Fundamentals of Math

Fundamentals of Mathematics

Example 1.5.2

[3(15 + 3) + 27]/3

1. Start with the inner most parenthesis 15 plus 3 which equals 18. In this case, we do not drop the parenthesis yet because we still need to distribute (a.k.a. multiply) the 3 – the parenthesis themselves turn into an operator.

[3(18) + 27]/3

2. Next me move to the multiplication 3 times 18 which equals 54.

[54 + 27]/3

3. Since the sum of 54 and 27 is inside a bracket you do that before dividing by 3 – the sum is 81.

4. Lastly, you can divide by 3. The quotient of 81 divide by 3 is 27.

81/3

The solution is 27.

27

Fractional form, "everything on top" over "everything on the bottom", means all the stuff "on top" is divided by all the stuff "on the bottom". And "all the stuff" or "everything" implies those expressions go together. Think of it as the top part of the fraction has a master set of parentheses around it and the bottom part has its own master set of parentheses around it. It's just that, when fractions are typeset vertically (as opposed to typed out horizontally like in an email or a Math forum) we don't include the grouping symbols (a.k.a. parenthesis). The vertical layout actually makes things easier to understand.

If you have a problem written in a fractional form, then it is implied that you should do everything up top and everything on the bottom before dividing.

Example 1.5.3

(6 + 2 2 ) × 7 2 × 15 − 25

1. Remember, we do everything up top and everything on the bottom before dividing.

a. Top: Exponents first 2 2 = 2 × 2 = 4 b. Bottom: Multiply first, 2 × 15 = 30

(6 + 4) × 7 30 − 25

2. Again, simplify the top and bottom completely before dividing. a. Top : 6 + 4 = 10 b. Bottom: 30 − 25 = 5 3. We still must finish simplifying the top before we divide. a. Top: 10 × 7 = 70 b. Bottom: ---

10 × 7 5

70 5 14

4. Now you can divide, 70 ÷ 5 = 14 .

The solution is 14.

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