Decimals and Fractions

By the end of this section, you will be able to: convert fractions to decimals, order decimals and fractions, simplify expressions using the order of operations, and find the circumference and area of circles.

Convert fractions to decimals

In Decimals, we learned to convert decimals to fractions. Now we will do the reverse — convert fractions to decimals. Remember that the fraction bar indicates division. So $\tfrac{4}{5}$ can be written $4 \div 5$ or $5\overline{\smash{)}\,4}$ . This means that we can convert a fraction to a decimal by treating it as a division problem.

Convert a fraction to a decimal. To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

Example. Write the fraction $\tfrac{3}{4}$ as a decimal.

A fraction bar means division, so we can write the fraction $\tfrac{3}{4}$ using division, $4\overline{\smash{)}\,3}$ . Divide:

\begin{array}{r} \phantom{4\,\overline{\smash{)}\,}}0.75 \\ 4\,\overline{\smash{)}\,3.00} \\ \phantom{4\,\overline{\smash{)}\,}}\underline{28}\phantom{0} \\ \phantom{4\,\overline{\smash{)}\,}}\phantom{0}20 \\ \phantom{4\,\overline{\smash{)}\,}}\phantom{0}\underline{20} \\ \phantom{4\,\overline{\smash{)}\,}}\phantom{00}0 \end{array}

So the fraction $\tfrac{3}{4}$ is equal to $0.75$ .

Write the fraction as a decimal: $\tfrac{1}{4}$

Write the fraction as a decimal: $\tfrac{3}{8}$

Example. Write the fraction $-\tfrac{7}{2}$ as a decimal.

The value of this fraction is negative. After dividing, the value of the decimal will be negative. We do the division ignoring the sign, and then write the negative sign in the answer. Divide $7$ by $2$ :

\begin{array}{r} \phantom{2\,\overline{\smash{)}\,}}3.5 \\ 2\,\overline{\smash{)}\,7.0} \\ \phantom{2\,\overline{\smash{)}\,}}\underline{6}\phantom{.0} \\ \phantom{2\,\overline{\smash{)}\,}}10 \\ \phantom{2\,\overline{\smash{)}\,}}\underline{10} \\ \phantom{2\,\overline{\smash{)}\,}}\phantom{0}0 \end{array}

So, $-\tfrac{7}{2} = -3.5$ .

Write the fraction as a decimal: $-\tfrac{9}{4}$

Write the fraction as a decimal: $-\tfrac{11}{2}$

Repeating decimals

So far, in all the examples converting fractions to decimals the division resulted in a remainder of zero. This is not always the case. Let’s see what happens when we convert the fraction $\tfrac{4}{3}$ to a decimal. First, notice that $\tfrac{4}{3}$ is an improper fraction — its value is greater than $1$ , so the equivalent decimal will also be greater than $1$ .

We divide $4$ by $3$ :

\begin{array}{r} \phantom{3\,\overline{\smash{)}\,}}1.333\ldots \\ 3\,\overline{\smash{)}\,4.000} \\ \phantom{3\,\overline{\smash{)}\,}}\underline{3}\phantom{.000} \\ \phantom{3\,\overline{\smash{)}\,}}\phantom{0}10 \\ \phantom{3\,\overline{\smash{)}\,}}\phantom{0}\underline{9}\phantom{0} \\ \phantom{3\,\overline{\smash{)}\,}}\phantom{00}10 \\ \phantom{3\,\overline{\smash{)}\,}}\phantom{00}\underline{9}\phantom{0} \\ \phantom{3\,\overline{\smash{)}\,}}\phantom{000}10 \\ \phantom{3\,\overline{\smash{)}\,}}\phantom{000}\underline{9}\phantom{0} \\ \phantom{3\,\overline{\smash{)}\,}}\phantom{0000}1 \end{array}

No matter how many more zeros we write, there will always be a remainder of $1$ , and the threes in the quotient will go on forever. The number $1.333\ldots$ is called a repeating decimal — the “ $\ldots$ ” means the pattern repeats.

