Of all the math topics that raise dread, fear, anxiety, and confusion, few do so as consistently and as potently as do fractions. I have my own personal theory why fractions are hard. The trouble is: think about when you learned fractions — maybe the third, fourth, and/or fifth grades. That's when fractions are usually taught, but there are two problems with that. First of all, that's before the tsunami of puberty hits you and virtually obliterated all previously held logical connections in your head. More importantly, fractions, like many other topics in math, involve sophisticated patterns, but in the fourth grade, no one is capable of abstraction, so instead you are just taught to reproduce patterns mechanically, and relying on mechanical repetition has severe limitations: similarly looking things become conflated, and when you get confused, you basically don't know what to do. Many people simply give up at that point.

The solution is re-approach those mechanical procedures, but with understanding. When you understand why you do each thing, then (a) you can remember it much better, and (b) in a moment of confusion you can figure out what to do. I will lead you through fractions from the ground up.

A fraction is a way of showing division. The fraction **numerator**, and the bottom of a fraction is called the **denominator**.

The fraction

That, visually, represents

Notice that if you have the fractions **Cancelling is division**. That's a big idea — thus, when you have

First of all, let's address the common mistake: when you add fractions, you can't simply add across in the numerator and denominator (this is the mistake people make when they mechanically perform the rule for multiplication with addition instead!)

You may dimly remember that **you only can add and subtract fractions when you have a common denominator**. That's true, but why is that true? Believe it or not, the basis of this fact is none other than the Distributive Law,

When the denominators are not the same —

The same thing works for subtraction:

This is the easiest of all fractions rules. To multiply fractions, multiply across in the numerators and denominators.

What's a little tricky about multiply is what you can cancel. If you are multiply two fractions, of course you can cancel any numerator with its own denominator, but you can also cancel one numerator with another denominator. Sometimes, that is called "cross-cancelling", which I think is a **you can cancel any numerator with any denominator**. Here's a horrendous multiplication problem that simplifies elegantly with the liberal use of cancelling.

First of all, multiplying by

Notice, as always, cancel **before** you multiply. Dividing a fraction by a number follows the same pattern:

Notice, this is really the same idea as: dividing by

Another word for a fraction is a **ratio**: ratios and fractions are exactly the same thing. A proportion is when you have two ratios, two fractions, set equal to each other. For example,

One legitimate move is to **cross-multiply**, although doing so here would violate the ultra-strategic dictum: cancel before you multiply. And it's precisely this issue, what can you cancel and what can't cancel in a proportion, that causes endless confusion. Let's look at the general proportion

First of all, as always, you can cancel any numerator with its own denominator — you can cancel common factors in

The following are highly tempting but complete illegal ways to cancel in proportions:

The trouble is, folks mechanically memorize the cancelling pattern for multiplying fractions — or even worse, they learn an utterly useless term like "cross-cancelling" — and then they mechanically apply that pattern when there's an equal sign between the two fractions instead of a multiplication sign. This is a major mistake, and any time a proportion appears on the QUANT, the test-maker is expecting a large flock of test-takers to fall into this trap.

Let's solve the proportion we wrote above, with proper cancelling:

Notice, in that last step, to isolate

I hope this refresher has clear up some fractions concepts for you. The best way to cement a new mathematical understand: practice, practice, practice!

**Q1**. If

A.

B.

C.

D.

E.

We're asked to find the value of:

Let's first combine these two fractions into one. We can obtain a common denominator of

From here, we can substitute in the values from the prompt that

And now, we convert the above into a mixed fraction:

**Answer = A**

**Q2**. If

A.

B.

C.

D.

E.

**Answer = E**

**Q3**. To reach her destination, Jeanette must drive 90 miles. If she drives 5 miles every 7 minutes, how much time will it take her to reach her destination?

A. 2 hours and 2 minutes

B. 2 hours and 6 minutes

C. 2 hours and 10 minutes

D. 2 hours and 12 minutes

E. 2 hours and 15 minutes

**Answer = B**