Appendix D

### Mathematics Review

1. Order of Arithmetic Operations

Certain arithmetic operations take precedence over others. In completing problems with a series of operations the following guidelines apply:

1. Addition or subtraction may occur in any order.

Example: 4 + 8 − 7 + 3 = 8 or 8 + 3 + 4 − 7 = 8

2. Multiplication or division must be completed before addition or subtraction.

Example: 48 ÷ 6 + 2 = 10

Example: 4 + (2/3)(1/2) = 4 1/3

3. Any quantity above a division line, under a division line or a radical sign Display Formula$(000)$, or within parentheses or brackets must be treated as one number.

Example: Display Formula$32-25=11$

Example: 2(5 + 3 − 4) = 8

Example: Display Formula$9+23=113$

2. Fractions, Decimals, and Percents

1. To add (or subtract) fractions, the denominator in each term must be the same. (Choose the lowest common denominator for each term. Multiply each term by the common denominator and then add [or subtract].)

Example: Display Formula$34+53=2912=2⁢512$

(lowest common denominator = 12)

Solution: Display Formula$(34×1212)+(53×1212)=912+2012=2912=2⁢512$

Example: Display Formula$cdx+xc=c2d+x2xc$

(lowest common denominator = xc)

Solution:

Display Formula$(cdx·xcxc)+(xc·xcxc)=c2dxc+x2xc=c2d+x2xc$

2. To multiply fractions, multiply the numerators by each other and the denominators by each other.

Example: Display Formula$38·23=624=14$

Example: Display Formula$pq(pq)=p2q=p2$

3. To divide fractions, invert the divisor and multiply.

Example: Display Formula$38÷92=38×29=672=112$

Example: Display Formula$nr÷st=nr×ts=ntrs$

Example: Display Formula$(1a+1b)÷(1a-1b)=b+aab·abb-a=b+ab-a$

4. To convert a fraction to a percentage divide the numerator by the denominator and multiply by 100.

Example: Display Formula$38=0.375×100=37.5%$

Note: To convert a percentage to a decimal, move the decimal point two places to the left.

5. When dividing by a decimal, divide by the integer and add sufficient zeros to move the decimal point the appropriate number of digits to the right.

Example: 36 ÷ 0.04 = 900 or

36 ÷ 4 = 9 plus ...

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