Repeating decimal. A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.

How do you know how many “repeats” to write? Instead of writing $1.333\ldots$ we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal $1.333\ldots$ is written $1.\overline{3}$ . The line above the $3$ tells you that the $3$ repeats endlessly. So $1.333\ldots = 1.\overline{3}$ .

For other decimals, two or more digits might repeat. The table below shows some more examples of repeating decimals.

Repeating decimal	Repeating digit(s)
$1.333\ldots = 1.\overline{3}$	$3$ is the repeating digit
$4.1666\ldots = 4.1\overline{6}$	$6$ is the repeating digit
$4.161616\ldots = 4.\overline{16}$	$16$ is the repeating block
$0.271271271\ldots = 0.\overline{271}$	$271$ is the repeating block

Example. Write $\tfrac{43}{22}$ as a decimal.

Divide $43$ by $22$ :

\begin{array}{r} \phantom{22\,\overline{\smash{)}\,}}1.95454\ldots \\ 22\,\overline{\smash{)}\,43.00000} \\ \phantom{22\,\overline{\smash{)}\,}}\underline{22}\phantom{.00000} \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{0}210 \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{0}\underline{198}\phantom{00} \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{00}120 \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{00}\underline{110}\phantom{0} \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{000}100 \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{000}\underline{88}\phantom{0} \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{0000}120 \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{0000}\underline{110}\phantom{0} \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{00000}100 \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{00000}\underline{88} \\ \phantom{22\,\overline{\smash{)}\,}}\phantom{000000}\ldots \end{array}

Notice that the differences of $120$ and $100$ repeat, so there is a repeat in the digits of the quotient — $54$ will repeat endlessly. The first decimal place in the quotient, $9$ , is not part of the pattern. So,

\frac{43}{22} = 1.9\overline{54}

Write as a decimal: $\tfrac{27}{11}$ . This is a repeating decimal — enter it rounded to 4 decimal places.

Write as a decimal: $\tfrac{51}{22}$ . This is a repeating decimal — enter it rounded to 4 decimal places.

It is useful to convert between fractions and decimals when we need to add or subtract numbers in different forms. To add a fraction and a decimal, for example, we would need to either convert the fraction to a decimal or the decimal to a fraction.

Example. Simplify: $\tfrac{7}{8} + 6.4$ .

Change $\tfrac{7}{8}$ to a decimal by dividing $7$ by $8$ , which gives $0.875$ . Then add:

0.875 + 6.4 = 7.275

Simplify: $\tfrac{3}{8} + 4.9$

Simplify: $5.7 + \tfrac{13}{20}$

Order decimals and fractions

In Decimals, we compared two decimals and determined which was larger. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals.

Example. Order $\tfrac{3}{8} \_\_ 0.4$ using $<$ or $>$ .

Convert $\tfrac{3}{8}$ to a decimal: $0.375$ . Compare $0.375$ to $0.4$ : $0.375 < 0.4$ . Rewrite with the original fraction:

\frac{3}{8} < 0.4

Order each of the following pairs of numbers, using $<$ or $>$ : $\tfrac{17}{20}$ __ $0.82$ . Enter the full inequality, e.g. $\tfrac{1}{2}<0.6$ .

Order each of the following pairs of numbers, using $<$ or $>$ : $\tfrac{3}{4}$ __ $0.785$ . Enter the full inequality, e.g. $\tfrac{1}{2}<0.6$ .

When ordering negative numbers, remember that larger numbers are to the right on the number line, and any positive number is greater than any negative number.

Example. Order $-0.5 \_\_ -\tfrac{3}{4}$ using $<$ or $>$ .

Convert $-\tfrac{3}{4}$ to a decimal: $-0.75$ . Compare $-0.5$ to $-0.75$ : $-0.5 > -0.75$ . Rewrite the inequality with the original fraction:

-0.5 > -\frac{3}{4}

Order each of the following pairs of numbers, using $<$ or $>$ : $-\tfrac{5}{8}$ __ $-0.58$ . Enter the full inequality, e.g. $-\tfrac{1}{2}<-0.4$ .

