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Decimals and Fractions

Convert fractions to decimals

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 \(54.\) This means that we can convert a fraction to a decimal by treating it as a division problem.

Example

Try it.

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.
Divide.
So the fraction \(\frac{3}{4}\) is equal to \(0.75.\)
Example

Try it.

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 \(2.\)
So, \(-\frac{7}{2}=-3.5.\)

Condensed — the full section is in OpenStax Prealgebra 2e.

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

Try it.

Order \(\frac{3}{8}__0.4\) using \(<\) or \(\text{>.}\)

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

Try it.

Order \(-0.5___-\frac{3}{4}\) using \(<\) or \(\text{>.}\)

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

Try it.

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.\(0.61,\frac{13}{20},\frac{11}{16}\)

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

Try it.

Simplify the expressions:

  1. ⓐ \(\ 7(18.3-21.7)\)
  2. ⓑ \(\ \frac{2}{3}(8.3-3.8)\)
Solution
ⓐ
\(7(18.3-21.7)\)
Simplify inside parentheses.\(7(-3.4)\)
Multiply.\(-23.8\)
ⓑ
\(\frac{2}{3}(8.3-3.8)\)
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

Try it.

Simplify each expression:

  1. ⓐ \(\ 6\div 0.6+(0.2)4-{(0.1)}^{2}\)
  2. ⓑ \(\ {(\frac{1}{10})}^{2}+(3.5)(0.9)\)
Solution
ⓐ
\(6\div 0.6+(0.2)4-{(0.1)}^{2}\)
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.\(10.79\)
ⓑ
\({(\frac{1}{10})}^{2}+(3.5)(0.9)\)
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.\(3.16\)

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 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 \(\text{\pi }.\) 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 \(C=\ \text{\pi }\text{d}.\)

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 \(\text{\pi .}\)

\[\begin{array}{l} \\ A=\ \text{\pi }{\text{r}}^{2} \\ A=\ \text{\pi }\text{(}{10}^{2}\text{)} \\ A=\pi \cdot 100\end{array}\]

We write \(\pi\) after the \(100.\) So the exact value of the area is \(A=100\text{\pi }\) square inches.

\[\begin{array}{lll}A & = & 100\text{\pi } \\ \\ & \approx & 100\cdot 3.14 \\ & \approx & 314\ \text{square inches}\end{array}\]

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

  • 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.
  • Properties of Circles \(r\) 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 \(\pi {r}^{2}\). \(\ A=\pi {r}^{2}\)

Decimals and Fractions

Convert Fractions to Decimals

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

Try it.

\(\frac{2}{5}\)

Solution

0.4

Try it.

\(\frac{4}{5}\)

Try it.

\(-\frac{3}{8}\)

Solution

−0.375

Try it.

\(-\frac{5}{8}\)

Try it.

\(\frac{17}{20}\)

Solution

0.85

Try it.

\(\frac{13}{20}\)

Try it.

\(\frac{11}{4}\)

Solution

2.75

Try it.

\(\frac{17}{4}\)

Try it.

\(-\frac{310}{25}\)

Solution

−12.4

Try it.

\(-\frac{284}{25}\)

Try it.

\(\frac{5}{9}\)

Solution

\(0.\overset{-}{5}\)

Try it.

\(\frac{2}{9}\)

Try it.

\(\frac{15}{11}\)

Solution

\(1.\overset{\text{—}}{36}\)

Try it.

\(\frac{18}{11}\)

Try it.

\(\frac{15}{111}\)

Solution

\(0.\overset{\text{—}}{135}\)

Try it.

\(\frac{25}{111}\)

In the following exercises, simplify the expression.

Try it.

\(\frac{1}{2}+6.5\)

Solution

7

Try it.

\(\frac{1}{4}+10.75\)

Try it.

\(2.4+\frac{5}{8}\)

Solution

3.025

Try it.

\(3.9+\frac{9}{20}\)

Try it.

\(9.73+\frac{17}{20}\)

Solution

10.58

Try it.

