A fraction can be written in a number of different ways and still mean the same (have the same quantity). Lets look at the fraction 3/4. If we have a pizza that we cut up into 4 slices and we eat one of those pieces we get a pizza that resembles the picture below:
The pizza now has 3 of the original 4 pieces remaining or 3/4 of the pizza. Lets say we have to feed more people with the same pizza. We decided instead to cut it up into 16 pieces. This time 4 pieces of pizza are eaten or 4/16 of the pizza. They are smaller pieces, but they still taste just as good. After we eat those 4 pieces we get a pizza that looks like the image below.
When looking at these two images side by side we can see in the pizza where we ate 1/4 of the pizza we have the same amount remaining as in the pizza where we ate 4/16 of the pizza. With the first pizza we have 3/4 of the pizza left and in the second we have 12/16 of the pizza left.
We can say that 3/4 and 12/16 are equivalent fractions. We also can say that 3/4 is the reduced form of 12/16 since the denominator of 4 is less than that of 16.
3/4 is the most we can reduce this fraction because there is no way to represent 3/4 with a smaller integer denominator that will also have a smaller integer numerator.
How to Reduce a Fraction
Let's mathematically reduce the fraction of 12/16 from above.
Mathematically it's very simple to reduce a fraction. The first step is to find the Greatest Common Factor or (GCF) of the numerator and denominator. In this case the GCF of 12 and 16 is 4. We then divide the fraction by a form of 1 that includes the GCF, 4 in this case, in the numerator and denominator. We can do this because dividing any number by 1 always results in that number.
Now we have a division problem on our hands, however if we solve this using the normal way of dividing fractions we will get a fraction that is even less reduced than our original, so what we need to do is rearrange our problem a little. When multiplying or dividing fractions, you can combine the numerators and denominators into a single fraction as so:
We can then solve the numerator division problem (12 ÷ 4 = 3) as well as the denominator division problem (16 ÷ 4 = 4).
We now have a 3 for the numerator and a 4 for the denominator or a reduced fraction result of 3/4 which matches our pizza above. We know that we are done reducing the fraction because the GCF of 3 and 4 is 1. Once we hit that point we cannot reduce any further.