Order matters. It definitely matters that you put your socks on first and then your shoes. The same goes for doing algebra. Check out this example to see why.
You have found a real bargain at your favorite store. Jeans were on sale for $10, and you found three t-shirts for $8 each. The computer is down, so the cashier has calculated the total by hand. The cashier says, "That will be $104." You know that seems like a lot of money, so you ask the cashier about the calculation. The cashier tells you that the order of the calculations doesn't matter, and that he always calculates from left to right.
Let's look at the calculations involved. Is the cashier correct?
Your Calculations
You did the following calculation in your head when you calculated the cost of the shirts and jeans: 3 · 8 + 10 =
The correct answer is $34. 3 times 8 is 24. 24 plus 10 is 34
This time the answer you calculated is correct when going left to right, but is this always true?
The Cashier's Calculations
The cashier wrote the following calculation:
10 + 3 · 8 =
Perform these operations from left to right to see what happens.
10 plus 3 equals 13; then multiply 13 times 8 which equals 104. When going left to right in this case, the cashier is incorrect.
Therefore, calculating from left to right is not always correct.
Using several methods of grouping, we could have a myriad of different solutions. Notice how we use the same numbers from the problem in the chart below and get different results depending on how you group and the order in which you operate. In each of these examples, the operation inside the parentheses is done first.
| Example | Solution |
| (3 · 8) + 10 | 24 + 10 = 34 |
| 3 · (8 + 10) | 3 · 18 = 54 |
| (10 + 3) · 8 | 13 · 8 = 104 |
| 10 + (3 · 8) | 10 + 24 = 34 |
In considering the original problem of one pair of jeans for $10 and three t-shirts for $8 each, you have two groupings: jeans and t-shirts. The jeans amount is written 1·10 = 10 and the t-shirt amount is written 3·8 = 24. The last step is adding the two amounts together, 10 + 24 = 34. So the problem is written 1·10 + 3·8, and you multiply first and then add. This is following what is called the Order of Operations.
| Order of Operations | |
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1. Perform any operations within grouping symbols first. |
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2. Simplify terms involving exponents. |
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3. Multiply or divide from left to right. |
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4. Add or subtract from left to right. |
