Fractions

The most basic operations involving fractions.

Mixing With Whole Numbers

When you have a whole number, and a fraction, sometimes performing operations with both can be difficult, an easy trick to solve these kinds of problems is to take the whole number, and put a 1 under it, it represents the same number, but now you have an denominator to work with. Here is an example:

$5\times {\frac {1}{3}}={\frac {5}{1}}\times {\frac {1}{3}}$

Addition & Subtraction

In addition and subtraction, all fractions must have the same lowest common denominator (LCD) before you can perform any operations on them, so lets try that here:

$\left({\frac {3}{4}}-{\frac {2}{3}}\right)+\left({\frac {1}{2}}+{\frac {1}{3}}\right)$

The easiest way to get the LCD is to multiply the numerator and denominator on each side by the denominator of the other side. In much more complicated fractions, this is the easiest method of getting started, with less complex fractions, you can just eyeball the numbers and figure them out. We're going to start with the fractions in the first set of parenthesis.

${\frac {3(*3)}{4(*3)}}-{\frac {2(*4)}{3(*4)}}={\frac {9}{12}}+{\frac {8}{12}}$

Then on the second parenthesis.

${\frac {1(*3)}{2(*3)}}-{\frac {1(*2)}{3(*2)}}={\frac {3}{6}}+{\frac {2}{6}}$

Which leaves us with:

$\left({\frac {9}{12}}-{\frac {8}{12}}\right)+\left({\frac {3}{6}}+{\frac {2}{6}}\right)$

Now lets solve, first the left side:

$\left({\frac {9}{12}}-{\frac {8}{12}}\right)={\frac {1}{12}}$

Then the right side:

$\left({\frac {3}{6}}+{\frac {2}{6}}\right)={\frac {5}{6}}$

Now we are left with:

${\frac {1}{12}}+{\frac {5}{6}}$

Now, since we know that 6 goes into 12 twice, we can simply multiply the right side by 2 to get both fractions with the same LCD.

${\frac {1}{12}}+{\frac {10}{12}}$

Solving gives us:

${\frac {11}{12}}$

Since this fraction has 11 which is an odd number, we can't reduce it down any further, so this is the final answer.

And that's addition and subtraction of fractions.

Multiplication

Multiplying fractions is actually pretty simple, it just invovles multiplying the numerator of one fraction by the numerator of the other fraction. Then repeat this same process for the denominator of each fraction. Here we start with a fraction.

${\frac {1}{2}}+\left({\frac {2}{3}}\times {\frac {3}{4}}\right)-\left({\frac {4}{5}}\times {\frac {2}{4}}\right)$

We're going to start with the left set of parenthesis first:

$\left({\frac {2}{3}}\times {\frac {3}{4}}\right)={\frac {2*3=6}{3*4=12}}={\frac {1}{2}}$

Then lets work through the second set of parenthesis:

$\left({\frac {4}{5}}\times {\frac {2}{4}}\right)={\frac {4*2=8}{5*4=20}}={\frac {8}{20}}={\frac {4}{10}}$

Now we have this problem to solve, lets find the LCD:

$\left({\frac {1}{2}}+{\frac {1}{2}}-{\frac {4}{10}}\right)=\left({\frac {5}{10}}+{\frac {5}{10}}-{\frac {4}{10}}\right)={\frac {6}{10}}={\frac {3}{5}}$

And that's our answer for multiplication.

Division

Dividing fractions is very similiar to multipling, simply because it just involves flipping the numerator and denominator places on the divisor fraction. An example:

${\frac {1}{2}}\div {\frac {3}{5}}={\frac {1}{2}}\times {\frac {5}{3}}={\frac {1*5=5}{2*3=6}}={\frac {5}{6}}$

And that's about it for fractions and divison.