\(6.29+\frac{21}{40}\)

Order Decimals and Fractions

In the following exercises, order each pair of numbers, using \(<\) or \(\text{>.}\)

Try it.

\(\frac{1}{8}___0.8\)

Solution

<

Try it.

\(\frac{1}{4}___0.4\)

Try it.

\(\frac{2}{5}___0.25\)

Solution

>

Try it.

\(\frac{3}{5}___0.35\)

Try it.

\(0.725\ ___\ \frac{3}{4}\)

Solution

<

Try it.

\(0.92\ ___\ \frac{7}{8}\)

Try it.

\(0.66\ ___\ \frac{2}{3}\)

Solution

<

Try it.

\(0.83\ ___\ \frac{5}{6}\)

Try it.

\(-0.75___-\frac{4}{5}\)

Solution

>

Try it.

\(-0.44___-\frac{9}{20}\)

Try it.

\(-\frac{3}{4}___-0.925\)

Solution

>

Try it.

\(-\frac{2}{3}___-0.632\)

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

Try it.

\(\frac{3}{5},\frac{9}{16},0.55\)

Solution

\(0.55,\frac{9}{16},\frac{3}{5}\)

Try it.

\(\frac{3}{8},\frac{7}{20},0.36\)

Try it.

\(0.702,\frac{13}{20},\frac{5}{8}\)

Solution

\(\frac{5}{8},\frac{13}{20},0.702\)

Try it.

\(0.15,\frac{3}{16},\frac{1}{5}\)

Try it.

\(-0.3,-\frac{1}{3},-\frac{7}{20}\)

Solution

\(-\frac{7}{20},-\frac{1}{3},-0.3\)

Try it.

\(-0.2,-\frac{3}{20},-\frac{1}{6}\)

Try it.

\(-\frac{3}{4},-\frac{7}{9},-0.7\)

Solution

\(-\frac{7}{9},-\frac{3}{4},-0.7\)

Try it.

\(-\frac{8}{9},-\frac{4}{5},-0.9\)

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

Try it.

\(10(25.1-43.8)\)

Solution

−187

Try it.

\(30(18.1-32.5)\)

Try it.

\(62(9.75-4.99)\)

Solution

295.12

Try it.

\(42(8.45-5.97)\)

Try it.

\(\frac{3}{4}(12.4-4.2)\)

Solution

6.15

Try it.

\(\frac{4}{5}(8.6+3.9)\)

Try it.

\(\frac{5}{12}(30.58+17.9)\)

Solution

20.2

Try it.

\(\frac{9}{16}(21.96-9.8)\)

Try it.

\(10\div 0.1+(1.8)4-{(0.3)}^{2}\)

Solution

107.11

Try it.

\(5\div 0.5+(3.9)6-{(0.7)}^{2}\)

Try it.

\((37.1+52.7)\div (12.5\div 62.5)\)

Solution

449

Try it.

\((11.4+16.2)\div (18\div 60)\)

Try it.

\({(\frac{1}{5})}^{2}+(1.4)(6.5)\)

Solution

9.14

Try it.

\({(\frac{1}{2})}^{2}+(2.1)(8.3)\)

Try it.

\(-\frac{9}{10}\cdot \frac{8}{15}+0.25\)

Solution

−0.23

Try it.

\(-\frac{3}{8}\cdot \frac{14}{15}+0.72\)

Mixed Practice

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

Try it.

\(3\frac{1}{4}-6.5\)

Solution

−3.25

Try it.

\(5\frac{2}{5}-8.75\)

Try it.

\(10.86\div \frac{2}{3}\)

Solution

16.29

Try it.

\(5.79\div \frac{3}{4}\)

Try it.

\(\frac{7}{8}(103.48)+1\frac{1}{2}(361)\)

Solution

632.045

Try it.

\(\frac{5}{16}(117.6)+2\frac{1}{3}(699)\)

Try it.

\(3.6(\frac{9}{8}-2.72)\)

Solution

−5.742

Try it.

\(5.1(\frac{12}{5}-3.91)\)

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.

