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Visualize Fractions

Understand the meaning of fractions

Understand the Meaning of Fractions

Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts.

In math, we write \(\frac{1}{2}\) to mean one out of two parts.

On Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has \(\frac{1}{4}\) of the pizza.

On Wednesday, the family invites some friends over for a pizza dinner. There are a total of \(12\) people. If they share the pizza equally, each person would get \(\frac{1}{12}\) of the pizza.

A fraction is a way to represent parts of a whole. The denominator \(b\) represents the number of equal parts the whole has been divided into, and the numerator \(a\) represents how many parts are included. The denominator, \(b,\) cannot equal zero because division by zero is undefined.

In , the circle has been divided into three parts of equal size. Each part represents \(\frac{1}{3}\) of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.

What does the fraction \(\frac{2}{3}\) represent? The fraction \(\frac{2}{3}\) means two of three equal parts.

Example

Try it.

Name the fraction of the shape that is shaded in each of the figures.

Solution

We need to ask two questions. First, how many equal parts are there? This will be the denominator. Second, of these equal parts, how many are shaded? This will be the numerator.

ⓐ
\(\begin{array}{llll}\text{How many equal parts are there?} & & & \text{There are eight equal parts}\text{.} \\ \text{How many are shaded?} & & & \text{Five parts are shaded}\text{.}\end{array}\)

Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is \(\frac{5}{8}.\)

ⓑ
\(\begin{array}{llll}\text{How many equal parts are there?} & & & \text{There are nine equal parts}\text{.} \\ \text{How many are shaded?} & & & \text{Two parts are shaded}\text{.}\end{array}\)

Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is \(\frac{2}{9}.\)

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

Model Improper Fractions and Mixed Numbers

In (b), you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, \(\frac{1}{5},\) so altogether you had eight fifths, which we can write as \(\frac{8}{5}.\) The fraction \(\frac{8}{5}\) is one whole, \(1,\) plus three fifths, \(\frac{3}{5},\) or \(1\frac{3}{5},\) which is read as one and three-fifths.

The number \(1\frac{3}{5}\) is called a mixed number. A mixed number consists of a whole number and a fraction.

Fractions such as \(\frac{5}{4},\frac{3}{2},\frac{5}{5},\) and \(\frac{7}{3}\) are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as \(\frac{1}{2},\frac{3}{7},\) and \(\frac{11}{18}\) are proper fractions.

Example

Try it.

Name the improper fraction modeled. Then write the improper fraction as a mixed number.

Solution

Each circle is divided into three pieces, so each piece is \(\frac{1}{3}\) of the circle. There are four pieces shaded, so there are four thirds or \(\frac{4}{3}.\) The figure shows that we also have one whole circle and one third, which is \(1\frac{1}{3}.\) So, \(\frac{4}{3}=1\frac{1}{3}.\)

Example

Try it.

Draw a figure to model \(\frac{11}{8}.\)

Solution

The denominator of the improper fraction is \(8.\) Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have \(11\) eighths. We must shade three of the eight parts of another circle.

So, \(\frac{11}{8}=1\frac{3}{8}.\)

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

Convert between Improper Fractions and Mixed Numbers

In , we converted the improper fraction \(\frac{11}{6}\) to the mixed number \(1\frac{5}{6}\) using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the \(11\) pieces were left. We saw that \(\frac{11}{6}\) made one whole group of six sixths plus five more sixths, showing that \(\frac{11}{6}=1\frac{5}{6}.\)

The division expression \(\frac{11}{6}\) (which can also be written as \(611\)) tells us to find how many groups of \(6\) are in \(11.\) To convert an improper fraction to a mixed number without fraction circles, we divide.

Example

Try it.

Convert \(\frac{11}{6}\) to a mixed number.

Solution

	\(\frac{11}{6}\)
Divide the denominator into the numerator.	Remember \(\frac{11}{6}\) means \(11\div 6\).

Identify the quotient, remainder and divisor.
Write the mixed number as \(\text{quotient}\ \frac{\text{remainder}}{\text{divisor}}\).	\(1\frac{5}{6}\)
So, \(\frac{11}{6}=1\frac{5}{6}\)

Example

Try it.

Convert the improper fraction \(\frac{33}{8}\) to a mixed number.

