Math Core

$\dfrac{3}{4}$ , $0.75$ and $75\%$ are three names for the same number. Fractions are best for exact multiplication and division, decimals are best for comparing and for calculators, and percents are best for describing parts of a whole. Being able to switch quickly between them, including for negative numbers, is a skill you'll use all through algebra.

Fraction to decimal

A fraction is a division, so divide the numerator by the denominator.

\frac{3}{8} = 3 \div 8 = 0.375

Sometimes the division ends, as with $\dfrac{3}{8}$ . That's a terminating decimal. Other times a digit or block of digits repeats forever, like $\dfrac{2}{3} = 0.666\ldots$ That's a repeating decimal, and you show the repeating part with a bar:

\frac{2}{3} = 0.\overline{6} \qquad \frac{5}{11} = 0.\overline{45} \qquad \frac{1}{6} = 0.1\overline{6}

In $0.1\overline{6}$ only the 6 repeats: $0.1666\ldots$

Definition

Rational number

A rational number is any number that can be written as a fraction $\dfrac{a}{b}$ of integers, with $b \ne 0$ . Every rational number is either a terminating or a repeating decimal. Integers, fractions, and negatives of them are all rational: $-5 = \dfrac{-5}{1}$ and $-0.25 = -\dfrac{1}{4}$ .

Tip

You can tell whether a fraction terminates without dividing. Simplify it first, then look at the denominator. If its only prime factors are 2s and 5s (like 4, 8, 20, 25, 40), the decimal terminates. Any other prime factor (3, 7, 11, ...) makes it repeat. So $\dfrac{7}{40}$ terminates but $\dfrac{7}{30}$ repeats.

Worked example: Negative fractions to decimals

Write each as a decimal.

$-\dfrac{7}{20}$
$-2\dfrac{4}{9}$

Solutions.

$7 \div 20 = 0.35$ , so $-\dfrac{7}{20} = -0.35$ .
The whole part is 2. For the fraction, $4 \div 9 = 0.444\ldots$ So $-2\dfrac{4}{9} = -2.\overline{4}$ .

The sign simply comes along. Convert the positive number, then put the negative sign in front.

Decimal to fraction

Read the decimal's place value, write it as a fraction over 10, 100, 1000, ..., and simplify.

$0.6$ is six tenths: $\dfrac{6}{10} = \dfrac{3}{5}$ .
$0.45$ is forty-five hundredths: $\dfrac{45}{100} = \dfrac{9}{20}$ .
$-1.125$ is negative one and 125 thousandths: $-1\dfrac{125}{1000} = -1\dfrac{1}{8} = -\dfrac{9}{8}$ .

For a repeating decimal, memorize the common ones: $0.\overline{3} = \dfrac{1}{3}$ , $0.\overline{6} = \dfrac{2}{3}$ , and in general $0.\overline{d} = \dfrac{d}{9}$ for a single repeating digit $d$ . (In Algebra 1 you'll learn a method that works for any repeating decimal.)

Percents

Percent means "per hundred," so $n\% = \dfrac{n}{100}$ .

Converting with percents

Decimal to percent: multiply by 100 (move the decimal point 2 places right) and add $\%$ . So $0.375 = 37.5\%$ .
Percent to decimal: divide by 100 (move the decimal point 2 places left). So $8\% = 0.08$ .
Fraction to percent: change to a decimal first, then to a percent.
Percent to fraction: write over 100 and simplify.

Worked example: All three forms

Complete each row.

fraction	decimal	percent
$\dfrac{3}{5}$	?	?
?	?	$140\%$
?	$-0.04$	?

Solutions.

$\dfrac{3}{5} = 3 \div 5 = 0.6 = 60\%$ .
$140\% = 1.40 = 1.4$ , and $\dfrac{140}{100} = \dfrac{7}{5}$ . Percents over 100 mean more than one whole.
$-0.04 = -\dfrac{4}{100} = -\dfrac{1}{25}$ , and as a percent, $-4\%$ . Negative percents appear when something decreases.

Common mistake

Watch out for small percents. $0.5\%$ is not $0.5$ . Move the decimal point two places left: $0.5\% = 0.005$ , which is half of one percent. Likewise, $0.3$ is $30\%$ , not $3\%$ .

Comparing and ordering rational numbers

To order numbers written in different forms, convert them all to decimals. Remember that on the number line, negative numbers get smaller as you move left, so $-0.8$ is less than $-0.75$ .

Worked example: Ordering

Order from least to greatest: $-\dfrac{3}{4}$ , $-0.8$ , $0.7$ , $\dfrac{2}{3}$ .

As decimals: $-0.75$ , $-0.8$ , $0.7$ , $0.666\ldots$

The negatives come first. $-0.8$ is farther left than $-0.75$ . Then $0.666\ldots$ is less than $0.7$ . So the order is:

-0.8,\ -\frac{3}{4},\ \frac{2}{3},\ 0.7

-0.8 < -3/4 < 2/3 < 0.7

Practice

Practice 1

Write $\dfrac{5}{8}$ as a decimal.

Type your answer

Practice 2

Which is $-\dfrac{5}{6}$ written as a decimal?

Practice 3

Which is $0.36$ written as a fraction in simplest form?

Practice 4

Write $7\%$ as a decimal.

Type your answer

Practice 5

Write $-2\dfrac{3}{4}$ as a decimal.

Type your answer

Practice 6

Write $\dfrac{17}{40}$ as a percent.

Type your answer

Practice 7

Which fraction has a repeating decimal?

Practice 8

Which list is in order from least to greatest?