so many parts b. In other words, if we have the ratio 2 oil:5 gas, that means for every 2 parts of oil, there should be 5 parts gas in the mixture, which is super important for any type of engine, because without the right mix we could have an unwanted boom!


The next idea is that of proportions, which look like equalities, but are a bit trickier. Proportions describe a relationship where two ratios are proportional. If two ratios are proportional, then we can set them equal to each other and solve for any unknown quantities, and we will find the relationship between the two.

Let's do this example for some clarification:

The ratio of students to professors at a local college is 15:1 (15 students per professor). If there are 675 students, how many professors are there?

  professor = 1
  student 15

 

Since the ratio and the fact are proportional, we can set them equal to each other to figure out the number of professors:

  ratio of professor or 1 = x or total number of professors
  student 15 675 students

 

  • But how do we solve?

Well, we have to cross-multiply the numbers (multiply times the opposite side of the equal sign), so we end up with 675 * 1 = 15 * x.

I bet you can solve it from here.

  • How many professors are there?

x = 45, so there are 45 professors at the college.

That is an example of basic proportions.

  • What happens if we don't have an example that is 1:some number?
  • How do we find out how many of b there are to each a?

We try to find the unit rate, which is a lot easier than it sounds.

Let's look at the Nathan's Hot Dog eating contest.

  • How many hot dogs did the winner from 2016 eat per minute during the 10 minute contest?

Well that's easy, Joey Chestnut ate 70 hot dogs in 10 minutes. We can write that as 70/10, but we want to know how many hot dogs were eaten in one minute. We will be trying to find our x number of hot dogs in one minute, so we have:

  hot dogs   70 = x   hot dogs
  minutes 10 1 minutes

 

Remember to keep the same units on the same side of the division line for a correct answer.

Now, we just solve for x by cross multiplying, and we get 7 hots dogs per minute.

  • But wait, shouldn't it be 7 hot dogs per 10 minutes?

No, because we always use the units from the same side as the variable in our answer. Plus, that doesn't make much sense when you think about it — how can he eat 7 hot dogs and 70 hot dogs in the same amount of time?


  • Now we know how to find proportions and ratios, but is there any place we are going to actually use this?
  • Have you ever looked at a map?

Be it paper or from Google, maps need ratios to work because, without using ratios and being able to scale the map, we would have to walk around with a piece of paper the size of the area we are trying to navigate! I don't think there are any tablets or phones that are quite big enough to show us the entire world, so ratios and scale models will probably help, right?

If we look at this Map of New York (ironically from the University of Texas), we can see the scale in the bottom-left corner. Grab a ruler and print a copy of this map. We are going to measure the distance from Niagara County to Orange county.

I measured and came up with an answer of 9 centimeters. When I set my ruler on the the scale, I found that there are 100 miles in every 3.5 centimeters (You may get slightly different measurements, but as long as your end answer is close, you are fine.).

Let's make our proportion:

  centimeters   3.5 = 9   centimeters
  miles 100 x miles

 

  • Do you think you can solve this?

I got an answer of about 260 miles; how about you?

When we check against a survey of New York, we find out that it is about 300 miles across all of New York, so that's about right!

If you would like, you can try to figure out the distance between some more points on the map and check it against a website and see how close you are! If you'd like some online practice, check out Map Scale from Mr. Nussbaum.

Continue on to the Got It? section to practice your skills!

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