# What is the easiest method for adding and subtracting rational numbers

## Main Question or Discussion Point

What is the easiest method for adding and subtracting rational numbers

Mark44
Mentor
What is the easiest method for adding and subtracting rational numbers
You need to be more specific. Do you mean rational numbers represented as fractions, like $\frac{1}{2} + \frac{3}{5}$ or rational numbers represented as decimal fractions, like .5 + .6?

represented as fractions

What is the easiest method for adding and subtracting rational numbers
1. Make sure the denominators (bottom numbers) of the two fractions you're adding/subtracting are the same

2. Add/Subtract the numerators (the top numbers) and put the answer over the common denominator you've found

And if needed,

3. Simplify the fraction by dividing both top and bottom by the same amount, to keep the value of the fraction the same

So, the standard way of adding / subtracting numbers is to first identify a least common denominator". So, say I had:
$\frac{1}{2} + \frac{3}{7}$
Well, what is the smallest number that is a multiple of both 2 and 7? It's 14. So now, how do we write the original fractions with denominators of 14? Well, we multiply 2 by 7 to get 14, so we do the same to the numberator, 1. Likewise, we multiply 7 by 2 to get 14, so we multiply 3 by 2 to get 6. So we may write
$\frac{1}{2} + \frac{3}{7} = \frac{7}{14} + \frac{6}{14} = \frac{13}{14}$
Which doesn't reduce. So, very roughly speaking,
$\frac{a}{bc} + \frac{d}{ef} = \frac{(aef)+(bcd)}{bcef}$

Mark44
Mentor
So, the standard way of adding / subtracting numbers is to first identify a least common denominator". So, say I had:
$\frac{1}{2} + \frac{3}{7}$
Well, what is the smallest number that is a multiple of both 2 and 7? It's 14. So now, how do we write the original fractions with denominators of 14? Well, we multiply 2 by 7 to get 14, so we do the same to the numberator, 1. Likewise, we multiply 7 by 2 to get 14, so we multiply 3 by 2 to get 6. So we may write
$\frac{1}{2} + \frac{3}{7} = \frac{7}{14} + \frac{6}{14} = \frac{13}{14}$
To elaborate on what AMenendez is saying, we are multiplying each fraction by 1 in some form so as to get the denominator we want.

$\frac{1}{2} + \frac{3}{7} = \frac{1}{2} \cdot \frac{7}{7} + \frac{3}{7} \cdot \frac{2}{2} = \frac{7}{14} + \frac{6}{14} = \frac{13}{14}$
You can always multiply by 1 without changing the underlying value of an expression. Once the denominators are the same, you just add the numerators using that common denominator.
AMenendez said:
Which doesn't reduce. So, very roughly speaking,
$\frac{a}{bc} + \frac{d}{ef} = \frac{(aef)+(bcd)}{bcef}$