Decimals and Fractions

5.3 Decimals and Fractions

The topics covered in this section are:

  1. Convert fractions to decimals
  2. Order decimals and fractions
  3. Simplify expressions using the order of operations
  4. Find the circumference and area of circles

5.3.1 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 $\frac{4}{5}$ can be written $4 \div 5$ or 5)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 1

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

Solution

A fraction bar means division, so we can write the fraction $\frac{3}{4}$ using division.A division problem is shown. 3 is on the inside of the division sign and 4 is on the outside.
Divide.A division problem is shown. 3.00 is on the inside of the division sign and 4 is on the outside. Below the 3.00 is a 28 with a line below it. Below the line is a 20. Below the 20 is another 20 with a line below it. Below the line is a 0. Above the division sign is 0.75.
So the fraction $\frac{3}{4}$ is equal to $0.75$.

Example 2

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

Solution

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.$- \frac{7}{2}$
Divide $7$ by $2$.A division problem is shown. 7.0 is on the inside of the division sign and 2 is on the outside. Below the 7 is a 6 with a line below it. Below the line is a 10. Below the 10 is another 10 with a line below it. Below the line is a 0. 3.5 is written above the division sign.
So, $- \frac{7}{2} = -3.5$.

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 $\frac{4}{3}$ to a decimal. First, notice that $\frac{4}{3}$ is an improper fraction. Its value is greater than $1$. The equivalent decimal will also be greater than $1$.

We divide $4$ by $3$.

A division problem is shown. 4.000 is on the inside of the division sign and 3 is on the outside. Below the 4 is a 3 with a line below it. Below the line is a 10. Below the 10 is a 9 with a line below it. Below the line is another 10, followed by another 9 with a line, followed by another 10, followed by another 9 with a line, followed by a 1. Above the division sign is 1.333...

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 $1.333…$ is called a repeating decimal. Remember that the “$…$” means that the pattern repeats.

REPEATING DECIMAL

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 $1.333…$ we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal $1.333…$ is written $1. \stackrel{-}{3}$. The line above the $3$ tells you that the $3$ repeats endlessly. So $1.333… = 1.\stackrel{-}{3} $.

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

$1.333…=1. \stackrel{-}{3}$$3$ is the repeating digit
$4.1666…=4. \stackrel{-}{6}$$6$ is the repeating digit
$4.161616…=4. \stackrel{—–}{16}$$16$ is the repeating block
$0.271271271…=0. \stackrel{——}{271}$$271$ is the repeating block

Example 3

Write $\frac{43}{22}$ as a decimal.

Solution

Divide $43$ by $22$.

A division problem is shown. 43.00000 is on the inside of the division sign and 22 is on the outside. Below the 43 is a 22 with a line below it. Below the line is a 210 with a 198 with a line below it. Below the line is a 120 with 110 and a line below it. Below the line is 100 with 88 and a line below it. Below the line is 120 with 110 and a line below it. Below the line is 100 with 88 and a line below it. Below the line is an ellipses. There are arrows pointing to the 120s saying 120 repeats. There are arrows pointing to the 100s saying 100 repeats. There are arrows pointing to the 88s saying, in red, “The pattern repeats, so the numbers in the quotient will repeat as well.” The quotient is shown above the division sign. It is 1.95454.

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,

$\frac{43}{22} = 1.9 \stackrel{—–}{54}$

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 4

Simplify: $\frac{7}{8} + 6.4$.

Solution

$\frac{7}{8} + 6.4$
Change $\frac{7}{8}$ to a decimal..$0.875+6.4$
Add.$7.275$

5.3.2 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 5

Order $\frac{3}{8}$ ____ $0.4$ using $<$ or $>$.

Solution

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

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 6

Order $-0.5$ ____ $- \frac{3}{4}$ using $<$ or $>$.

Solution

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

Example 7

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

Solution

$\frac{13}{20} , 0.61, \frac{11}{16}$
Convert the fractions to decimals.$0.65, 0.61, 0.6875$
Write the smallest decimal number first.$0.61$, _____, _____
Write the next larger decimal number in the middle place.$0.61, 0.65$, _____
Write the last decimal number (the larger) in the third place.$0.61, 0.65, 0.6875$
Rewrite the list with the original fractions.$0.61, \frac{13}{20} , \frac{11}{16}$

5.3.3 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 8

Simplify the expressions:

  1. $7(18.3-21.7)$
  2. $\frac{2}{3} (8.3-3.8)$
Solution

Part 1.
$7(18.3-21.7)$
Simplify inside parentheses.$7(-3.4)$
Multiply.$-23.8$
Part 2.
$\frac{2}{3} (8.3-3.8)$
Simplify inside parentheses.$\frac{2}{3} (4.5)$
Write $4.5$ as a fraction.$\frac{2}{3} ( \frac{4.5}{1} )$
Multiply.$\frac{9}{3}$
Simplify.$3$