Solution

	\(\frac{33}{8}\)
Divide the denominator into the numerator.	Remember, \(\frac{33}{8}\) means \(833\).
Identify the quotient, remainder, and divisor.
Write the mixed number as quotient \(\frac{\text{remainder}}{\text{divisor}}\).	\(4\frac{1}{8}\)
	So, \(\frac{33}{8}=4\frac{1}{8}\)

In , we changed \(1\frac{4}{5}\) to an improper fraction by first seeing that the whole is a set of five fifths. So we had five fifths and four more fifths.

\[\frac{5}{5}+\frac{4}{5}=\frac{9}{5}\]

Where did the nine come from? There are nine fifths—one whole (five fifths) plus four fifths. Let us use this idea to see how to convert a mixed number to an improper fraction.

Example

Try it.

Convert the mixed number \(4\frac{2}{3}\) to an improper fraction.

Solution

	\(4\frac{2}{3}\)
Multiply the whole number by the denominator.
The whole number is 4 and the denominator is 3.
Simplify.
Add the numerator to the product.
The numerator of the mixed number is 2.
Simplify.
Write the final sum over the original denominator.
The denominator is 3.	\(\frac{14}{3}\)

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

Model Equivalent Fractions

Let’s think about Andy and Bobby and their favorite food again. If Andy eats \(\frac{1}{2}\) of a pizza and Bobby eats \(\frac{2}{4}\) of the pizza, have they eaten the same amount of pizza? In other words, does \(\frac{1}{2}=\frac{2}{4}?\) We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza.

Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of and extend it to include eighths, tenths, and twelfths.

Start with a \(\frac{1}{2}\) tile. How many fourths equal one-half? How many of the \(\frac{1}{4}\) tiles exactly cover the \(\frac{1}{2}\) tile?

Since two \(\frac{1}{4}\) tiles cover the \(\frac{1}{2}\) tile, we see that \(\frac{2}{4}\) is the same as \(\frac{1}{2},\) or \(\frac{2}{4}=\frac{1}{2}.\)

How many of the \(\frac{1}{6}\) tiles cover the \(\frac{1}{2}\) tile?

Since three \(\frac{1}{6}\) tiles cover the \(\frac{1}{2}\) tile, we see that \(\frac{3}{6}\) is the same as \(\frac{1}{2}.\)

So, \(\frac{3}{6}=\frac{1}{2}.\) The fractions are equivalent fractions.

Example

Try it.

Use fraction tiles to find equivalent fractions. Show your result with a figure.

ⓐ How many eighths equal one-half?
ⓑ How many tenths equal one-half?
ⓒ How many twelfths equal one-half?

Solution

ⓐ It takes four \(\frac{1}{8}\) tiles to exactly cover the \(\frac{1}{2}\) tile, so \(\frac{4}{8}=\frac{1}{2}.\)

ⓑ It takes five \(\frac{1}{10}\) tiles to exactly cover the \(\frac{1}{2}\) tile, so \(\frac{5}{10}=\frac{1}{2}.\)

Suppose you had tiles marked \(\frac{1}{20}.\) How many of them would it take to equal \(\frac{1}{2}?\) Are you thinking ten tiles? If you are, you’re right, because \(\frac{10}{20}=\frac{1}{2}.\)

We have shown that \(\frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{4}{8},\frac{5}{10},\frac{6}{12},\) and \(\frac{10}{20}\) are all equivalent fractions.

Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to \(\frac{1}{2}.\) For example, \(\frac{2}{4},\frac{3}{6},\) and \(\frac{4}{8}\) are all equivalent to \(\frac{1}{2}.\) When we lined up the fraction tiles, it took four of the \(\frac{1}{8}\) tiles to make the same length as a \(\frac{1}{2}\) tile. This showed that \(\frac{4}{8}=\frac{1}{2}.\) See .

We can show this with pizzas, too. (a) shows a single pizza, cut into two equal pieces with \(\frac{1}{2}\) shaded. (b) shows a second pizza of the same size, cut into eight pieces with \(\frac{4}{8}\) shaded.

This is another way to show that \(\frac{1}{2}\) is equivalent to \(\frac{4}{8}.\)

How can we use mathematics to change \(\frac{1}{2}\) into \(\frac{4}{8}?\) How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:

These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.

So, we say that \(\frac{1}{2},\frac{2}{4},\frac{3}{6},\) and \(\frac{10}{20}\) are equivalent fractions.

Example

Try it.

