When can you cancel in a fraction?
Updated
The short answer
You can cancel only a factor that multiplies the whole numerator and the whole denominator. In (x + 6)/6 the 6 on top is added to x, so it can't be canceled, and (x + 6)/6 equals x/6 + 1.
What it looks like
| Wrong | Right |
|---|---|
| (x + 6)/6 = x + 1 | (x + 6)/6 = x/6 + 1 |
| (x² + 3x)/x = x² + 3 | (x² + 3x)/x = x(x + 3)/x = x + 3 |
The second right answer holds for every x except 0, where the original fraction isn't defined. In the first row, the part that went wrong is the x, which never got divided by 6.
Plain numbers show why the first one fails. (6 + 4)/2 = 10/2 = 5, but canceling the 2 against the 6 alone gives 3 + 4 = 7. Paul Dawkins makes the same point with (8 − 3)/2: you can cancel something only if it multiplies the whole numerator and the whole denominator.
In alai, an iPad app that reads your handwriting as you work, the tutor, Ally, asks about the first line that doesn't follow from the one before.
Why it happens
Canceling is a shortcut for dividing the top and the bottom by the same number. In (2 × 7)/(2 × 5), dividing both by 2 leaves 7/5, and the value doesn't change. That works because the 2 multiplies everything on its line.
In (x + 6)/6, the 6 on top is added to x. Dividing the top by 6 means dividing both of its terms, x/6 + 6/6, which is x/6 + 1. Crossing out the two 6s divides only part of the top. The rule “cancel what matches on the top and the bottom” is easy to remember, and easy to stretch to terms it doesn't cover.
How to catch it yourself
Before you cross anything out, check whether it multiplies the whole top and the whole bottom. If the top is a sum, try to factor it first. When you can pull the same factor out of every term, you can cancel it. When you can't, the fraction stays as it is, or you split it into separate fractions. Then point at the x in your answer and ask yourself:
Did the x get divided by 6 too?
A number works as a check here too. With x = 6, (6 + 6)/6 = 2, while x + 1 = 7.
Try these
Simplify each one. Assume the bottom isn't 0.
- (4x + 8)/4
- (x + 10)/5
- (x² − 9)/(x − 3)
Answers
- x + 2, because 4x + 8 = 4(x + 2)
- x/5 + 2, because the 5 divides both terms: x/5 + 10/5
- x + 3, because x² − 9 = (x − 3)(x + 3)
How alai helps
In Homework help, when you pause after writing a step, Ally reads it. When a line doesn't follow from the one before, Ally underlines it and asks one short question, then waits while you fix it yourself. Ally also remembers what tripped you up in past sessions, so a later session can start from there.