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Order of Operations
Suppose you are given the problem and come up with an answer of 17 and your friend is given the same problem and comes up with an answer of 25. How can two people be given the same problem and arrive at two different answers? It is because to get the answer of 17, you used order of operations and your friend got 25 by doing the problem as it was written and NOT following the order of operations. If you were to do this problem on a graphing calculator, it would give you the correct answer of 17 because it has order of operations built into it.

Order of operations is simply a way to standardize the way arithmetic problems are done. Most people remember the order of operations using the letters P E M D A S which stand for
  1. Parentheses
  2. Exponents
  3. Multiplication
  4. Division
  5. Addition
  6. Subtraction
Although this is a convenient way to remember the order of operations, it can be a little misleading.
Whenever you see parentheses, those should be dealt with first. However, other grouping symbols such as brackets, braces, and absolute value signs act like parentheses also. If you have more than one set of grouping symbols, you should do the innermost grouping first and then work your way outward.

Exponents are next in the order of operations. After clearing all the grouping symbols, you should work with the exponents. If there are exponents within the grouping symbols, you should address them when working with the parentheses.

Multiplication and division are actually on the same level of importance in order of operations. So when remembering the letters PEMDAS, keep in mind that you should do multiplication and division in the order they appear in the problem. So if division is first and then multiplication, you must do the division first.

As with the multiplication and division, addition and subtraction are on the same level in working with order of operations. It is not addition then subtraction, it is whatever order they appear in the problem.


Let’s Practice and work through some examples to see how order of operations works in problems.
We will use the notation to indicate multiplication instead of  so there won’t be any confusion with the variable x.

There are no grouping symbols or exponents, so we will find which is 40 and then which is 24. Now the problem is which gives a final answer of 16. You can outline this step by step implementation below.
There are two sets of grouping symbols here so we will work with the innermost () first and then work within the brackets. Finally, we will multiply the answer in the brackets by 5. Again, we’ll look at each part step by step.
In this problem, the parentheses are worked first, then the division, followed by subtraction and then finally multiplying by 9.
In this problem, we will approach the numerator and the denominator separately, doing order of operations on each one separately and then simplify the final answer if necessary.

Examples
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S Taylor

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