Fractions, Decimals, and Percents
Fractions, Decimals, and Percents
Fractions, decimals, and percents are three different ways of writing the exact same value. Being able to jump quickly between the three forms — without a calculator — is one of the most useful skills in competition math, since a problem might hand you a percent but expect a fraction answer, or vice versa.
Converting Between Forms
Fraction to decimal: divide the numerator by the denominator. For example, \frac{3}{8} = 3 \div 8 = 0.375.
Decimal to fraction: use the place value of the last digit as the denominator, then simplify. For example, 0.375 has three digits after the decimal point, so 0.375 = \frac{375}{1000} = \frac{3}{8} after dividing top and bottom by 125.
Decimal to percent: multiply by 100 and attach a percent sign (equivalently, shift the decimal point two places right). So 0.375 = 37.5\%.
Percent to fraction: write the percent over 100 and simplify: 37.5\% = \frac{37.5}{100} = \frac{375}{1000} = \frac{3}{8}.
It pays to memorize a few common conversions outright, since they appear constantly: \frac{1}{2} = 0.5 = 50\%, \frac{1}{4} = 0.25 = 25\%, \frac{1}{5} = 0.2 = 20\%, \frac{1}{3} = 0.\overline{3} = 33.\overline{3}\%, and \frac{1}{8} = 0.125 = 12.5\%.
Comparing Fractions
To compare two fractions, give them a common denominator and then compare numerators, or convert both to decimals. For example, to compare \frac{5}{7} and \frac{3}{4}, use the common denominator 28: \frac{5}{7} = \frac{20}{28} and \frac{3}{4} = \frac{21}{28}, so \frac{3}{4} > \frac{5}{7}. A quick shortcut called cross- multiplication does the same job without fully rebuilding the fractions: for \frac{a}{b} vs. \frac{c}{d} with positive denominators, compare a \cdot d against b \cdot c.
Basic Fraction Arithmetic
- Adding/subtracting: get a common denominator first, then add or subtract the numerators. \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}
- Multiplying: multiply numerators together and denominators together, then simplify. \frac{2}{3} \times \frac{9}{10} = \frac{18}{30} = \frac{3}{5}
- Dividing: multiply by the reciprocal of the second fraction. \frac{2}{3} \div \frac{4}{9} = \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}
Example
Which is larger, 62.5\% or \frac{5}{8}?
Step 1: Convert the fraction to a percent by first finding its decimal form: \frac{5}{8} = 5 \div 8 = 0.625.
Step 2: Convert that decimal to a percent: 0.625 = 62.5\%.
Step 3: Compare: 62.5\% equals \frac{5}{8} exactly — they are the same value written two different ways, so neither is larger.
Key Takeaways
- Fractions, decimals, and percents are three notations for the same value; convert by dividing (fraction to decimal) and multiplying/dividing by 100 (decimal to percent).
- Memorize common fraction-decimal-percent triples to save time on contests.
- Compare fractions using a common denominator or cross-multiplication.
- Add/subtract fractions with a common denominator; multiply straight across; divide by multiplying by the reciprocal.