5.3 Decimals and Fractions

Learning objectives

Learning objectives

• Convert fractions to decimals

• Order decimals and fractions

• Simplify expressions using the order of operations

• 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 $\frac{4}{5}$ can be written $4\mathrm{÷}5$ or $54.$ This means that we can convert a fraction to a decimal by treating it as a division problem.

Example 5.28

Solution

1. Rewrite the fraction as division. Set $\frac{3}{4}=3.00\div4$, keeping a decimal point in the dividend so the place values are explicit. The displayed long-division setup shows $3.00$ divided by $4$ with the quotient placed above the division line.
2. Perform the first decimal-place step. Bring down one zero so $30$ tenths are divided by $4$, obtaining $7$ tenths with remainder $2$ tenths. Thus the first digit after the decimal point is $7$.
3. Check the remaining places. Regroup the $2$ remaining tenths as $20$ hundredths, and divide $20\div4=5$ with no remainder, so the quotient terminates after two decimal places.
4. State the decimal result. The long division shown with $28$ above $20$, then $20$ above $0$, agrees with the result $\frac{3}{4}=0.75$.

Takeaway: Writing the dividend with a decimal point makes the place-value steps in the division visible and confirms the quotient.

Example 5.29

Solution

1. Separate the sign from the fraction. Rewrite $-7/2$ as $(-1)(7/2)$, so the magnitude can be found first and the negative sign applied afterward.
2. Divide to find the magnitude. Compute $7.0\div2.0=3.5$; the displayed long division first gives $7\div2=3$ with remainder $1$, then brings down the zero so $10\div2=5$ with no remainder.
3. Apply the negative sign. Since the quotient $3.5$ is positive, multiplying by $-1$ gives $-3.5$.
4. State the decimal result. The result is $-7/2=-3.5$, which differs from the positive quotient shown in the division picture because the original fraction carries a negative sign.

Takeaway: Find the positive quotient by long division and then use the external negative sign to determine the answer's direction.

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 $3.$

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

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 $1.\overset{-}{3}.$ The line above the $3$ tells you that the $3$ repeats endlessly. So $\mathrm{1.333…}=1.\overset{-}{3}$

For other decimals, two or more digits might repeat. Table 5.5 shows some more examples of repeating decimals.

Table

5.5

Example 5.30

Solution

1. Begin the long division. Dividing $43$ by $22$ gives $1$ with remainder $21$, so the whole-number part of the quotient is $1$.
2. Find the tenths digit. The remainder $21$ is ten times $210$, and $210\div 22=9$ with remainder $12$, so the first decimal digit is $9$.
3. Continue the quotient. The remainder $12$ is ten times $120$, and $120\div 22=5$ with remainder $10$, giving the next digit $5$.
4. Detect the repeating cycle. Multiplying the remainder $10$ by ten again gives $100$, and $100\div 22=4$ with remainder $12$, which repeats the earlier remainder.
5. Write the repeating decimal. The repeating block is $54$, so $\frac{43}{22}=1.9\overline{54}$.

Takeaway: Long division repeats after the remainder $12$ returns, producing the block $54$ in $1.9\overline{54}$.

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 5.31

Solution

1. Convert the fraction to a decimal. Divide $7$ by $8$ using long division. The displayed work shows $70÷8=8$ with remainder $6$, then $60÷8=7$ with remainder $4$, and finally $40÷8=5$ with remainder $0$, confirming that $7/8=0.875$.
2. Align the decimal places. Rewrite the addition as $0.875+6.400$, so the decimal points and the ones, tenths, and hundredths columns line up exactly.
3. Add the fractional places. The thousandths column gives $5+0=5$, the hundredths column gives $7+0=7$, and the tenths column gives $8+4=12$, so carry $1$ to the ones column.
4. Complete the addition. The ones column then gives $1+6=7$, producing $6.400+0.875=7.275$.
5. State the simplified result. The expression simplifies to $7/8+6.4=7.275$, agreeing with the quotient $0.875$ shown in the long-division picture.

Takeaway: Converting the fraction to a decimal and aligning place values makes the combined addition both visible and exact.

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 5.32

Solution

1. Convert the fraction to a decimal. Since $8$ has eight equal parts in one whole, $\frac{3}{8}=0.375$.
2. Use a common form. The comparison becomes $0.375<0.4$ after rewriting $0.4$ as $0.400$ if desired.
3. Compare the tenths digits. Both numbers have tenths digit $3$, so the hundredths digits must be compared next.
4. Compare the hundredths digits. The hundredths digit $7$ in $0.375$ is less than the hundredths digit $0$ in $0.400$.
5. Restore the original fraction. The ordered statement is therefore $\frac{3}{8}<0.4$.

Takeaway: Converting $\frac{3}{8}$ to $0.375$ shows that it is less than $0.4$ because $375$ is less than $400$.

