Question

How do you combine `\frac { x + 8} { 3x } + \frac { 5x + 4} { 2x } + \frac { 7x + 2} { 6x }`?

Answer 1

`(4x+5)/x`

Explanation:

We ask ourselves: What is the least common multiple of `3x` `2x` and `6x`?

As we try to see whether one of the term is the LCM, we see that `3x`,`2x`, and `6x` are all divisible by `6x` itself.

We now turn the fractions' denominator to `6x`

Using the fact that `a/b*c/c=a/b`, we can turn `(x+8)/(3x)` to a fraction of `6x` by multiplying it by `2/2`

=>`(x+8)/(3x)*2/2=(2(x+8))/(2(3x))`
=>`(2x+16)/(6x)`

Similarly, we can turn `(5x+4)/(2x)` to a fraction of `6x` by multiplying it by `3/3`

=>`(5x+4)/(2x)*3/3=(3(5x+4))/(3(2x))`

=>`(15x+12)/(6x)`

We now have:
`(2x+16)/(6x)+(15x+12)/(6x)+(7x+2)/(6x)`

We now use the fact that `a/b+c/b+d/b=(a+c+d)/b`
to turn our expression to: `((2x+16)+(15x+12)+(7x+2))/(6x)`

We combine like terms.

=>`(24x+30)/(6x)`

=>`(6(4x+5))/(6x)`

=>`(cancel6(4x+5))/(cancel6x)`

=>`(4x+5)/x`

That is our answer!