Lesson 3: Decimals and Fractions

New Concept

Master the connection between fractions and decimals. You will learn to convert fractions to decimals, a key skill for ordering numbers, simplifying complex expressions, and calculating the circumference and area of circles.

What’s next

Get ready to apply this concept! Next, you’ll work through interactive examples of converting, ordering, and calculating with fractions and decimals.

Property

To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction. The fraction bar indicates division, so a fraction like $\frac{a}{b}$ can be written as $a \div b$.

Examples

• To write $\frac{3}{4}$ as a decimal, we divide 3 by 4. The calculation $3.00 \div 4$ gives us $0.75$. So, $\frac{3}{4} = 0.75$.

• To convert the improper fraction $\frac{9}{5}$ to a decimal, we divide 9 by 5. The calculation $9.0 \div 5$ results in $1.8$. So, $\frac{9}{5} = 1.8$.

Property

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

Examples

• To convert $\frac{2}{3}$ to a decimal, dividing 2 by 3 gives $0.666...$. The digit 6 repeats endlessly, so we write it as $0.\overline{6}$.

• For the fraction $\frac{5}{11}$, dividing 5 by 11 gives $0.454545...$. The block of digits 45 repeats, so we write it as $0.\overline{45}$.

Property

To compare a decimal to a fraction, first convert the fraction to a decimal. Then, compare the two decimal numbers. When ordering negative numbers, remember that the number closer to zero on the number line is the larger number.

Examples

• To order $\frac{3}{8}$ and $0.4$, first convert $\frac{3}{8}$ to a decimal, which is $0.375$. Since $0.375 < 0.4$, we know that $\frac{3}{8} < 0.4$.

• To order $-\frac{1}{2}$ and $-0.55$, convert $-\frac{1}{2}$ to $-0.5$. On a number line, $-0.5$ is to the right of $-0.55$, so $-0.5 > -0.55$. Therefore, $-\frac{1}{2} > -0.55$.

Property

The order of operations (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) applies to expressions with decimals and fractions.

Examples

• To simplify $8(4.5 - \frac{1}{2})$, first solve the parenthesis: $4.5 - 0.5 = 4$. Then multiply: $8(4) = 32$.

• In the expression $(\frac{1}{5})^2 + (0.5)(1.2)$, first handle the exponent: $(\frac{1}{5})^2 = 0.2^2 = 0.04$. Then multiply: $(0.5)(1.2) = 0.6$. Finally, add: $0.04 + 0.6 = 0.64$.

Property

For a circle with radius $r$ and diameter $d$:
The circumference is $C = 2\pi r$ or $C = \pi d$.
The area is $A = \pi r^2$.
The number $\pi$ (pi) is a constant used for all circles. For exact answers, leave the symbol $\pi$ in the solution. For approximate answers, use $\pi \approx 3.14$ or $\pi \approx \frac{22}{7}$.

Examples

• For a circle with a radius of 5 cm, the approximate circumference is $C \approx 2(3.14)(5) = 31.4$ cm, and the approximate area is $A \approx 3.14(5^2) = 78.5$ square cm.

• A circle has a diameter of 20 feet, which means its radius is 10 feet. The exact area is $A = \pi (10^2) = 100\pi$ square feet.