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A fraction represents a smaller part of a whole and is usually written as a numerator, which represents the smaller part, written over a denominator, which represents the whole. To express the fraction as a single number, the quotient, we divide the numerator by the denominator.
There are three main kinds of fractions:

• Proper fractions

  The numerator is smaller than the denominator. $$\frac{1}{4}$$ is a proper fraction.



• Improper fractions

  The numerator is larger than the denominator. $$\frac{5}{4}$$ is an improper fraction.



• Mixed fractions

  A whole number combined with a proper fraction. $$2\frac{3}{4}$$ is a mixed fraction.

It is important to note that improper fractions and mixed fractions can be used to express the same values. For example: $$\frac{5}{4}=1\frac{1}{4}$$.
When doing operations with fractions, it is usually easier to first convert any integers and/or mixed fractions into improper fractions:

• To convert an integer into an improper fraction, simply place the integer over $$1$$. For example, $$3$$ would become $$\frac{3}{1}$$.
• To convert a mixed fraction into an improper fraction, multiply the denominator (bottom number) by the whole number (number in front or to the left of the fraction), add the product to the numerator (top number), and write the sum over the original numerator. For example, in converting $$2\frac{3}{4}$$ to an improper fraction, we would multiply the denominator, $$4$$, by the whole number, $$2$$, to get $$8$$. We would then add this to the numerator, $$3$$, to get $$11$$, which we would place over the original denominator, $$4$$, to get $$\frac{11}{4}$$.

Adding and subtracting fractions

The general rule for adding fractions is: $$\frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{bc}{bd}=\frac{ad+bc}{bd}$$
The general rule for subtracting fractions is: $$\frac{a}{b}-\frac{c}{d}=\frac{ad}{bd}-\frac{bc}{bd}=\frac{ad-bc}{bd}$$
There are 4 steps to adding and subtracting fractions:

1. Simplify the fractions by reducing them, if possible. Divide the numerator (top number) and the denominator (bottom number) by their greatest common factor (gcf). The gcf of a set of numbers is the highest number that can divide evenly into all numbers in the set with no remainder. For example, $$3$$ is the largest number by which $$3$$ and $$9$$ can be evenly divided, so we can divide the numerator and denominator of $$\frac{3}{9}$$ by $$3$$ to reduce it to $$\frac{1}{3}$$. Another example is $$\frac{4}{16}$$, which would reduce to $$\frac{1}{4}$$.


2. Find the fractions' common denominator. There are two ways to find the common denominator:
   1. Multiply the top and bottom of each fraction by the denominator of the other fraction. For example, $$\frac{1}{3}+\frac{1}{4}=\frac{1·4}{3·4}+\frac{1·3}{4·3}=\frac{1·4}{12}+\frac{1·3}{12}=\frac{4}{12}+\frac{3}{12}$$
   2. Find the least common denominator. To do this, we find the least common multiple (lcm) of the denominators and use it as the common denominator. There are two ways to find the lcm: listing numbers' multiples (solver coming soon!) and by prime factorization.


3. Add or subtract the numerators. At this point, the fractions should have the same denominator, meaning we can simply add or subtract the numerators and write the result over the denominator we found in the previous steps. For example, $$\frac{4}{12}+\frac{3}{12}$$ would become $$\frac{7}{12}$$.


4. Simplify the resulting fraction by reducing, if possible, as described above in step 1. If the result was $$\frac{4}{8}$$, for example, we would reduce it to $$\frac{1}{2}$$.

Multiplying fractions

The general rule for multiplying fractions is: $$\frac{a}{b}·\frac{c}{d}=\frac{a·c}{b·d}$$
There are 4 steps to multiplying fractions:

1. Simplify the fractions by reducing them, if possible. Divide the numerator (top number) and the denominator (bottom number) by their greatest common factor (gcf). The gcf of a set of numbers is the highest number that can divide evenly into all numbers in the set with no remainder. For example, $$3$$ is the largest number by which $$3$$ and $$9$$ can be evenly divided, so we can divide the numerator and denominator of $$\frac{3}{9}$$ by $$3$$ to reduce it to $$\frac{1}{3}$$. Another example is $$\frac{4}{16}$$, which would reduce to $$\frac{1}{4}$$.


2. Multiply the numerators (top numbers). For example, $$\frac{2}{3}·\frac{3}{5}$$ would become $$\frac{6}{3·5}$$


3. Multiply the denominators (bottom numbers). For example, $$\frac{6}{3·5}$$ would become $$\frac{6}{15}$$.


4. Simplify the resulting fraction by reducing, if possible, as described above in step 1. If the result was $$\frac{4}{8}$$, for example, we would reduce it to $$\frac{1}{2}$$.

Dividing Fractions

Dividing fractions is very similar to multiplying fractions but includes an extra step, in which we swap the numerator and denominator of the divisor—the number by which we will divide the other fraction—to find its reciprocal. From here we simply multiply the fractions together.

The general rule for dividing fractions is: $$\frac{a}{b}:\frac{c}{d}=\frac{a}{b}·\frac{d}{c}=\frac{a·d}{b·c}$$
There are 5 steps to dividing fractions:

1. Simplify the fractions by reducing them, if possible. Divide the numerator (top number) and the denominator (bottom number) by their greatest common factor (gcf). The gcf of a set of numbers is the highest number that can divide evenly into all numbers in the set with no remainder. For example, $$3$$ is the largest number by which $$3$$ and $$9$$ can be evenly divided, so we can divide the numerator and denominator of $$\frac{3}{9}$$ by $$3$$ to reduce it to $$\frac{1}{3}$$. Another example is $$\frac{4}{16}$$, which would reduce to $$\frac{1}{4}$$.


2. Flip the fraction we are dividing by (the divisor) so its numerator is on the bottom and its denominator is on the top. For example, $$\frac{3}{4}:\frac{1}{3}$$ would become $$\frac{3}{4}·\frac{3}{1}$$.
   
3. Multiply the numerators (top numbers). For example, $$\frac{2}{3}·\frac{3}{5}$$ would become $$\frac{6}{3·5}$$


4. Multiply the denominators (bottom numbers). For example, $$\frac{6}{3·5}$$ would become $$\frac{6}{15}$$.


5. Simplify the resulting fraction by reducing, if possible, as described above in step 1. If the result was $$\frac{4}{8}$$, for example, we would reduce it to $$\frac{1}{2}$$.