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 5.33

Solution

1. Convert the negative fraction. The value $-\frac{3}{4}$ is equivalent to $-0.75$, so compare $-0.5$ with $-0.75$.
2. Compare their distances from zero. The first number is $0.5$ from zero, while the second is $0.75$ from zero.
3. Apply the negative-number rule. The negative number farther from zero is smaller, so $-0.75<-0.5$.
4. Check with a number line. The point for $-0.5$ lies to the right of the point for $-0.75$, confirming the ordering.
5. Restore the original fraction. The required inequality is $-0.5>-\frac{3}{4}$.

Takeaway: Because $-0.5$ is closer to zero than $-0.75$, it lies farther right and satisfies $-0.5>-\frac{3}{4}$.

Example 5.34

Solution

1. Convert the first fraction. Dividing $13$ by $20$ gives $\frac{13}{20}=0.65$, so the three values are now $0.65$, $0.61$, and $0.6875$.
2. Convert the second fraction. Dividing $11$ by $16$ gives $\frac{11}{16}=0.6875$, allowing every value to be compared by place value.
3. Find the smallest value. The number $0.61$ is less than both $0.65$ and $0.6875$, so it comes first.
4. Find the largest value. Comparing $650$ with $6875$ in thousandths shows that $0.6875$ is greater than $0.65$.
5. Restore the original forms. The order from smallest to largest is $0.61<\frac{13}{20}<\frac{11}{16}$.

Takeaway: The decimal equivalents $0.65$ and $0.6875$ establish the order $0.61<\frac{13}{20}<\frac{11}{16}$.

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 5.35

Solution

1. Simplify the first parenthesis. Subtracting inside $7(18.3-21.7)$ gives $18.3-21.7=-3.4$.
2. Multiply the first result. The expression becomes $7(-3.4)=-23.8$, since multiplying a positive number by a negative number gives a negative product.
3. Simplify the second parenthesis. Subtracting inside $\frac{2}{3}(8.3-3.8)$ gives $8.3-3.8=4.5$.
4. Write the decimal as a fraction. The second expression becomes $\frac{2}{3}\left(\frac{4.5}{1}\right)=\frac{2}{3}\left(\frac{9}{2}\right)$.
5. Multiply and simplify. Canceling common factors gives $\frac{2}{3}\cdot\frac{9}{2}=\frac{9}{3}=3$.

Takeaway: Evaluating each parenthesis first gives the two distinct results $-23.8$ and $3$ for the expressions.

Example 5.36

Solution

1. Square the first exponent. In $6\div0.6+(0.2)4-(0.1)^2$, the square $(0.1)^2$ equals $0.01$.
2. Divide and multiply. The expression becomes $10+(0.2)4-0.01$, and the product $(0.2)4$ equals $0.8$.
3. Finish the first expression. Adding and subtracting from left to right gives $10+0.8-0.01=10.79$.
4. Simplify the second expression. The square $(\frac{1}{10})^2$ is $\frac{1}{100}=0.01$, and the product $(3.5)(0.9)$ equals $3.15$.
5. Add the second expression. Its value is $0.01+3.15=3.16$, so the two results are $10.79$ and $3.16$.

Takeaway: Applying powers, then multiplication and division, and finally addition and subtraction produces $10.79$ and $3.16$.

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 Figure 5.6.

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 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.

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 $\mathrm{π.}$

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 $\pi\approx{}3.14.$

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

Example 5.37

Solution

1. Choose the circumference formula. For a radius of $10$ centimeters, the circumference is found with $C=2\pi r$.
2. Substitute the given values. Using $\pi\approx3.14$ and $r=10$ gives $C\approx2(3.14)(10)$ centimeters.
3. Multiply the factors. Since $2(3.14)=6.28$ and $6.28(10)=62.8$, the circumference is approximately $62.8$ centimeters.
4. Choose the area formula. The area is found with $A=\pi r^2$, so substitute the values to obtain $A\approx(3.14)(10)^2$.
5. Evaluate the area. Squaring the radius gives $100$, and $3.14(100)=314$, so the area is approximately $314$ square centimeters.

Takeaway: With radius $10$ and $\pi\approx3.14$, the circle's measurements are $62.8$ centimeters around and $314$ square centimeters inside.

Example 5.38

Solution

1. Choose the circumference formula. The formula is $C=2\pi r$, and the radius is $42.5$ centimeters.
2. Substitute the given values. Using $\pi\approx3.14$ gives $C\approx2(3.14)(42.5)$ centimeters.
3. Multiply for the circumference. The product $2(3.14)(42.5)=266.9$, so the circumference is approximately $266.9$ centimeters.
4. Substitute into the area formula. The formula $A=\pi r^2$ gives $A\approx(3.14)(42.5)^2$ square centimeters.
5. Square and multiply. Since $42.5^2=1806.25$ and $3.14(1806.25)=5671.625$, the area is approximately $5671.625$ square centimeters.

