Ok, fractions are pretty cool, if you think about it. For one thing, they can save you a lot of time and effort. With quite a few math problems, all you need to do is write a number in its fractional equivalent, and presto! Suddenly you’re halfway to your solution. That is, IF you’ve got your Math Etiquette down: i.e., you’ve memorized the facts you need to memorize, and have a holistic grasp of how numbers work. (Wondering what this “Math Etiquette” magic is? Well, since I invented the term, allow me to explain it, and why it’s going to skyrocket your SAT no-calculator Math score (or ACT score)!
Because at the end of the day, a fraction is just a schmancy way of doing division! It lets you carry out division problems without all the messy decimals that might otherwise be involved in your answer.
Consider, for instance, the fraction 2/9.
There’s a variety of ways that we can conceptualize this fraction. The overarching theme that unites them, though, is that a fraction is a mathematical representation of a relationship between two numbers (that's what the slash signifies!). Keeping that baseline fact in mind is empowering: because now YOU get to decide which of those several fraction “mindsets” best suits the particular problem you’re working on. This is a recipe for saving time and energy, which is what Math Etiquette is all about. Some ways that you can think about that core relationship follow below the Table of Contents below.
ARTICLE CONTENTS
Ways to think about Fractions
Fractions as Division
When we consider our example fraction this way, we're just representing the division without doing the math: 2 is divided by 9. For an example, imagine you were given this problem: There are 2 chocolate cakes. There are 9 of us. How much chocolate cake do we each get? Answer: 2/9 of a chocolate cake, which is way easier to picture and way more succinct than 0.22222... (yes, it's a repeating decimal! Not a fun kind of division math!) chocolate cakes!
Fractions as a "Part-to-Whole" Relationship
This framework 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 _____. Let’s apply that to this problem: for every 9 friends total, 2 know how to make a chocolate cake. Notice the units are different, and the numerator “chocolate cake bakers” is a subset of the denominator, which would be “total friends.” So we’d get “2 chocolate cake bakers per 9 friends.”
Fractions as 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 chocolate cake bakers to every 9 cookie makers.
Fractions as Rates
This is like a part-to-whole relationship, but one that compares two unrelated units. The numerator is NOT a subset of the denominator, nor is it a sister category to the denominator. Ex: Leslie can bake 2 chocolate cakes in 9 hours. One unit is “chocolate cakes” while the other is “hours.” “Chocolate cakes” is not a type of “hour,” and “chocolate cakes” and “hours” are not both types of some other thing. In this case, we are using unrelated units to connect the number of cakes to time.
Fractions as Proportions
This just means taking ANY one of the above 4 situations and making them equal to another situation of the same type. Regardless of which situation it is, you simply cross multiply, and your answer emerges!
So, if Leslie makes 2 chocolate cakes in 9 hours, how long will it take her to make 5 chocolate cakes? And how would we set up that equation?
Five fraction hacks
So now that you know WHEN you’ll use fractions, here are my top 5 tricks for HOW to manipulate them with ease and accuracy:
1) When you encounter 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 comes particularly in handy on the SAT No-Calculator Math section, but will save you loads 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 (despite being a mega math nerd). So…don’t force yourself into having to consider a thorny question like “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 you have to multiply AND do division by hand on the SAT No-Calculator Math, this small problem ends up gobbling up multiple minutes…and you might commit an error or two along the way!
Consider, alternatively, approaching this same problem with Kristina’s Math Etiquette Method. Cancel out and simplify as you go, like this:
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 notice that the 5 and the 15 reduce neatly, and so you cancel them, too. Then you're left with an easy math problem you can do in your head!
Pay attention to these 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 add (+) or subtract (–) fractions with unlike denominators, make a “bow tie”!
Here's how we go about this one:
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 two 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:
Yikes. Sure, you'll eventually cancel out those Cs if you go about it the long way. But why do that when you could be DONE in one step? After all, we're trying to save time and trouble (and therefore points) here!
5) If your variable is in the denominator, swap it out with its diagonal!
In other words:
See? One diagonal swap, and you’re finished! This is one of my favorite tricks because it saves several 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, stress, and potential 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 No-Calculator Math section of the current, paper SAT, of course, but it will help you wherever fractions are found. Happy fraction-ing!
P.S.: If you’d like to strengthen any of these skills with the Math Enthusiast Extraordinaire herself, you can book an appointment here to work with me directly on your SAT/ACT prep! Or, if you’re more of an independent studier, check out my complete Math Cram Plan ebooks: I’ve got versions for both the SAT and the ACT.
