Fractions, decimals and percents without a calculator

One number, three spellings

A fraction, a decimal and a percent are three ways of writing the same value, and without a calculator the right spelling decides how long a question takes. $37.5\%$ of 168 is awkward as a decimal and quick as $\frac{3}{8}$ of 168. These equivalents are worth knowing without thinking:

Fraction	Decimal	Percent
$\frac{1}{3}$, $\frac{2}{3}$	$0.\overline{3}$, $0.\overline{6}$	$33\frac{1}{3}\%$, $66\frac{2}{3}\%$
$\frac{1}{6}$, $\frac{5}{6}$	$0.1\overline{6}$, $0.8\overline{3}$	$16\frac{2}{3}\%$, $83\frac{1}{3}\%$
$\frac{1}{8}$, $\frac{3}{8}$, $\frac{5}{8}$, $\frac{7}{8}$	$0.125$, $0.375$, $0.625$, $0.875$	$12.5\%$, $37.5\%$, $62.5\%$, $87.5\%$
$\frac{1}{9}$	$0.\overline{1}$	$11\frac{1}{9}\%$
$\frac{1}{12}$	$0.08\overline{3}$	$8\frac{1}{3}\%$
$\frac{1}{20}$, $\frac{1}{25}$	$0.05$, $0.04$	$5\%$, $4\%$

One more swap saves time: $x\%$ of $y$ equals $y\%$ of $x$, because both are $\frac{xy}{100}$. So $16\%$ of 75 is $75\%$ of 16, which is 12.

Comparing fractions

To compare $\frac{a}{b}$ and $\frac{c}{d}$ with positive denominators, cross-multiply: $\frac{a}{b} > \frac{c}{d}$ exactly when $ad > bc$. This is the same as writing both over the common denominator $bd$, without writing it. Two quicker tools often finish the job first:

• Benchmarks. $\frac{5}{11}$ is below $\frac{1}{2}$ and $\frac{6}{11}$ is above it, so neither needs arithmetic to be placed against $\frac{1}{2}$.
• Distance from 1. $\frac{7}{8}$ and $\frac{9}{10}$ fall short of 1 by $\frac{1}{8}$ and $\frac{1}{10}$. The smaller shortfall wins, so $\frac{9}{10}$ is larger.

Decimals by hand

To multiply decimals, multiply the digits as whole numbers, then count decimal places. $0.4 \times 0.025$ is $4 \times 25 = 100$ with $1 + 3 = 4$ places, so $0.0100 = 0.01$. To divide, move both decimal points the same number of places until the divisor is a whole number; this multiplies top and bottom by the same power of ten, so the value does not change: $\frac{0.36}{0.009} = \frac{360}{9} = 40$. When the places get confusing, write each number as digits times a power of ten and add the exponents.

Which fractions terminate?

A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5. $\frac{7}{40}$ terminates, since $40 = 2^3 \times 5$; $\frac{7}{30}$ does not, since 30 contains 3. Reduce first: $\frac{21}{56} = \frac{3}{8} = 0.375$ terminates even though 56 contains 7.

Percents as multipliers

A rise of $p\%$ multiplies by $1 + \frac{p}{100}$ and a fall of $p\%$ by $1 - \frac{p}{100}$. Successive changes multiply, and to undo a change you divide by its multiplier. A percent change is always measured from the value the change starts from:

$$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$$

Profit and discount questions are chains of multipliers. A cost marked up 50% and then discounted 20% sells for $1.5 \times 0.8 = 1.2$ times the cost, a profit of 20% of the cost, not the 30% that subtracting the percents suggests.

Fractions of what is left

"Spent $\frac{1}{5}$, then $\frac{3}{4}$ of the remainder" takes the second fraction of a smaller amount. Track what is left after each step: $\frac{4}{5}$ remains after the first step and $\frac{1}{4}$ of that after the second, so $\frac{4}{5} \times \frac{1}{4} = \frac{1}{5}$ of the original remains. Writing the remaining fraction at each step avoids the most common slip, taking every fraction of the original amount.

Estimate before you choose

Answer choices on problem-solving questions are often spread out enough that a rough value picks the right one. Rounding $\frac{29}{61}$ to just under $\frac{1}{2}$, or $0.198$ to $0.2$, is legitimate whenever the options are far apart; when they are close, do the exact arithmetic.