Find three fractions equivalent to \(\frac{2}{5}.\)

Solution

To find a fraction equivalent to \(\frac{2}{5},\) we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by \(2,3,\) and \(5.\)

So, \(\frac{4}{10},\frac{6}{15},\) and \(\frac{10}{25}\) are equivalent to \(\frac{2}{5}.\)

Example

Try it.

Find a fraction with a denominator of \(21\) that is equivalent to \(\frac{2}{7}.\)

Solution

To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in \(21.\)

Since we can multiply \(7\) by \(3\) to get \(21,\) we can find the equivalent fraction by multiplying both the numerator and denominator by \(3.\)

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

Locate Fractions and Mixed Numbers on the Number Line

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values.

Let us locate \(\frac{1}{5},\frac{4}{5},3,3\frac{1}{3},\frac{7}{4},\frac{9}{2},5,\) and \(\frac{8}{3}\) on the number line.

We will start with the whole numbers \(3\) and \(5\) because they are the easiest to plot.

The proper fractions listed are \(\frac{1}{5}\) and \(\frac{4}{5}.\) We know proper fractions have values less than one, so \(\frac{1}{5}\) and \(\frac{4}{5}\) are located between the whole numbers \(0\) and \(1.\) The denominators are both \(5,\) so we need to divide the segment of the number line between \(0\) and \(1\) into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as \(\frac{1}{5},\frac{2}{5},\frac{3}{5},\) and \(\frac{4}{5}.\)

Now plot points at \(\frac{1}{5}\) and \(\frac{4}{5}.\)

The only mixed number to plot is \(3\frac{1}{3}.\) Between what two whole numbers is \(3\frac{1}{3}?\) Remember that a mixed number is a whole number plus a proper fraction, so \(3\frac{1}{3}>3.\) Since it is greater than \(3,\) but not a whole unit greater, \(3\frac{1}{3}\) is between \(3\) and \(4.\) We need to divide the portion of the number line between \(3\) and \(4\) into three equal pieces (thirds) and plot \(3\frac{1}{3}\) at the first mark.

Finally, look at the improper fractions \(\frac{7}{4},\frac{9}{2},\) and \(\frac{8}{3}.\) Locating these points will be easier if you change each of them to a mixed number.

\[\frac{7}{4}=1\frac{3}{4},\ \frac{9}{2}=4\frac{1}{2},\ \frac{8}{3}=2\frac{2}{3}\]

Example

Try it.

Locate and label the following on a number line: \(\frac{3}{4},\frac{4}{3},\frac{5}{3},4\frac{1}{5},\) and \(\frac{7}{2}.\)

Solution

Start by locating the proper fraction \(\frac{3}{4}.\) It is between \(0\) and \(1.\) To do this, divide the distance between \(0\) and \(1\) into four equal parts. Then plot \(\frac{3}{4}.\)

Next, locate the mixed number \(4\frac{1}{5}.\) It is between \(4\) and \(5\) on the number line. Divide the number line between \(4\) and \(5\) into five equal parts, and then plot \(4\frac{1}{5}\) one-fifth of the way between \(4\) and \(5\).

Now locate the improper fractions \(\frac{4}{3}\) and \(\frac{5}{3}\).

It is easier to plot them if we convert them to mixed numbers first.

\[\frac{4}{3}=1\frac{1}{3},\ \frac{5}{3}=1\frac{2}{3}\]

Divide the distance between \(1\) and \(2\) into thirds.

Next let us plot \(\frac{7}{2}.\) We write it as a mixed number, \(\frac{7}{2}=3\frac{1}{2}\). Plot it between \(3\) and \(4.\)

The number line shows all the numbers located on the number line.

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

Order Fractions and Mixed Numbers

We can use the inequality symbols to order fractions. Remember that \(a>b\) means that \(a\) is to the right of \(b\) on the number line. As we move from left to right on a number line, the values increase.

Example

Try it.

