Question

How do you simplify `3\times ( \frac { 5} { 18} + \frac { 1} { 6} ) + \frac { 4} { 5}`?

Answer 1

`32/15 =` `2 2/15.`

Explanation:

Let's look at the problem:
`color(red)(3*(5/18 + 1/6) + 4/5)`

According to PEMDAS or the Order of Operations, we need to solve the problem in the parentheses, `5/18 + 1/6.`

First, we need to make the denominators the same. Since 18 is a multiple of 6, we just need to change `color(green)(1/6)` to a denominator with 18. So we would do:

`3*(5/18+(1*3)/(6*3))+4/5`.

`darr`

`1/6 = 3/18`

Now, let's plug`3/18` into the problem:

`color(red)(3*( 5/18 + 3/18) + 4/5)`
Or,
`color(red)(3*((5+3)/18)+4/5)`
Thus, the problem is :

`color(red)(3*( 8/18) + 4/5)`
Here, we need to do `3 * 8/18`. Because 3 is equal to `3/1`, the problem becomes:

`color(red)[(3/1 * 8/18)]color(red)+color(red)(4/5)`

Now let's solve, `3*8=24, and 1*18=18`,

`color(red)[(24/18)]color(red)(+4/5)`
Since the top and bottom of `color(red)(24/18)` can both be divided by 2,
`darr`
`=color(red)([cancel24^12]/[cancel18^9])`
`darr`
`=color(red)(cancel12^4/cancel9^3)`

Now, the last step is to do `color(blue)(4/3 + 4/5)`. Just like before, we need to change the denominators to be the same. As 3 and 5's LCM is 15 , we'll need to change the two fractions denominator to 15. So the fractions become:

`color(blue)(4/3 = 20/15)`

`color(blue)(4/5=12/15)`

Finally, the last step is to add:

`color(red)(20/15 + 12/15 = 32/15)`

Then simplify:

`32/15 = 2 2/15`

So `3*(5/18 + 1/6) + 4/5` is equal to `2 2/15.`

Answer 2

`32/15 = 2 2/15`

Explanation:

Given: `3xx(5/18+1/6)+4/5`

Multiply and divide have priority over add or subtract but remembering that brackets group things

Consider `(5/18+1/6)`

`color(green)(5/18+[1/6color(red)(xx1)])`

`color(green)(5/18+[1/6color(red)(xx3/3)])`

`color(green)(5/18+3/18=8/18`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting it all back together")`

`3xx8/18+4/5`

This this is the same as:

`3/1xx8/18+4/5`

`(3xx8)/(1xx18)+4/5`

`24/18+4/5`

5 will divide exactly into any number ending in 5 or 0. So we need to multiply the 18 until we have a 0 as the last digit.

Known that `5xx8=40` giving the 0 so `5xx 18= 90` again giving the 0

`color(green)([24/18color(red)(xx1)]color(white)("d")+color(white)("d")[4/5color(red)(xx1)]`

`color(green)([24/18color(red)(xx5/5)]+[4/5color(red)(xx18/18)]`

`color(green)(color(white)("dd")120/90color(white)("ddd")+color(white)("ddd")72/90 color(white)("ddd")=color(white)("d") 192/90color(white)("d")=color(white)("d")32/15)`