Order each of the following pairs of numbers, using $<$ or $>$ : $-0.53$ __ $-\tfrac{11}{20}$ . Enter the full inequality, e.g. $-\tfrac{1}{2}<-0.4$ .

Example. Write the numbers $\tfrac{13}{20}, 0.61, \tfrac{11}{16}$ in order from smallest to largest.

Convert the fractions to decimals: $0.65, 0.61, 0.6875$ . Write the smallest decimal number first, then the next larger, then the largest: $0.61, 0.65, 0.6875$ . Rewrite the list with the original fractions:

0.61, \frac{13}{20}, \frac{11}{16}

Write each set of numbers in order from smallest to largest: $\tfrac{7}{8}, \tfrac{4}{5}, 0.82$

Write each set of numbers in order from smallest to largest: $0.835, \tfrac{13}{16}, \tfrac{3}{4}$ . Enter as decimals separated by commas, e.g. $0.1, 0.2, 0.3$ .

Simplify expressions using the order of operations

The order of operations introduced in Use the Language of Algebra also applies to decimals. Do you remember what the phrase “Please excuse my dear Aunt Sally” stands for?

Example. Simplify the expressions: (a) $7(18.3 - 21.7)$ (b) $\tfrac{2}{3}(8.3 - 3.8)$ .

(a) Simplify inside parentheses: $7(-3.4)$ . Multiply: $-23.8$ .

(b) Simplify inside parentheses: $\tfrac{2}{3}(4.5)$ . Write $4.5$ as a fraction: $\tfrac{2}{3}\left(\tfrac{4.5}{1}\right)$ . Multiply: $\tfrac{9}{3}$ . Simplify: $3$ .

Simplify: $8(14.6 - 37.5)$

Simplify: $\left(\tfrac{3}{5}\right)(9.6 - 2.1)$

Example. Simplify each expression: (a) $6 \div 0.6 + (0.2)4 - (0.1)^2$ (b) $\left(\tfrac{1}{10}\right)^2 + (3.5)(0.9)$ .

(a) Simplify exponents: $6 \div 0.6 + (0.2)4 - 0.01$ . Divide: $10 + (0.2)4 - 0.01$ . Multiply: $10 + 0.8 - 0.01$ . Add: $10.8 - 0.01$ . Subtract: $10.79$ .

(b) Simplify exponents: $\tfrac{1}{100} + (3.5)(0.9)$ . Multiply: $\tfrac{1}{100} + 3.15$ . Convert $\tfrac{1}{100}$ to a decimal: $0.01 + 3.15$ . Add: $3.16$ .

Simplify: $9 \div 0.9 + (0.4)3 - (0.2)^2$

Simplify: $\left(\tfrac{1}{2}\right)^2 + (0.3)(4.2)$

Find the circumference and area of circles

The properties of circles have been studied for over $2{,}000$ years. All circles have exactly the same shape, but their sizes are affected by the length of the radius, a line segment from the center to any point on the circle. A line segment that passes through a circle’s center connecting two points on the circle is called a diameter. The diameter is twice as long as the radius.

The size of a circle can be measured in two ways. The distance around a circle is called its circumference. Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number. The value of this number is pi, symbolized by the Greek letter $\pi$ (pronounced “pie”). However, the exact value of $\pi$ cannot be calculated since the decimal never ends or repeats.

If we want the exact circumference or area of a circle, we leave the symbol $\pi$ in the answer. We can get an approximate answer by substituting $3.14$ as the value of $\pi$ . We use the symbol $\approx$ to show that the result is approximate, not exact.

Properties of circles. If $r$ is the length of the radius and $d$ is the length of the diameter of a circle, then:

the circumference is $2\pi r$ , so $C = 2\pi r$ ;

the area is $\pi r^2$ , so $A = \pi r^2$ .