Example 8

Simplify the expressions:

  1. $6 \div 0.6 + (0.2)4-(0.1)^{2}$
  2. $( \frac{1}{10} )^{2} + (3.5)(0.9)$
Solution

Part 1.
$6 \div 0.6 + (0.2)4 – (0.1)^{2}$
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.$10.79$
Part 2.
$( \frac{1}{10} )^{2} + (3.5)(0.9)$
Simplify exponents.$\frac{1}{100} + (3.5)(0.9)$
Multiply.$\frac{1}{100} + 3.15$
Convert $\frac{1}{100}$ to a decimal.$0.01 + 3.15$
Add.$3.16$

5.3.4 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. See the figure below.

The size of a circle can be measured in two ways. The distance around a circle is called its circumference.

A circle is shown. A dotted line running through the widest portion of the circle is labeled as a diameter. A dotted line from the center of the circle to a point on the circle is labeled as a radius. Along the edge of the circle is the 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 Greek letter $\pi$ (pronounced pie). However, the exact value of $\pi$ cannot be calculated since the decimal never ends or repeats (we will learn more about numbers like this in The Properties of Real Numbers.)

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

A circle is shown. A line runs through the widest portion of the circle. There is a red dot at the center of the circle. The half of the line from the center of the circle to a point on the right of the circle is labeled with an r. The half of the line from the center of the circle to a point on the left of the circle is also labeled with an r. The two sections labeled r have a brace drawn underneath showing that the entire segment is labeled d.

$r$ is the length of the radius.

$d$ is the length of the diameter.

The circumference is $2 \pi r$.$C=2 \pi r$
The area is $\pi r^{2}$$A= \pi r^{2}$

Since the diameter is twice the radius, another way to find the circumference is to use the formula $C=2 \pi r$.

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 $\pi$.

$A= \pi r^{2}$

$A= \pi (10^{2})$

$A= \pi \cdot 100$

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 $\pi \approx 3.14$.

$A = 100 \pi$

$\approx 100 \cdot 3.14$

$\approx 314$ square inches

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

Example 9

A circle has radius $10$ centimeters. Approximate its…

  1. circumference
  2. area
Solution

Part 1. Find the circumference when $r=10$.
Write the formula for circumference.$C=2 \pi r$
Substitute $3.14$ for $\pi$ and $10$ for $r$.$C \approx 2(3.14)(10)$
Multiply.$C \approx 62.8$ centimeters
Part b. Find the area when $r=10$.
Write the formula for area.$A= \pi r^{2}$
Substitute $3.14$ for $\pi$ and $10$ for $r$.$A \approx (3.14)(10)^{2}$
Multiply.$A \approx 314$ square centimeters

Example 10

A circle has radius $42.5$ centimeters. Approximate its…

  1. circumference
  2. area
Solution

Part 1. Find the circumference when $r=42.5$.
Write the formula for circumference.$C=2 \pi r$
Substitute $3.14$ for $\pi$ and $42.5$ for $r$.$C \approx 2(3.14)(42.5)$
Multiply.$C \approx 266.9$ centimeters
Part b. Find the area when $r=42.5$.
Write the formula for area.$A= \pi r^{2}$
Substitute $3.14$ for $\pi$ and $42.5$ for $r$.$A \approx (3.14)(42.5)^{2}$
Multiply.$A \approx 5671.625$ square centimeters

Approximate $\pi$ with a Fraction

Convert the fraction $\frac{22}{7}$ 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 $\pi$π. When we have a circle with radius given as a fraction, we can substitute $\frac{22}{7}$ for $\pi$ instead of $3.14$. And, since $\frac{22}{7}$ is also an approximation of $\pi$, we will use the $\approx$ symbol to show we have an approximate value.

Example 11

A circle has radius $\frac{14}{15}$ meter. Approximate its…

  1. circumference
  2. area
Solution

Part 1. Find the circumference when $r= \frac{14}{15}$.
Write the formula for circumference.$C=2 \pi r$
Substitute $\frac{22}{7}$ for $\pi$ and $\frac{14}{15}$ for $r$.$C \approx 2( \frac{22}{7} )( \frac{14}{15} )$
Multiply.$C \approx \frac{88}{15}$ meters
Part b. Find the area when $r= \frac{14}{15} $.
Write the formula for area.$A= \pi r^{2}$
Substitute $ \frac{22}{7} $ for $\pi$ and $\frac{14}{15}$ for $r$.$A \approx ( \frac{22}{7} )( \frac{14}{15})^{2}$
Multiply.$A \approx \frac{616}{225}$ square meters
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