Try it.

\(\text{radius}=\text{5 in.}\)

Solution

  1. ⓐ 31.4 in
  2. ⓑ 78.5 sq.in.

Try it.

\(\text{radius}=\text{20 in.}\)

Try it.

\(\text{radius}=\text{9 ft.}\)

Solution

  1. ⓐ 56.52.ft.
  2. ⓑ 254.34 sq.ft.

Try it.

\(\text{radius}=\text{4 ft.}\)

Try it.

\(\text{radius}=\text{46 cm}\)

Solution

  1. ⓐ 288.88 cm
  2. ⓑ 6644.24 sq.cm

Try it.

\(\text{radius}=\text{38 cm}\)

Try it.

\(\text{radius}=\text{18.6 m}\)

Solution

  1. ⓐ 116.808 m
  2. ⓑ 1086.3144 sq.m

Try it.

\(\text{radius}=\text{57.3 m}\)

Try it.

\(\text{radius}=\frac{7}{10}\ \text{mile}\)

Solution

  1. ⓐ \(\ \frac{22}{5}\ \text{mile}\\)
  2. ⓑ \(\ \frac{77}{50}\ \text{sq.mile}\)

Try it.

\(\text{radius}=\frac{7}{11}\ \text{mile}\)

Try it.

\(\text{radius}=\frac{3}{8}\ \text{yard}\)

Solution

  1. ⓐ \(\ \frac{33}{14}\ \text{yard}\\)
  2. ⓑ \(\ \frac{99}{224}\ \text{sq.yard}\)

Try it.

\(\text{radius}=\frac{5}{12}\ \text{yard}\)

Try it.

\(\text{diameter}=\frac{5}{6}\ \text{m}\)

Solution

  1. ⓐ \(\ \frac{55}{21}\ \text{m}\\)
  2. ⓑ \(\ \frac{275}{504}\ \text{sq.m}\)

Try it.

\(\text{diameter}=\frac{3}{4}\ \text{m}\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Divide: \(0.24\div 8.\)
    If you missed this problem, review .

    答えを明らかにしろ

    \(0.03\)

  2. Order \(0.64__0.6\) using \(<\) or \(\text{>.}\)
    If you missed this problem, review .

    答えを明らかにしろ

    \(>\)

  3. Order \(-0.2__-0.1\) using \(<\) or \(\text{>.}\)
    If you missed this problem, review .

    答えを明らかにしろ

    \(<\)

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

    答えを明らかにしろ
    A fraction bar means division, so we can write the fraction \(\frac{3}{4}\) using division.
    Divide.
    So the fraction \(\frac{3}{4}\) is equal to \(0.75.\)

  5. Write the fraction as a decimal: \(\frac{1}{4}.\)

    答えを明らかにしろ

    0.25

  6. Write the fraction as a decimal: \(\frac{3}{8}.\)

    答えを明らかにしろ

    0.375

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

    答えを明らかにしろ
    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 \(2.\)
    So, \(-\frac{7}{2}=-3.5.\)

  8. Write the fraction as a decimal: \(-\frac{9}{4}.\)

    答えを明らかにしろ

    −2.25

  9. Write the fraction as a decimal: \(-\frac{11}{2}.\)

    答えを明らかにしろ

    −5.5

  10. Write \(\frac{43}{22}\) as a decimal.

    答えを明らかにしろ

    Divide \(43\) by \(22.\)

    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,

    \[\frac{43}{22}=1.9\overset{\text{—}}{54}\]

  11. Write as a decimal: \(\frac{27}{11}.\)

    答えを明らかにしろ

    \(2.\overset{\text{—}}{45}\)

  12. Write as a decimal: \(\frac{51}{22}.\)

    答えを明らかにしろ

    \(2.3\overset{\text{—}}{18}\)

  13. Simplify: \(\frac{7}{8}+6.4.\)

    答えを明らかにしろ
    \(\frac{7}{8}+6.4\)
    Change \(\frac{7}{8}\) to a decimal.\(0.875+6.4\)
    Add.\(7.275\)

