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Reducing (Simplifying) Fractions

Reducing fractions is simply taking a fraction and seeing if it can be written with smaller numbers. This is done so you don't end up with huge numbers that aren't as easily understood as smaller ones - and if you have to perform another calculation on the fraction, you are not doing additional calculations on larger numbers, increasing the chances of making a mistake.

For example, which fraction is easier to read and understand:
987/1974 or ½? If you had to multiply it by 3/2, it's much, much easier to use 1/2 than 987/1974. So let's get down to it!

Steps to Reduce Fractions:
Find a common multiple in both numbers.
For example, if both end in an even number, you can divide the numerator and denominator by 2.
If the sum of the digits are divisible by 3, then you can divide by 3.
(For example, if the fraction is 27/132, you add up the digits in 27, 2+7=9, and 132, 1+3+2=6. Since both 9 and 6 are divisible by 3, then you can divide both numerator and denominator by 3 to reduce.)
If both numbers end in 0 or 5, you can divide by 5.
If both numbers end in 0, you can divide by 10.

Its usually best to start with the largest possible number to divide both numerator and denominator by.

Let's try a few easy ones:

In the first one, since both the numerator and denominator are even, we can divide them by 2:
Numerator: 2 ÷ 2 = 1
Denominator: 4 ÷ 2 = 2
Neither (1 or 2) can be further reduced, making the solution: ½

In the second example, both the numerator and denominator are even, so we could divide them by two, which would give us 6/9.
Numerator: 12. 12÷2 =6
Denominator: 18. 18÷2=9

However, the numerator and denominator are both divisible by 3 also:
Numerator: 12. 12÷3=4
Denominator: 18. 18÷3=6

Now we have 4/6.
We can see that both numerator and denominator are even, and thus divisible by 2:
Numerator: 4. 4÷2=2
And our answer is 2/3.
(Note: If we would have continued after dividing by 2, we would use 6 ÷ 9. Since both are divisible by 3, we could have:
6 ÷ 3 = 2 and 9 ÷3 = 3, for the same answer of 2/3.)

Lastly, 4/9.
Numerator and denominator are not both even, so we can't divide by 2. Only the denominator is divisible by 3, so we cannot divide numerator and denominator by 3.. They don't end in 5, so we can't divide by 5. In fact, we can't divide by anything in common - so that means this is already reduced as much as it can be!

The final example brings us to an important point - and that is to keep attempting to find a common number to divide by until you hit the number itself. If you do not find a number that's and you're already halfway to the number that you're hoping to divide - stop! It's already reduced to its simplest form.

Reducing Larger Fractions
Of course, there are more difficult situations to deal with. Can you simplify 143/187? Not by using 2, 3, or 5! There are several ways to determine this one, but the one that we find easiest is our Subtraction method.

1. Using this method, simply subtract the numerator from the denominator. That gives you: 187-142 = 44.
2. Next, find all the multiples of 44. Here they are: 1, 2, 4, 11, 44. The multiples of 44 are simply another way of saying the numbers can divide into 44 evenly.
3. The next step is to include the highest multiple that is not equal to the number itself. Basically, this means to pick the second largest number. In this case, that would be 11.
4. Using that number, divide the numerator and denominator of the original fraction:
Numerator: 143. 143 ÷ 11 = 13.
Denominator: 187 ÷ 11 = 17.

So can reduce 143/187 down to 13/17. Both of these are prime numbers, so we know we cannot reduce further. This is just one of the many things you can do in math that relies only on the basic fundamentals (addition, subtraction, multiplication, and division).

Find another math lesson at the Math Lesson and Tutorial Center!

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