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In what follows, it will sometimes be useful to remember that fractions can indicate division. For instance, 1/3 can mean "one divided by three", as well as "one part out of three parts". In fact, let's cut to the chase; memorize this sentence: "Fractions are division."
You know that any number, divided by itself, is just 1. You use this fact when you reduce fractions. If you can convert part of a given fraction into being a multiplied form of 1, then you can ignore this part, because multiplying by 1 doesn't change anying.
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For instance, here's how you would find and use a form of 1 to reduce the fraction 4/8 to lowest terms (that is, to simplest form):
To be very clear, the point of finding the common factor (in this case, the 4's) is to allow you to convert part of the fraction to 1. Since 4/4 = 1, then what I did above was the following:
Warning: Note how I switched from a fraction with products (in the numerator and denominator):
...to a product of fractions:
This switch is okay as long as you're multiplying:
...but it is very much NOT if you're adding. For instance:
The left-hand side above, being a fraction containing addition, is equal to 5/6, while the right-hand side above, being an addition containing fractions, is equal to 1 1/2, so the two expressions are not at all the same value. Just remember: For fractions, multiplying is way easier than adding. Now, to get back to business...
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In addition to the canceling method I used above (with the pink 1's), you may also have seen either of the following "shorthands" for cancellation:
Any of these formats is fine. The two shorthand methods are probably simplest for your handwritten homework; the format I'd used above is easier for typesetting.
If you have a regular (scientific, business, etc.) calculator that can handle fractions, then you can enter the fraction and then hit the "equals" button to get the reduced fraction. If you have a graphing calculator with a fraction command, then you can enter the fraction as a division (because 4/8 means "four divided by eight"), and then convert to fraction form. Check your calculator's owners manual for specifics.
If your calculator can't handle fractions, or if the denominator is too large for the calculator to handle, then you'll need to do the reduction by hand. (And you'll need the concept and methodology of fraction-reduction in later algebra courses.)
Remember that if "everything" cancels out of, say, the numerator, then there still remains a factor of 1. Everything is always multiplied by 1, but we don't usually notice this. However, if all the non-trivial factors (that is, all the factors that aren't 1) get cancelled by matching factors on the other side of the fraction line, then you've still got that 1; the fraction does not become headless.
I'll grab my calculator and some scrap paper, and factor the numerator (top number) and denominator (bottom number). A quick shorthand for getting the prime factorization of each of these numbers is demonstrated below, in the stacked division (by prime numbers) of 2940:
To find the factorization, I just read off the prime factors from around the outside of the upside-down division. From the above, I can see that 2940 factors as 2×2×3×5×7×7.
Next, I'll factor the denominator, being the number 3150:
So 3150 factors as 2×3×3×5×5×7.
Now I can reduce the fraction by canceling off the common factors:
So, after cancelling all the factors that were duplicated in (that is, were common to) the numerator and denominator, my simplified form is:
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