What Are Fractions?
Fractions are everywhere. You might have
\(\frac{1}{2}\)
(half) a pizza, or
\(\frac{1}{4}\)
(one quarter) of an apple. Yet what exactly are fractions, and what can we do with them?
A fraction is any portion of a whole. It has two parts: The numerator, which goes on top and shows how many “pieces” of something we have, and the denominator, which goes on the bottom and shows how many pieces in total make up the whole.
Think about an apple pie. If we have
\(\frac{1}{2}\)
the pie on our plate, we have
\(1\)
piece (the numerator) that makes up half of the whole. If we had
\(2\)
pieces (the denominator), we would have the whole pie.
Simplifying Fractions
Sometimes, it is possible to make larger fractions smaller (simpler) using division. As long as both the numerator and denominator can be divided evenly by the same number — a common factor — you can make the fraction smaller.
Example:
\(\frac{12}{16}\)
Both numbers can be divided by
\(2\)
, which gives
\(\frac{6}{8}\)
. Both numbers can be divided by
\(2\)
again, which gives a final answer of
\(\frac{3}{4}\)
.
Adding Fractions
It’s also possible to add fractions together. If they have the same denominator, we simply add the numerators together.
Example:
\(\frac{1}{5}+\frac{2}{5}\)
Solution:
\(\frac{3}{5}\)
But what happens if the denominators aren’t the same? There are two methods to make them the same and make addition possible.
Method 1: Lowest Common Multiple
Example:
\(\frac{2}{5}+\frac{3}{7}\)
Solution: List the common multiples of both denominators:
\(5\)
— 10, 15, 20, 25, 30, 35
\(7\)
— 14, 21, 28, 35, 42, 49
The lowest common multiple is 35. Multiply
\(\frac{2}{5}\)
by
\(7\)
to get
\(\frac{14}{35}\)
, then multiply
\(\frac{3}{7}\)
by
\(5\)
to get
\(\frac{15}{35}\)
. Now we can add them together and we get
\(\frac{29}{35}\)
.
Method 2: Prime Factors
Example:
\(\frac{2}{9}+\frac{3}{15}\)
Solution: First, list the prime factors of both denominators:
\(9\)
— 3, 3
\(15\)
— 3, 5
Next, find the union of these primes, which means that, for each prime, find the most number of times it occurs in each list. Then, multiply them together. This gives us
\(3\times 3\times 5=45\)
. This allows us to write the equation as
\(\frac{10}{45}+\frac{9}{45}\)
. Now, we can add the fractions together to give
\(\frac{19}{45}\)
.
What's Next
Ready to try some more fraction addition? Enter the question into our pre-algebra fraction calculator to see our solution steps, then try some other practice problems to tackle more challenges like square roots, prime factorization or completing the square. You can also upgrade to Cymath Plus for even more help and explanations.
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Pre-Algebra: