Quantitative Reasoning · 3 min read
Fractions, Decimals, and Percents
Fractions
A fraction names part of a whole: the numerator counts the parts and the denominator tells how many equal parts make the whole. A proper fraction has a numerator smaller than its denominator (3/4); an improper fraction does not (7/4); a mixed number combines a whole number and a fraction (1 3/4). Simplify by dividing the numerator and denominator by their greatest common factor: 24/36 = 2/3.
| Operation | Procedure | Example |
|---|---|---|
| Addorsubtract | Rewritewiththeleastcommondenominator then , | 2/3 + 1/4 = 8/12 + 3/12 = 11/12; 5/6 − 1/4 = 10/12 − |
| combinenumerators | 3/12=7/12 | |
| Multiply | Multiplynumeratorsandmultiplydenominators; | 3/4×2/5=6/20=3/10 |
| simplify | ||
| Divide | Multiplybythereciprocalofthedivisor | 3/4÷2/5=3/4×5/2=15/8=17/8 |
| Improper mixed → | Divide;theremainderbecomesthenumerator | 17/5=32/5 |
| Mixed improper → | Whole×denominator+numerator overthe , | 42/3=(4×3+2)/3=14/3 |
| denominator |
Decimals
Each place to the right of the decimal point is a power of ten: tenths, hundredths, thousandths. To add or subtract, line up the decimal points (3.45 + 2.7 = 6.15). To multiply, multiply as whole numbers and count the total decimal places in the factors (2.5 × 1.4 = 3.50). 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). Convert a fraction to a decimal by dividing (5/8 = 0.625) and a decimal to a fraction with place value (0.125 = 125/1000 = 1/8).
Percents
Percent means "per hundred." Move the decimal point two places left to change a percent to a decimal (8% = 0.08; 125% = 1.25) and two places right to change a decimal to a percent (0.06 = 6%). To change a percent to a fraction, write it over 100 and simplify (37.5% = 375/1000 = 3/8).
| Fraction | Decimal | Percent | Fraction | Decimal | Percent |
|---|---|---|---|---|---|
| 1/2 | 05 . | 50% | 1/5 | 02 . | 20% |
| 1/3 | 0333 . … | 33⅓% | 1/8 | 0125 . | 125% . |
| 2/3 | 0666 . … | 66⅔% | 3/8 | 0375 . | 375% . |
| 1/4 | 025 . | 25% | 1/10 | 01 . | 10% |
| 3/4 | 075 . | 75% | 1/20 | 005 . | 5% |
The Three Percent Problems
| Find | Method | Example | ||
|---|---|---|---|---|
| Apercentofanumber | part=percent×whole | 35%of80=035×80=28 . | ||
| Whatpercentonenumberisof | percent=part÷whole×100 | 15is(15÷60)×100=25%of60 | ||
| another | ||||
| Thewhole | whole=part÷percent | 1 If 24 is 40% of a number , the number is 24 ÷ 0 .40 = 60 |
Percent Change and Successive Percents
percent change = (new − original) ÷ original × 100
A positive result is an increase and a negative result is a decrease. Always divide by the original value.
A price that rises from $50 to $65 increases by 15 ÷ 50 = 30%. A population that falls from 800 to 720 decreases by 80 ÷ 800 = 10%. To apply a change directly, multiply by a growth or decay factor: a 20% discount multiplies the price by 0.80, and a 6% tax multiplies it by 1.06.
A $120 jacket is discounted 25%, then 8% sales tax is added.
120 × 0.75 = 90.00 price after discount
90 × 1.08 = 97.20 price after tax
Final cost: $97.20
A 25% decrease followed by a 25% increase does not return to the original value: 100 × 0.75 × 1.25 = 93.75. Successive percents multiply; they do not add.
Identify the whole before computing any percent.
Read carefully: some questions ask for the amount of change, others for the new value.
Percent change always divides by the original amount, not the new one.
Use multipliers (0.80, 1.06) to combine discounts, taxes, and tips in one step.
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