  14. Simplify: \(\frac{3}{8}+4.9.\)

    答えを明らかにしろ

    5.275

  15. Simplify: \(5.7+\frac{13}{20}.\)

    答えを明らかにしろ

    6.35

  16. Order \(\frac{3}{8}__0.4\) using \(<\) or \(\text{>.}\)

    答えを明らかにしろ
    \(\frac{3}{8}__0.4\)
    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.\(\frac{3}{8}<0.4\)

  17. Order each of the following pairs of numbers, using \(<\) or \(\text{>.}\)

    \(\frac{17}{20}__0.82\)

    答えを明らかにしろ

    >

  18. Order each of the following pairs of numbers, using \(<\) or \(\text{>.}\)

    \(\frac{3}{4}__0.785\)

    答えを明らかにしろ

    <

  19. Order \(-0.5___-\frac{3}{4}\) using \(<\) or \(\text{>.}\)

    答えを明らかにしろ
    \(-0.5___-\frac{3}{4}\)
    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.\(-0.5>-\frac{3}{4}\)

  20. Order each of the following pairs of numbers, using \(<\) or \(\text{>:}\)

    \(-\frac{5}{8}__-0.58\)

    答えを明らかにしろ

    <

  21. Order each of the following pairs of numbers, using \(<\) or \(\text{>:}\)

    \(-0.53__-\frac{11}{20}\)

    答えを明らかにしろ

    >

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

    答えを明らかにしろ
    \(\frac{13}{20},0.61,\frac{11}{16}\)
    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.\(0.61,\frac{13}{20},\frac{11}{16}\)

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

    答えを明らかにしろ

    \(\frac{4}{5},0.82,\frac{7}{8}\)

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

    答えを明らかにしろ

    \(\frac{3}{4},\frac{13}{16},0.835\)

  25. Simplify the expressions:

    1. ⓐ \(\ 7(18.3-21.7)\)
    2. ⓑ \(\ \frac{2}{3}(8.3-3.8)\)
    答えを明らかにしろ
    ⓐ
    \(7(18.3-21.7)\)
    Simplify inside parentheses.\(7(-3.4)\)
    Multiply.\(-23.8\)
    ⓑ
    \(\frac{2}{3}(8.3-3.8)\)
    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.\(3\)

  26. Simplify: ⓐ \(\ 8(14.6-37.5)\) ⓑ \(\ \frac{3}{5}(9.6-2.1)\text{.}\)

    答えを明らかにしろ

    1. ⓐ −183.2
    2. ⓑ 4.5

  27. Simplify: ⓐ \(\ 25(25.69-56.74)\) ⓑ \(\ \frac{2}{7}(11.9-4.2)\text{.}\)

    答えを明らかにしろ

    1. ⓐ −776.25
    2. ⓑ 2.2

  28. Simplify each expression:

    1. ⓐ \(\ 6\div 0.6+(0.2)4-{(0.1)}^{2}\)
    2. ⓑ \(\ {(\frac{1}{10})}^{2}+(3.5)(0.9)\)
    答えを明らかにしろ
    ⓐ
    \(6\div 0.6+(0.2)4-{(0.1)}^{2}\)
    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.\(10.79\)
    ⓑ
    \({(\frac{1}{10})}^{2}+(3.5)(0.9)\)
    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.\(3.16\)

  29. Simplify: \(9\div 0.9+(0.4)3-{(0.2)}^{2}.\)

    答えを明らかにしろ

    11.16

  30. Simplify: \({(\frac{1}{2})}^{2}+(0.3)(4.2)\text{.}\)

    答えを明らかにしろ

    1.51

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

    答えを明らかにしろ
    ⓐ Find the circumference when \(r=10.\)
    Write the formula for circumference.\(C=2\text{\pi }\text{r}\)
    Substitute 3.14 for \(\ \pi\) and 10 for ,\(r\).\(C\approx 2(3.14)(10)\)
    Multiply.\(C\approx 62.8\ \text{centimeters}\)
    ⓑ Find the area when \(r=10.\)
    Write the formula for area.\(A=\ \text{\pi }{\text{r}}^{2}\)
    Substitute 3.14 for \(\pi\) and 10 for \(r\).\(A\approx (3.14){\text{(}10\text{)}}^{2}\)
    Multiply.\(A\approx 314\ \text{square centimeters}\)

  32. A circle has radius \(50\) inches. Approximate its ⓐ circumference and ⓑ area.

    答えを明らかにしろ

    1. ⓐ 314 in.
    2. ⓑ 7850 sq. in.

