Quantitative · 3 min read
Decimals, Fractions, and Percents · Operations
Decimals, fractions, and percents are three ways of writing the same values, and the SSAT expects you to move between them quickly.
Working with Decimals
Read decimals by place value: 0.7 is seven tenths, 0.45 is forty-five hundredths, and 0.123 is one hundred twenty-three thousandths. To compare decimals, line up the decimal points and add trailing zeros if needed: 0.45 = 0.450 and 0.450 > 0.405.
| Operation | Method | Example |
|---|---|---|
| Addorsubtract | Lineupthedecimalpoints thenaddorsubtractaswith , | 325+14=465 . . . |
| wholenumbers | ||
| Multiply | Multiplyaswholenumbers thencountthetotaldecimal , | 2 .5 × 1 .4: 25 × 14 = 350; two places → 3 .50 |
| placesinbothfactors | ||
| Divide | Movethedecimalpointinthedivisortomakeitawhole | 456÷012=456÷12=38 . . |
| number;movethedividend'spointthesamenumberof | ||
| places |
Understanding Fractions
A fraction represents part of a whole: the numerator (top) counts the parts taken, and the denominator (bottom) is the number of equal parts in the whole. A proper fraction is less than 1 (3/4), an improper fraction is 1 or greater (7/4), and a mixed number combines a whole number and a proper fraction (1 3/4).
| Skill | Method | Example | ||
|---|---|---|---|---|
| Improper mixed → | Divide;thequotientisthewholenumberandthe | 17/5: 17 ÷ 5 = 3 R 2 → 3 2/5 | ||
| remainderisthenewnumerator | ||||
| Mixed improper → | Multiplythewholenumberbythedenominator add , | 4 2/3: (4 × 3) + 2 = 14 → 14/3 | ||
| thenumerator keepthedenominator , | ||||
| Equivalentfractions | Multiplyordividethenumeratoranddenominatorby | 2/3=4/6=6/9=8/12 | ||
| thesamenonzeronumber | ||||
| Simplify | DividethenumeratoranddenominatorbytheirGCF | 12/18: GCF 6 2/3 → | ||
| Compare(common | Rewritewithacommondenominatorandcompare | 3/4=9/12and5/6=10/12 so3/4< , | 5/6 | |
| denominator) | numerators | |||
| Compare (cross-multiply) | Fora/bandc/d comparea×dwithb×c , | 3/4vs5/6:18<20 so3/4<5/6 , | ||
| Addorsubtract | Useacommondenominator thencombinethe , | 1/4+2/3=3/12+8/12=11/12 | ||
| numerators | ||||
| Multiply | Multiplynumeratorsandmultiplydenominators | 2/3×4/5=8/15 | ||
| Divide Multiplybythereciprocalofthedivisor onvertingAmongDecimals Fractions andPercents , , ercentmeans"perhundred"so45%=45/100=045 , . . | 3/4÷2/5=3/4×5/2=15/8=17/8 | 1 |
Converting Among Decimals, Fractions, and Percents
Percent means "per hundred," so 45% = 45/100 = 0.45.
| Conversion | Method | Example |
|---|---|---|
| Decimal percent → | Multiplyby100(movethepointtwoplacesright) | 0375=375% . . |
| Percent decimal → | Divideby100(movethepointtwoplacesleft) | 62%=062 . |
| Fraction percent → | Dividetogetadecimal thenmultiplyby100 , | 3/8=0375=375% . . |
| Percent fraction → | Writethepercentover100andsimplify | 40%=40/100=2/5 |
| Decimal fraction → | Useplacevalue thensimplify , | 075=75/100=3/4 . |
| Fraction decimal → | Dividethenumeratorbythedenominator | 5/8=5÷8=0625 . |
| Fraction | 1/2 | 1/3 | 2/3 | 1/4 | 3/4 | 1/5 | 2/5 | 3/5 | 4/5 | 1/8 | 1/10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Decimal | 05 . | ≈0333 . | ≈0667 . | 025 . | 075 . | 02 . | 04 . | 06 . | 08 . | 0125 . | 01 . |
| Percent | 50% | 33⅓% | 66⅔% | 25% | 75% | 20% | 40% | 60% | 80% | 125% . | 10% |
Percent Problems
| ProblemType | Method | Example |
|---|---|---|
| Percentofanumber | Convertthepercenttoadecimaland | 25%of80=025×80=20 . |
| multiply | ||
| Whatpercentonenumberisof | Dividethepartbythewhole thenmultiply , | 12of50:12÷50=024=24% . |
| another | by100 | |
| Findingthewhole | Dividethepartbythepercent(asa | 15is30%ofwhat?15÷030=50 . |
| decimal) | ||
| Percentincrease | (new − original) ÷ original × 100 | $40to$50:10÷40=25%increase |
| Percentdecrease | (original − new) ÷ original × 100 | $80to$60:20÷80=25%decrease |
| Discount | Multiply the original price by (100% − | $60at20%off:080×$60=$48 . |
| discount) | ||
| Salestax | Multiplythepricebythetaxrateandaddit | $48with8%tax:$48+$384=$5184 . . |
| totheprice |
Mastery of the four operations and their properties is fundamental to every quantitative question.
| Property | Meaning | Example | |
|---|---|---|---|
| Commutative Associative Identity | Orderdoesnotchangeasumorproduct Groupingdoesnotchangeasumorproduct Adding0ormultiplyingby1leavesanumber unchanged | 5+3=3+5;4×3=3×4 1 (2+3)+4=2+(3+4);(2×3)×4=2×(3× 4) 7+0=7;9×1=9 |
| Property | Meaning | Example |
|---|---|---|
| Zeroproperty | Anynumbertimes0is0 | 15×0=0 |
| Distributive | a(b+c)=ab+ac | 3(4+5) =12+15=27 |
Subtraction and division are neither commutative nor associative: 8 − 3 = 5 but 3 − 8 = −5, and 20 ÷ 4 = 5 but 4 ÷ 20 = 0.2. Subtraction can be rewritten as adding the opposite (12 − 5 = 12 + (−5) = 7). Zero divided by any nonzero number is 0, but division by zero is undefined.
Operations with Negative Numbers
| Operation | Rule | Examples |
|---|---|---|
| Addingsamesigns | Addtheabsolutevaluesandkeepthesign | ( −5)+( −3) =−8 |
| Addingdifferentsigns | Subtractthesmallerabsolutevaluefromthelarger | 7+( −4) =3;( −9)+6=−3 |
| andkeepthesignofthelarger | ||
| Subtractinganegative | Sameasaddingapositive | 5 − ( −3) = 8; ( −4) − ( −7) = 3 |
| Multiplyingordividing | Samesignsgiveapositive;differentsignsgivea | ( −6)×( −2) =12;( −15)÷3=−5;8× |
| negative | ( −4) =−32 |
Order of Operations
➜
➜
➜
and other grouping symbols
and roots
left to right
left to right
Evaluate 3 + 4 × 2² − (6 − 1)
Parentheses: 6 − 1 = 5
Exponents: 2² = 4
Multiplication: 4 × 4 = 16
Add/subtract left to right: 3 + 16 − 5 = 14
Multiplication does not always come before division: they share a step and are done from left to right, as are addition and subtraction.
Mental Math and Estimation
Estimation gives a quick approximate answer: round to compatible values before calculating (489 × 21 ≈ 500 × 20 = 10,000). On a test without a calculator, estimation is the fastest way to eliminate answer choices that are clearly too large or too small.
Finished reading?
Lock it in with a quick Quantitative quiz.