Takeaway: For radius $42.5$ centimeters, the approximate circumference is $266.9$ centimeters and the area is $5671.625$ 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 $\pi{}.$ 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 5.39

Solution

1. Choose the circumference formula. With $r=\frac{14}{15}$ meter, the circumference is $C=2\pi r$.
2. Substitute the fraction for pi. Using $\pi\approx\frac{22}{7}$ gives $C\approx2(\frac{22}{7})(\frac{14}{15})$ meters.
3. Cancel and multiply. Canceling $14$ with $7$ yields $C\approx\frac{2(22)}{15}=\frac{88}{15}$ meters.
4. Substitute into the area formula. The area is $A\approx(\frac{22}{7})(\frac{14}{15})^2$ square meters.
5. Simplify the area product. Canceling gives $A\approx\frac{22(14)}{15(15)}=\frac{616}{225}$ square meters.

Takeaway: Using $\pi\approx\frac{22}{7}$ and radius $\frac{14}{15}$ meter gives $\frac{88}{15}$ meters of circumference and $\frac{616}{225}$ square meters of area.

Section 5.3 Exercises

Section 5.3 Exercises

Convert Fractions to Decimals

In the following exercises, convert each fraction to a decimal.

In the following exercises, simplify the expression.

Order Decimals and Fractions

In the following exercises, order each pair of numbers, using $<$ or $\mathrm{>.}$

In the following exercises, write each set of numbers in order from least to greatest.

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

Mixed Practice

In the following exercises, simplify. Give the answer as a decimal.

Find the Circumference and Area of Circles

In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle. If measurements are given in fractions, leave answers in fraction form.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Write the fraction as a decimal: $-\frac{9}{4}.$

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

3. Write as a decimal: $\frac{27}{11}.$

1. Write as a decimal: $\frac{51}{22}.$

2. Order each of the following pairs of numbers, using $<$ or $\mathrm{>.}$

$\frac{17}{20}\mathrm{\_\_}0.82$

3. Order each of the following pairs of numbers, using $<$ or $\mathrm{>.}$

$\frac{3}{4}\mathrm{\_\_}0.785$

1. Order each of the following pairs of numbers, using $<$ or $\mathrm{>:}$

$-\frac{5}{8}\mathrm{\_\_}-0.58$

2. Order each of the following pairs of numbers, using $<$ or $\mathrm{>:}$

$-0.53\mathrm{\_\_}-\frac{11}{20}$

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

1. Write each set of numbers in order from smallest to largest: $0.835,\frac{13}{16},\frac{3}{4}.$

2. Simplify: ⓐ $\,8(14.6-37.5)$ ⓑ $\,\frac{3}{5}(9.6-2.1).$

3. Simplify: ⓐ $\,25(25.69-56.74)$ ⓑ $\,\frac{2}{7}(11.9-4.2).$

1. A circle has radius $50$ inches. Approximate its ⓐ circumference and ⓑ area.

2. A circle has radius $100$ feet. Approximate its ⓐ circumference and ⓑ area.

3. A circle has radius $51.8$ centimeters. Approximate its ⓐ circumference and ⓑ area.

1. A circle has radius $26.4$ meters. Approximate its ⓐ circumference and ⓑ area.

2. A circle has radius $\frac{5}{21}$ meters. Approximate its ⓐ circumference and ⓑ area.

3. A circle has radius $\frac{10}{33}$ inches. Approximate its ⓐ circumference and ⓑ area.

1. $-\frac{3}{8}$

2. $-\frac{310}{25}$

3. $\frac{5}{9}$

1. $\frac{15}{11}$

2. $\frac{15}{111}$

3. $\frac{1}{8}\mathrm{\_\_\_}0.8$

1. $\frac{2}{5}\mathrm{\_\_\_}\mathrm{0.25}$

2. $0.725\,\mathrm{\_\_\_}\,\frac{3}{4}$

3. $0.66\,\mathrm{\_\_\_}\,\frac{2}{3}$

1. $-0.75\mathrm{\_\_\_}-\frac{4}{5}$

2. $-\frac{3}{4}\mathrm{\_\_\_}-0.925$

3. $\frac{3}{5},\frac{9}{16},0.55$

1. $0.702,\frac{13}{20},\frac{5}{8}$

2. $-0.3,-\frac{1}{3},-\frac{7}{20}$

3. $-\frac{3}{4},-\frac{7}{9},-0.7$

1. $10(25.1-43.8)$

2. $-\frac{9}{10}\cdot\frac{8}{15}+0.25$

3. $3\frac{1}{4}-6.5$

1. $3.6(\frac{9}{8}-2.72)$

2. $\mathrm{radius}=\mathrm{5 in.}$

3. $\mathrm{radius}=\mathrm{9 ft.}$

1. $\mathrm{radius}=\mathrm{46 cm}$

2. $\mathrm{radius}=\mathrm{18.6 m}$

3. $\mathrm{radius}=\frac{7}{10}\,\mathrm{mile}$

1. $\mathrm{radius}=\frac{3}{8}\,\mathrm{yard}$

2. $\mathrm{diameter}=\frac{5}{6}\,m$

3. $\mathrm{diameter}=\frac{3}{4}\,m$

1. Kelly wants to buy a pair of boots that are on sale for $\frac{2}{3}$ of the original price. The original price of the boots is $\text{\textdollar 84.99}.$ What is the sale price of the shoes?

2. Is it easier for you to convert a decimal to a fraction or a fraction to a decimal? Explain.