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How to Use PEMA to Guide Order of Operations in Arithmetic Problems


Sample Problem

[35/{7(5 + 12^2)}][1/2] - 1.5

The above is a nice complex arithmetic expression with one and only one correct value. However knowing the correct order of operations in solving such an expression is the only way to arrive at that one correct value. The Acronym PEMA will guide you to your answer.



M-Multiplication and Division

A-Addition and Subtraction

This is the order in which the operations should be performed, follow this guide and you'll be fine.


Solving It

[35/{7(5 + 12^2)}][1/2]-1.5

This looks intimidating but let's take it a step at a time.

First Parenthesis, as you can see there are a number of parenthesis within parenthesis (3 actually), we start by moving to the innermost set of parenthesis.

(5+12^2) Once we locate this starting point treat what's inside that set of parenthesis in the order designated by PEMA; we're dealing with the parenthesis (P) already, within that the next thing we see is an exponent (12^2)(E), so solve this and get 144.

(5+144) There is no multiplication or division(M) present here so move on to addition and subtraction(A).

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(note: You can do multiplication then division or division then multiplication during the M phase and addition then subtraction or subtraction then division during the A phase.) So,

(5+144)=(149) Let's plug this back into our original expression.

[35/{7(149)}][1/2]-1.5 Moving to the next outer set of parenthesis, we see we need to multiply.

7X149=1043 So plug this back into the expression.

(35/1043)(1/2)-1.5 We end up with this and see that we have fractions within each remaining set of parenthesis, so instead of dividing (which leaves us with ugly irrational numbers) we'll treat them as fractions that need to be multiplied together, so

(35/1043)(1/2)=35/2086 Plug this back in to the equation.

(35/2086)-(1.5) We only have one operation left, addition and subtraction, to do this we'll convert 1.5 to an improper fraction, find a common denominator, and subtract.

(35/2086)-(3/2) Remember to find a common denominator; determine what the lowest number both denominators divide into is, in this case it's easy 2086; and to adjust 3/2 to an equivalent fraction that we can work with; multiply the numerator by whatever number you needed to multiply the denominator by to get 2086, in this case 1043.

1043X3=3129 So the fraction equivalent to 3/2 is 3129/2086.

(35/2086)-(3129/2086) Now we subtract the numerators and leave the common denominator.

-3094/2086 Simplify by dividing by 2.

-1547/1043 Further simplify by dividing by 7.

-221/149 And there you have it. You could try converting it to a mixed number by dividing the numerator by the denominator, but if you try it you'll see you get an irrational number. So leave it as is.


Feel free to post any questions.

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