Since the diameter is twice the radius, another way to find the circumference is to use the formula $C = \pi d$ .

Suppose we want to find the exact area of a circle of radius $10$ inches. To calculate the area, we would evaluate the formula for the area when $r = 10$ inches and leave the answer in terms of $\pi$ :

A = \pi r^2 = \pi(10^2) = \pi \cdot 100

We write $\pi$ after the $100$ . So the exact value of the area is $A = 100\pi$ square inches. To approximate the area, we would substitute $\pi \approx 3.14$ :

A = 100\pi \approx 100 \cdot 3.14 \approx 314 \text{ square inches}

Remember to use square units, such as square inches, when you calculate the area.

Example. A circle has radius $10$ centimeters. Approximate its (a) circumference and (b) area.

(a) $C = 2\pi r \approx 2(3.14)(10) \approx 62.8$ centimeters.

(b) $A = \pi r^2 \approx (3.14)(10)^2 \approx 314$ square centimeters.

A circle has radius $50$ inches. Approximate its circumference. Use $3.14$ for $\pi$ .

A circle has radius $50$ inches. Approximate its area. Use $3.14$ for $\pi$ .

A circle has radius $100$ feet. Approximate its circumference. Use $3.14$ for $\pi$ .

Example. A circle has radius $42.5$ centimeters. Approximate its (a) circumference and (b) area.

(a) $C = 2\pi r \approx 2(3.14)(42.5) \approx 266.9$ centimeters.

(b) $A = \pi r^2 \approx (3.14)(42.5)^2 \approx 5671.625$ square centimeters.

A circle has radius $51.8$ centimeters. Approximate its circumference. Use $3.14$ for $\pi$ . Round to two decimal places.

A circle has radius $26.4$ meters. Approximate its area. Use $3.14$ for $\pi$ . Round to four decimal places.

Approximate pi with a fraction

Convert the fraction $\tfrac{22}{7}$ to a decimal. If you use your calculator, the decimal number will fill up the display and show $3.14285714$ . But if we round that number to two decimal places, we get $3.14$ , the decimal approximation of $\pi$ . When we have a circle with radius given as a fraction, we can substitute $\tfrac{22}{7}$ for $\pi$ instead of $3.14$ . And, since $\tfrac{22}{7}$ is also an approximation of $\pi$ , we will use the $\approx$ symbol to show we have an approximate value.

Example. A circle has radius $\tfrac{14}{15}$ meter. Approximate its (a) circumference and (b) area.

(a) $C = 2\pi r \approx 2\left(\tfrac{22}{7}\right)\left(\tfrac{14}{15}\right) \approx \tfrac{88}{15}$ meters.

(b) $A = \pi r^2 \approx \left(\tfrac{22}{7}\right)\left(\tfrac{14}{15}\right)^2 \approx \tfrac{616}{225}$ square meters.

A circle has radius $\tfrac{5}{21}$ meters. Approximate its circumference. Use $\tfrac{22}{7}$ for $\pi$ .

A circle has radius $\tfrac{10}{33}$ inches. Approximate its area. Use $\tfrac{22}{7}$ for $\pi$ .

Key terms

repeating decimal — a decimal in which the last digit or group of digits repeats endlessly. radius — a line segment from the center of a circle to any point on the circle. diameter — a line segment that passes through a circle’s center, connecting two points on the circle; equal to twice the radius. circumference — the distance around a circle, $C = 2\pi r$ . pi ( $\pi$ ) — the ratio of a circle’s circumference to its diameter, approximately $3.14$ or $\tfrac{22}{7}$ .

This section is adapted from Prealgebra 2e, Section 5.3: Decimals and Fractions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the long-division layouts as typeset math and the circle diagram as an accessible inline graphic; omitted the Be Prepared quiz, Manipulative Mathematics callout, Media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

Decimal Operations Solve Equations with Decimals