Math Etiquette III: Kristina’s Fraction Primer

Personally, I really love fractions. For one thing, they save a lot of work. Pretty frequently, writing a number in its fractional equivalent does half the math FOR you…that is, IF your Math Etiquette skills are up to par and you understand how numbers work.

Here's the bottom line of what a fraction is: it’s just a fancy way of doing division! It’s doing division without actually going to the trouble of finding the nasty decimal that might be your answer.

Consider the fraction 2/9.

There are a bunch of different ways we can think of this fraction. The most basic thing to remember about a fraction is that it's a mathematical representation of a relationship between two numbers (that's what the slash represents!). Keeping that in mind, and figuring out which way of thinking about that relationship is most helpful to you, will save you tons of time and energy, and it's a real Math Etiquette skill. Some ways that you can think about that central relationship are...

Division.

When we think of the fraction this way, we're just representing the division without doing the math: 2 is divided by 9. For an example, think about solving this problem: There are 2 cherry pies. There are 9 of us. How much cherry pie do we each get? Answer: 2/9 of a cherry pie, which is easier to picture and way more succinct than 0.22222... (yes, it's repeating! The best kind of division math to skip!) cherry pies!

 

Part to whole.

This way of thinking about fractions represents the relationship between a whole and a relevant part of that whole, usually with the words "for," "to," or "per." There are 2 ____ for every 9 _____. For example, think of this problem: For every 9 students total, 2 know how to make a cherry pie. Notice the units are different, and the numerator “cherry pie bakers” is a subset of the denominator, which would be “total students.” So we’d get “2 cherry pie bakers per 9 students.”

 

Ratios.

This way of thinking about the fraction relationship also uses the words “to” or “for,” but it uses them to compare two types of things that are on the same level. The numerator is NOT a subset of the denominator, but the numerator and denominator are both different categories of something else. For example: there are 2 cherry pie bakers to every 9 ice cream makers.

 

Rates.

This is like a part-to-whole relationship, but that compares two unrelated units. The numerator is NOT a subset of the denominator, nor is it a sister category to the denominator. (Ex: Hannah can bake 2 cherry pies in 9 hours. One unit is “cherry pies” while the other is “hours.” “Cherry pies” is not a type of “hour,” and “cherry pies” and “hours” are not both types of some other thing. In this case, we are using unrelated units to connect the number of pies to time.)

 

Proportions.

This is just ANY one of the above 4 situations set equal to another similar situation of the same type. Regardless of which situation it is, you cross multiply and are done!

So, if Hannah makes 2 cherry pies in 9 hours, how long will it take her to make 5 cherry pies? And how would we set up that equation?

 

So now that you know WHEN you’ll use fractions, here are my top 5 tricks to manipulate them with ease and accuracy:

1) When you see a fraction set equal to something else, CROSS MULTIPLY before doing anything else.

This also works when a fraction is set equal to a whole number. Just treat the whole number like a fraction with 1 in the denominator.

2) When multiplying fractions, always CROSS THINGS OUT before multiplying!

This is especially useful on the SAT No-Calculator Math section, but saves boatloads of time on the ACT as well. In general, we all learned our times tables up to roughly 12, yes? You probably didn’t learn your 24- or your 72-times tables, did you? I sure didn’t. So…don’t put yourself in a situation where you’d need to deal with “what times 24 equals 192?!”

You can accomplish this by always making numbers SMALLER before making them LARGER. Here's an example of how NOT to multiply fractions:

Because of multiplying by hand and long division, this simple problem ends up taking a few minutes…and you have plenty of opportunities for careless errors along the way!

Now, here's the Math Etiquette Method. Cancel out and simplify as you go, like this:

See? First you cancel out the 12 in the numerator with the 3 and 4 in the denominators to get rid of some extraneous terms. Then you can see that the 5 and the 15 reduce nicely, and cancel them too. Then you're left with an easy math problem you can do in your head!

 

Pay attention to the key points about working with fractions.

You can simplify ANY numerator with ANY denominator. 12 got canceled out with the 3 in its own fraction and the 4 in the denominator of the last fraction. 15 got simplified with the 5 in the denominator of the first fraction.

You are left with small numbers to multiply, which you should know how to do!

 

3) To quickly + or – fractions with unlike denominators, make a “bow tie”!

Here's how it works:

  • 1) Multiply the top left number by its diagonal. In this case, 2 x 8. This goes on the top left of your final fraction.
  • 2) Multiply the top right number by its diagonal. In this case, 3 x 7. This goes on the top right of your final fraction.
  • 3) Multiply the two denominators together. In this case, 7 x 8. This is your new denominator.
  • 4) Keep your + or – sign the same.
  • 5) Simplify if necessary!

 

4) If you divide 2 fractions with the same denominator, your new fraction uses just the numerators

In other words: 

See, the denominators (those Cs) cancel right out. This is a really cool shortcut that saves you a few steps of actually dividing the fractions. Here’s what it would have been the non-Math Etiquette way:

You'll eventually cancel out those Cs doing it the long way, too. But why do that when you could be DONE in one step? After all, we're trying to save time and trouble here!

 

5) If your variable is in the denominator, swap it out with its diagonal!

In other words: 

See? One diagonal swap, and done! I love this trick, because it saves a couple steps of cross multiplication. Otherwise, you would have had to do this:

Which just takes longer and is way less elegant.

 

Here's the bottom line: if you understand what fractions actually are and how they work, you can save yourself a ton of time, worry, and unnecessary mistakes on the SAT and ACT.

My Math Etiquette approach is all about giving you a better grasp on the foundational concepts of math so that you can figure out what you really need to know and what's just math busywork that you can handle quickly or skip altogether. This is particularly helpful on the new No-Calculator Math SAT section, of course, but it will help you wherever fractions are found. Happy fraction-ing!