Quantitative (Math) · 3 min read
Rational Numbers · Proportions and Ratios
Fractions
A fraction a/b represents part of a whole; a is the numerator and b is the denominator, which cannot be zero. Proper fractions have a numerator smaller than the denominator (3/4, 2/5). Improper fractions have a numerator equal to or greater than the denominator (5/3, 9/4). Mixed numbers combine a whole number and a proper fraction (2½, 3¾).
| Skill | Method | Example |
|---|---|---|
| Improper mixed → | Divide;theremainderbecomesthenew | 17/5:17÷5=3R2 so3⅖ , |
| numerator | ||
| Mixed improper → | Whole×denominator+numerator overthe , | 4⅔=(4×3+2)/3=14/3 |
| denominator | ||
| Equivalentfractions | Multiplyordividetopandbottombythesame | 2/3=4/6=6/9=8/12 |
| nonzeronumber | ||
| Simplifying | DividetopandbottombytheirGCF | 24/36(GCF12) =2/3 |
| Addorsubtract | Rewritewiththeleastcommondenominator then , | 2/3+3/4=8/12+9/12=17/12=15/12 |
| combinenumerators | ||
| Multiply | Multiplynumeratorsanddenominators then , | 3/4×2/5=6/20=3/10 |
| simplify | ||
| Divide | Multiplybythereciprocalofthedivisor | 3/4÷2/5=3/4×5/2=15/8=1⅞ |
Decimals
Decimals are fractions with denominators that are powers of 10, with place values of tenths, hundredths, thousandths, and so on. To convert a fraction to a decimal, divide: 3/8 = 0.375; some fractions repeat, as in 1/3 = 0.333… = 0.3̄. To convert a decimal to a fraction, write the digits over the matching power of 10 and simplify: 0.45 = 45/100 = 9/20.
Percents
A percent is a ratio out of 100; the symbol % means "per hundred."
| Conversion | Method | Example |
|---|---|---|
| Percent decimal → | Divideby100(movethepointtwoplacesleft) | 75%=075 . |
| Decimal percent → | Multiplyby100(movethepointtwoplaces | 032=32% . |
| right) | ||
| Percent fraction → | Writeover100andsimplify | 40%=40/100=2/5 |
| Fraction percent → | Divide thenmultiplyby100 , | 3/8=0375=375% . . |
| PercentProblem | Method | Example |
| Percentofanumber | Percent(asadecimal)×number | 35%of80=035×80=28 . |
| Whatpercentonenumberisof | Part÷whole×100 | 12of48:12÷48×100=25% |
| another | ||
| Findingthewhole | Part÷percent(asadecimal) | 24is30%ofwhat?24÷030=80 . |
| Percentincrease | (new − original) ÷ original × 100 | $50to$65:15÷50×100=30% |
| Percentdecrease | (original − new) ÷ original × 100 | $80to$60:20÷80×100=25% |
Memorize the common ones to save time without a calculator: 1/2 = 0.5 = 50%, 1/3 ≈ 0.333 ≈ 33⅓%, 1/4 = 0.25 = 25%, 1/5 = 0.2 = 20%, 1/8 = 0.125 = 12.5%, 1/10 = 0.1 = 10%, 3/4 = 0.75 = 75%, and 3/8 = 0.375 = 37.5%.
Ratios
A ratio compares two quantities and can be written a:b, a/b, or "a to b." Express ratios in simplest form by dividing both terms by their GCF: 12:18 = 2:3. A part-to-part ratio compares two parts of a group: a class of 15 boys and 12 girls has a boy-to-girl ratio of 15:12 = 5:4. A part-to-whole ratio compares one part with the total: boys to all students is 15:27 = 5:9.
Girls : boys = 4 : 5, and the class has 36 students.
Total parts = 4 + 5 = 9 One part = 36 ÷ 9 = 4
Girls = 4 × 4 = 16 Boys = 5 × 4 = 20
Proportions and Variation
A proportion states that two ratios are equal: a/b = c/d. Solve by cross-multiplying: ad = bc. Example: 3/5 = x/20 gives 5x = 60, so x = 12.
Scale Drawings, Maps, and Unit Rates
A scale of 1:50 means 1 unit on the drawing represents 50 units in reality. To find an actual measurement, set up a proportion with the scale. Example: on a map where 1 inch = 25 miles, two cities 3.5 inches apart are 3.5 × 25 = 87.5 miles apart.
A unit rate is a rate with a denominator of 1. If 8 apples cost $6, the unit rate is $6 ÷ 8 = $0.75 per apple. Unit rates make comparisons easy: Store A sells 5 pounds for $12 ($2.40 per pound) and Store B sells 3 pounds for $8 (about $2.67 per pound), so Store A is the better value.
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