Order each of the following pairs of numbers, using \(<\) or \(>:\)

ⓐ \(-\frac{2}{3}____-1\)
ⓑ \(-3\frac{1}{2}____-3\)
ⓒ \(-\frac{3}{7}____-\frac{3}{8}\)
ⓓ \(-2____\frac{-16}{9}\)

Solution

ⓐ \(-\frac{2}{3}>-1\)

ⓑ \(-3\frac{1}{2}<-3\)

ⓓ \(-2<\frac{-16}{9}\)

Key Concepts

Property of One
- Any number, except zero, divided by itself is one.
  \(\frac{a}{a}=1\), where \(a\ne 0\).
Mixed Numbers
- A mixed number consists of a whole number \(a\) and a fraction \(\frac{b}{c}\) where \(c\ne 0\).
- It is written as follows: \(a\frac{b}{c}\ c\ne 0\)
Proper and Improper Fractions
- The fraction \(\frac{a}{b}\) is a proper fraction if \(a
Convert an improper fraction to a mixed number.
1. Divide the denominator into the numerator.
2. Identify the quotient, remainder, and divisor.
3. Write the mixed number as quotient\(\frac{\text{remainder}}{\text{divisor}}\).
Convert a mixed number to an improper fraction.
1. Multiply the whole number by the denominator.
2. Add the numerator to the product found in Step 1.
3. Write the final sum over the original denominator.
Equivalent Fractions Property
- If \(a, b,\) and \(c\) are numbers where \(b\ne 0\), \(c\ne 0\), then \(\frac{a}{b}=\frac{a⋅c}{b⋅c}\).

Visualize Fractions

In the following exercises, name the fraction of each figure that is shaded.

Try it.

Solution

ⓐ \(\frac{1}{4}\)
ⓑ \(\frac{3}{4}\)
ⓒ \(\frac{3}{8}\)
ⓓ \(\frac{5}{8}\)

Try it.

In the following exercises, shade parts of circles or squares to model the following fractions.

Try it.

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

Solution

Try it.

\(\frac{1}{3}\)

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

Solution

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

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

Solution

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

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

Solution

Try it.

\(\frac{7}{10}\)

In the following exercises, use fraction circles to make wholes using the following pieces.

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\(3\) thirds

Solution

Try it.

\(8\) eighths

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\(7\) sixths

Solution

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\(4\) thirds

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\(7\) fifths

Solution

Try it.

\(7\) fourths

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

Try it.

Solution

ⓐ \(\frac{5}{4}=1\frac{1}{4}\)
ⓑ \(\frac{7}{4}=1\frac{3}{4}\)
ⓒ \(\frac{11}{8}=1\frac{3}{8}\)

Try it.

Solution

ⓐ \(\frac{11}{4}=2\frac{3}{4}\)
ⓑ \(\frac{19}{8}=2\frac{3}{8}\)

In the following exercises, draw fraction circles to model the given fraction.

Try it.

\(\frac{3}{3}\)

Try it.

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

Solution

Try it.

\(\frac{7}{4}\)

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\(\frac{5}{3}\)

Solution

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\(\frac{11}{6}\)

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\(\frac{13}{8}\)

Solution

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\(\frac{10}{3}\)

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\(\frac{9}{4}\)

Solution

In the following exercises, rewrite the improper fraction as a mixed number.

Try it.

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

Try it.

\(\frac{5}{3}\)

Solution

\(1\frac{2}{3}\)

Try it.

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

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\(\frac{13}{5}\)

Solution

\(2\frac{3}{5}\)

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\(\frac{25}{6}\)

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\(\frac{28}{9}\)

Solution

\(3\frac{1}{9}\)

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\(\frac{42}{13}\)

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\(\frac{47}{15}\)

Solution

\(3\frac{2}{15}\)

In the following exercises, rewrite the mixed number as an improper fraction.

Try it.

\(1\frac{2}{3}\)

Try it.

\(1\frac{2}{5}\)

Solution

\(\frac{7}{5}\)

Try it.

\(2\frac{1}{4}\)

Try it.

\(2\frac{5}{6}\)

Solution

\(\frac{17}{6}\)

Try it.

\(2\frac{7}{9}\)

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\(2\frac{5}{7}\)

Solution

\(\frac{19}{7}\)

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\(3\frac{4}{7}\)

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\(3\frac{5}{9}\)

Solution

\(\frac{32}{9}\)

In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.

Try it.

How many sixths equal one-third?

Try it.

How many twelfths equal one-third?

Solution

Try it.

How many eighths equal three-fourths?

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How many twelfths equal three-fourths?

Solution

Try it.

How many fourths equal three-halves?

Try it.

How many sixths equal three-halves?

Solution

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

Try it.

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

Try it.

\(\frac{1}{3}\)

Solution

Answers may vary. Correct answers include \(\frac{2}{6},\frac{3}{9},\frac{4}{12}.\)

Try it.

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

Try it.

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

Solution

Answers may vary. Correct answers include \(\frac{10}{12},\frac{15}{18},\frac{20}{24}.\)

Try it.

