Question

How do you simplify `15\div 5( 8- 6+ 3) \times 5`?

Answer 1

3

Explanation:

`15-:5(8-6+3)xx5`
`15-:5(5)xx5`
`15-:5(5)xx5`
`15-:25xx5`
`x/a =x*(1/a)`
`15*1/25*5`
`15/25*5`
`75/25`
= 3

Answer 2

It depends...

Explanation:

Given:

`15-:5(8-6+3)xx5`

I think we are all agreed that the content of the parentheses should be evaluated first. Subtraction and addition are given the same priority, so the expression in parentheses is to be evaluated from left to right:

`8-6+3 = 2+3 = 5`

Now we have:

`15-:5(5)xx5`

This is where it gets interesting. Here are three possibilities, in no particular order:

`color(white)()`
Possible interpretation 1 - "Historical"

Historically the obelus `-:` was used to express a division of everything on the left by everything on the right. In our example, that means that we have:

`15-:5(5)xx5 = 15/(5(5)xx5) = 15/(5xx5xx5) = 15/125 = 3/25`

`color(white)()`
Possible interpretation 2 - "Pure PEMDAS"

PEMDAS does not distinguish multiplication by juxtaposition from any other kind of multiplication. So in full we can write:

`15-:5(5)xx5 = 15-:5xx5xx5`

This is then evaluated from left to right (multiplication and division having the same priority). So we get:

`15-:5xx5xx5 = 3xx5xx5 = 15xx5 = 75`

`color(white)()`
Possible interpretation 3 - "Skewed PEMDAS"

This common practice gives higher priority to multiplication by juxtaposition. This is sometimes "justified" by people who claim that the "Parentheses first" includes any multiplier outside the parentheses. Such a justification seems spurious to me, but the visual proximity does suggest a higher priority.

Following this interpretation, we get:

`15-:5(5)xx5 = 15-:25xx5 = 3/5xx5 = 3`

`color(white)()`
Which is right?

They are all "right" or "wrong" or neither. The fact is that the given expression is ambiguous. Operator precedence rules are intended to clarify communication by providing agreed rules of interpretation.

They do not work well all of the time - especially if the particular rule set is not shared between the writer and reader.

It is best to use extra parentheses to make the meaning clear.

Answer 3

In addition to the other answers, here is an additional comment and warning about just applying `PEDMAS or BODMAS` without considering the meaning of an expression.

Consider the terms in the parentheses:

`(8-6+3)`

Following the [order of operations] (https://socratic.org/prealgebra/arithmetic-and-completing-problems/order-of-operations)at this point states that ADDITION is to be done first. However, many students will apply this incorrectly to do:

`(8" "-color(blue)(6+3)) = 8-color(blue)(9)" "` and then subtract to get `" "-1`

The negative sign applies ONLY to the `6` and not the sum of `6+3`
If this was intended it would be written as `" "(8-(6+3))`

A safer approach is to place the additions at the beginning and the subtracts at the end.

`(8-6+3)`

`=(8+3-6) = (11-6) = 5`

My approach for the calculation would now be

`15 div color(red)(5(5)) xx 5`

`=15 div color(red)(25) xx 5`

`=15/25 xx 5/1`

`=3`

However, this is one interpretation, as discussed by George C.

Answer 4

`15-:5(8-6+3)xx5=color(blue)(75`

Explanation:

Simplify:

`15-:5(8-6+3)xx5`

It is important to follow the order of operations:

Parentheses/Brackets, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right.

An acronym like PEMDAS or BODMAS can be helpful in learning to memorize the order of operations.

Simplify the parentheses.

`15-:5(color(red)8-color(red)6+3)xx5`

Simplify.

`15-:5(color(red)2+3)xx5`

Simplify the parentheses.

`15-:5(5)xx5`

Multiplication and division are equal, so they are performed from left to right. First do the division, and next multiply by `5xx5`.

`15/5xx5xx5`

Simplify.

`3xx5xx5`

Simplify.

`75`