Mathematics · 3 min read
Operations and Order of Operations · Fractions and Decimals
Addition and multiplication are commutative (order does not change the result) and associative (grouping does not change the result). Subtraction and division are neither. The distributive property links the two: a(b + c) = ab + ac, so 6(20 + 3) = 120 + 18 = 138. Division by zero is undefined.
When an expression contains several operations, apply the order of operations: Parentheses, Exponents, Multiplication and Division from left to right, then Addition and Subtraction from left to right (PEMDAS). Multiplication does not automatically come before division; whichever appears first from the left is done first.
| Expression | Steps | Result |
|---|---|---|
| 3 + 4 × 2 | Multiply first: 4 × 2 = 8; then 3 + 8 | 11 |
| 18 ÷ 3 × 2 | Left to right: 18 ÷ 3 = 6; 6 × 2 | 12 |
| (5 + 3)² − 10 ÷ 2 | Parentheses 8; exponent 64; 10 ÷ 2 = 5; 64 − 5 | 59 |
| 4 + 2(7 − 3)² ÷ 8 | 7 − 3 = 4; 4² = 16; 2 × 16 = 32; 32 ÷ 8 = 4; 4 + 4 | 8 |
Signed Numbers
- Adding numbers with the same sign: add and keep the sign (−7 + −5 = −12).
- Adding numbers with different signs: subtract the absolute values and keep the sign of the larger (−9 + 4 = −5).
- Subtracting: add the opposite (6 − (−3) = 6 + 3 = 9).
- Multiplying or dividing: like signs give a positive result, unlike signs a negative result (−8 × −3 = 24; 20 ÷ −4 = −5).
A fraction has a numerator (top) and a denominator (bottom). Proper fractions are less than 1; improper fractions have a numerator equal to or larger than the denominator; mixed numbers combine a whole number and a proper fraction. Simplify by dividing both parts by their greatest common factor (GCF): 12/18 ÷ 6/6 = 2/3.
| Operation | Rule | Example |
|---|---|---|
| Add or subtract | Find the least common denominator (LCD), rewrite, combine numerators | 1/4 + 1/6 = 3/12 + 2/12 = 5/12 |
| Mixed numbers | Convert to improper fractions first | 2 1/3 − 1 3/4 = 7/3 − 7/4 = 28/12 − 21/12 = 7/12 |
| Multiply | Multiply numerators and denominators; cancel first when possible | 2/3 × 9/10 = 18/30 = 3/5 |
| Divide | Multiply by the reciprocal of the divisor | 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8 |
| Compare | Cross-multiply or convert to decimals | 5/8 vs 3/5: 25 vs 24, so 5/8 is larger |
Decimals
Place values to the right of the decimal point are tenths, hundredths, and thousandths. Line up decimal points to add or subtract. To multiply, multiply as whole numbers, then count the total decimal places in both factors: 1.2 × 0.03 = 0.036 (three places). To divide, move the decimal point in the divisor to make it a whole number and move the dividend's decimal point the same number of places: 4.5 ÷ 0.15 = 450 ÷ 15 = 30.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/8 | 0.125 | 12.5% |
| 1/5 | 0.2 | 20% |
| 1/4 | 0.25 | 25% |
| 1/3 | 0.333… | 33.3% |
| 3/8 | 0.375 | 37.5% |
| 1/2 | 0.5 | 50% |
| 2/3 | 0.666… | 66.7% |
| 3/4 | 0.75 | 75% |
| 7/8 | 0.875 | 87.5% |
Rounding
Look at the digit to the right of the place you are rounding to: 5 or greater rounds up, 4 or less leaves the digit unchanged. 2.347 rounded to the nearest hundredth is 2.35; rounded to the nearest tenth it is 2.3. Round only at the final step of a multi-step problem so early rounding does not change the answer.
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