Understanding Basic Elementary Fractions

What Is a Fraction?

Most things in the world can be divided into parts. Pizzas can be divided into slices, and oranges can be divided into segments. Days are divided into hours, minutes and seconds. Fractions are what we use to show that something is a part of a whole.

Fractions always have two numbers. They are written one on top of the other with a line in between, or side by side, like this: 1/2. The denominator is the number on the bottom of the fraction. In the fraction 1/2, the denominator is two. The fraction's numerator is the number on top. The numerator for 1/2 is one.

Remember that fractions are used to talk about parts of a whole. Here's how they work: the denominator tells you how many total parts the whole has, and the numerator tells you how many of those parts you have. For example, if you have a pizza that has eight slices total, and you take three of them, you have taken 3/8 of the pizza. Since there were eight total slices, and you took three, that means that there are five slices left in the box, or 5/8 of the pizza.

Adding and Subtracting Simple Fractions

Notice that, in the example above, you took 3/8 of the pizza slices and left 5/8 behind. Three plus five equals eight, so if you add those two fractions together, you get 8/8. Eight minus three equals five, which is the number of slices left after you took your pizza. When you add and subtract fractions that have the same denominator, you just change the numerator, while keeping the denominator the same.

Imagine you're in a class of 13 students. Seven of the students, or 7/13, have brown hair. Four of the students, or 4/13, have blond hair, and two students, or 2/13, have red hair. If you want to know what fraction of the students have either blond or red hair, you would add together 4/13 and 2/13. You would leave the denominator (13) the same, and add together four and two to get six. This tells you that 6/13 of the students have either red or blond hair. Here are some other examples:

• 1/6 + 4/6 = 5/6
• 7/11 - 2/11 = 5/11
• 1/31 + 9/31 = 10/31

Finding Equivalent Fractions

Imagine that you have a pie that's divided into four large slices. You take two of the four slices, or 2/4 of the pie, and leave behind two slices, or 2/4, for your friend. This means that you've taken half of the pie and left half of it in the pan, right? Whether you cut the pie into six slices and take three of them (3/6), or cut the pie into eight slices and take four (4/8), you've still taken half of the pie. Here's how a mathematician would write this:

1/2 = 2/4 = 3/6 = 4/8

All of these fractions are equal, because they all represent the same total amount of the pie. One slice is half of two slices, two slices is half of four slices and so on. Here are some other examples of equivalent fractions:

• 1/3 = 2/6 = 3/9 = 4/12
• 1/4 = 2/8 = 3/12 = 4/16
• 2/5 = 4/10 = 6/15 = 8/20

Finding the Simplest Fraction

In math, we like to keep things simple by using the smallest numbers possible to represent fractions. For instance, if we have half of a pie, we'd rather say that we have 1/2 of a pie than 4/8 of a pie. In order to write fractions with the smallest possible numbers, we reduce or simplify them. This means that we find a fraction that is equivalent to the one we're simplifying, but use the smallest possible numbers.

For example, if we wanted to simplify 4/8, we would reduce it to 1/2. We know that 4/8 is equal to 3/6 and 2/4 as well, but 1/2 uses lower numbers to represent the fraction. The fraction 3/12 is equal to 2/8 and 1/4, but we would reduce it to 1/4, since one and four are smaller than two and eight.

How to Reduce Fractions

To find the simplest version of a fraction, think of a number that both the numerator and denominator are divisible by, and divide them both by it. For example, if you're simplifying 5/15, you could divide the numerator and denominator both by five to get 1/3. There are no numbers that both one and three are divisible by, so 1/3 is the simplest version of 5/15.

Let's try to simplify a slightly more complex fraction, like 18/24. Both 18 and 24 are divisible by two, so we can reduce this fraction to 9/12. Both nine and 12 are divisible by three, so we can reduce this fraction further to 3/4. There aren't any numbers that divide evenly into both three and four, so 3/4 is the simplest version of 18/24.

Some larger fractions, like 11/15 and 13/20, can't be reduced at all, since there is no number that divides evenly into both the numerator and the denominator. This is also true of smaller fractions like 2/3 and 4/5.