How to simplify fractions by cancelling common factors

 
 
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Why cancelling lets us simplify fractions

We already understand that a fraction just represents “part of a whole”. If a baseball player gets three “at bats” in a game (gets an opportunity as a hitter three times), and out of those three chances they get a hit two times, then their success rate for that game is

2 hits3 chances=23\frac{2\text{ hits}}{3\text{ chances}}=\frac23

Or if I borrow five books from the library but read only two of them, then I’ve read two out of five books, or

2 books read5 books total=25\frac{2\text{ books read}}{5\text{ books total}}=\frac25

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What we want to be able to do now is learn to simplify fractions. For example, we know that 5050 is half of 100100, so if we see the fraction

50100\frac{50}{100}

then we want to be able to rewrite that as

12\frac12

Because remember, a fraction is really just a relationship between the numerator and the denominator. If I’m driving 100100 miles to visit my family, and I’ve already driven 5050 miles, then I know that I’m halfway there. It would be simpler for me to express my progress as 1/21/2 than as 50/10050/100, so we want to know how we go about changing 50/10050/100 into 1/21/2.

The reason we want to reduce fractions to lowest terms is that even though a fraction like

630945\frac{630}{945}

is actually the same as

23\frac23,

that isn’t obvious to us when we look at it since the numbers are so big. But if we simplify that larger fraction down to 2/32/3, then we’ll be able to easily tell that we have “22 out of 33 parts.”

 
 

How to simplify fractions by cancelling common factors from the numerator and denominator


 
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Simplifying a fraction to its lowest terms

Example

Simplify the fraction to lowest terms.

630945\frac{630}{945}

There are a couple ways to tackle this, but here’s a reliable way to go about simplifying a fraction to lowest terms. We can first find the prime factorizations of the numerator and the denominator.

prime factorization

What we get is

630945=3357233573\frac{630}{945}=\frac{3\cdot3\cdot5\cdot7\cdot2}{3\cdot3\cdot5\cdot7\cdot3}

Now we’ll group together the factors that are common to the numerator and denominator, so our fraction can be expressed like this:

630945=(3357)2(3357)3\frac{630}{945}=\frac{\left(3\cdot3\cdot5\cdot7\right)\cdot2}{\left(3\cdot3\cdot5\cdot7\right)\cdot3}

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Because remember, a fraction is really just a relationship between the numerator and the denominator.

What we want to remember at this point is that whenever a factor is common to the numerator and denominator of a fraction, that factor in the numerator “cancels against” the matching factor in the denominator. (It’s as if that factor just disappears from both places.) However, if the numerator is equal to the denominator, the fraction doesn’t disappear (we have to have something there), so the fraction simplifies to 11. For example,

33=1\frac33=1

1010=1\frac{10}{10}=1

6767=1\frac{67}{67}=1

Getting back to our problem, the numerator isn’t equal to the denominator, but the factors in (3357)(3\cdot3\cdot5\cdot7) occur in both of them, so those factors cancel and we’re left with just 2/32/3.

630945=(3357)2(3357)3=23\frac{630}{945}=\frac{\left(3\cdot3\cdot5\cdot7\right)\cdot2}{\left(3\cdot3\cdot5\cdot7\right)\cdot3}=\frac{2}{3}

Therefore, because we were able to cancel all those common factors, we can say that

630945=23\frac{630}{945}=\frac23

 
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