  33. A circle has radius \(100\) feet. Approximate its ⓐ circumference and ⓑ area.

    答えを明らかにしろ

    1. ⓐ 628 ft.
    2. ⓑ 31,400 sq. ft.

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

    答えを明らかにしろ
    ⓐ Find the circumference when \(r=42.5.\)
    Write the formula for circumference.\(C=2\text{\pi }\text{r}\)
    Substitute 3.14 for \(\pi\) and 42.5 for \(r\)\(C\approx 2(3.14)(42.5)\)
    Multiply.\(C\approx 266.9\ \text{centimeters}\)
    ⓑ Find the area when \(r=42.5\).
    Write the formula for area.\(A=\ \text{\pi }{\text{r}}^{2}\)
    Substitute 3.14 for \(\pi\) and 42.5 for \(r\).\(A\approx (3.14){\text{(}42.5\text{)}}^{2}\)
    Multiply.\(A\approx 5671.625\ \text{square centimeters}\)

  35. A circle has radius \(51.8\) centimeters. Approximate its ⓐ circumference and ⓑ area.

    答えを明らかにしろ

    1. ⓐ 325.304 cm
    2. ⓑ 8425.3736 sq. cm

  36. A circle has radius \(26.4\) meters. Approximate its ⓐ circumference and ⓑ area.

    答えを明らかにしろ

    1. ⓐ 165.792 m
    2. ⓑ 2188.4544 sq. m

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

    答えを明らかにしろ
    ⓐ Find the circumference when \(r=\frac{14}{15}.\)
    Write the formula for circumference.\(C=2\text{\pi }\text{r}\)
    Substitute \(\frac{22}{7}\) for \(\pi\) and \(\frac{14}{15}\) for \(r\).\(C\approx 2(\frac{22}{7})(\frac{14}{15})\)
    Multiply.\(C\approx \frac{88}{15}\ \text{meters}\)
    ⓑ Find the area when \(r=\frac{14}{15}.\)
    Write the formula for area.\(A=\ \text{\pi }{\text{r}}^{2}\)
    Substitute \(\frac{22}{7}\) for \(\pi\) and \(\frac{14}{15}\) for \(r\).\(A\approx (\frac{22}{7}){\text{(}\frac{14}{15}\text{)}}^{2}\)
    Multiply.\(A\approx \frac{616}{225}\ \text{square meters}\)

  38. A circle has radius \(\frac{5}{21}\) meters. Approximate its ⓐ circumference and ⓑ area.

    答えを明らかにしろ

    1. ⓐ \(\ \frac{220}{147}\ \text{m}\)
    2. ⓑ \(\ \frac{550}{3087}\ \text{sq. m}\)

  39. A circle has radius \(\frac{10}{33}\) inches. Approximate its ⓐ circumference and ⓑ area.

    答えを明らかにしろ

    1. ⓐ \(\ \frac{40}{21}\ \text{in.}\)
    2. ⓑ \(\ \frac{200}{693}\ \text{sq.in.}\)

  40. \(\frac{2}{5}\)

    答えを明らかにしろ

    0.4

Symbols used here

\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\approx
approximately equal
Equal to the precision shown, not exactly.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
a \bmod n
remainder
What is left after dividing a by n.
\%
per cent
Per hundred: 15% = 15/100.
a : b,\ \frac{a}{b}
ratio, fraction
a for every b; a divided by b.

How to: Decimals and Fractions

  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

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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Order of operationsFractionsPercentagesPowers and roots