\(\frac{2}{7}\)

Try it.

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

Solution

Answers may vary. Correct answers include \(\frac{10}{18},\frac{15}{27},\frac{20}{36}.\)

In the following exercises, plot the numbers on a number line.

Try it.

\(\frac{2}{3},\frac{5}{4},\frac{12}{5}\)

Try it.

\(\frac{1}{3},\frac{7}{4},\frac{13}{5}\)

Solution

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\(\frac{1}{4},\frac{9}{5},\frac{11}{3}\)

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\(\frac{7}{10},\frac{5}{2},\frac{13}{8},3\)

Solution

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\(2\frac{1}{3},-2\frac{1}{3}\)

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\(1\frac{3}{4},-1\frac{3}{5}\)

Solution

Try it.

\(\frac{3}{4},-\frac{3}{4},1\frac{2}{3},-1\frac{2}{3},\frac{5}{2},-\frac{5}{2}\)

Try it.

\(\frac{2}{5},-\frac{2}{5},1\frac{3}{4},-1\frac{3}{4},\frac{8}{3},-\frac{8}{3}\)

Solution

In the following exercises, order each of the following pairs of numbers, using \(<\) or \(>.\)

Try it.

\(-1\underset{__}{\ }-\frac{1}{4}\)

Try it.

\(-1\underset{__}{\ }-\frac{1}{3}\)

Solution

Try it.

\(-2\frac{1}{2}\underset{__}{\ }-3\)

Try it.

\(-1\frac{3}{4}\underset{__}{\ }-2\)

Solution

Try it.

\(-\frac{5}{12}\underset{__}{\ }-\frac{7}{12}\)

Try it.

\(-\frac{9}{10}\underset{__}{\ }-\frac{3}{10}\)

Solution

Try it.

\(-3\underset{__}{\ }-\frac{13}{5}\)

Try it.

\(-4\underset{__}{\ }-\frac{23}{6}\)

Solution

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.

Simplify: \(5\cdot 2+1.\)
If you missed this problem, review .

Αποκάλυψέ την.

\(11\)
Fill in the blank with \(<\) or \(>:-2\underset{__}{\ }-5\)
If you missed this problem, review .

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\(-2>-5\)
Name the fraction of the shape that is shaded in each of the figures.

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We need to ask two questions. First, how many equal parts are there? This will be the denominator. Second, of these equal parts, how many are shaded? This will be the numerator.
ⓐ
\(\begin{array}{llll}\text{How many equal parts are there?} & & & \text{There are eight equal parts}\text{.} \\ \text{How many are shaded?} & & & \text{Five parts are shaded}\text{.}\end{array}\)
Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is \(\frac{5}{8}.\)
ⓑ
\(\begin{array}{llll}\text{How many equal parts are there?} & & & \text{There are nine equal parts}\text{.} \\ \text{How many are shaded?} & & & \text{Two parts are shaded}\text{.}\end{array}\)
Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is \(\frac{2}{9}.\)
Name the fraction of the shape that is shaded in each figure:
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1. ⓐ \(\frac{3}{8}\)
2. ⓑ \(\frac{4}{9}\)
Name the fraction of the shape that is shaded in each figure:
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1. ⓐ \(\frac{3}{5}\)
2. ⓑ \(\frac{3}{4}\)
Shade \(\frac{3}{4}\) of the circle.

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The denominator is \(4,\) so we divide the circle into four equal parts ⓐ.
The numerator is \(3,\) so we shade three of the four parts ⓑ.
\(\frac{3}{4}\) of the circle is shaded.
Shade \(\frac{6}{8}\) of the circle.

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Shade \(\frac{2}{5}\) of the rectangle.

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Use fraction circles to make wholes with the following pieces: \(3\) thirds.

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Use fraction circles to make wholes with the following pieces: \(8\) eighths.

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Use fraction circles to make wholes using the following pieces:
1. ⓐ \(3\) halves
2. ⓑ \(8\) fifths
3. ⓒ \(7\) thirds
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ⓐ \(3\) halves make \(1\) whole with \(1\) half left over.
ⓑ \(8\) fifths make \(1\) whole with \(3\) fifths left over.
ⓒ \(7\) thirds make \(2\) wholes with \(1\) third left over.
Use fraction circles to make wholes with the following pieces: \(5\) thirds.

Αποκάλυψέ την.
Use fraction circles to make wholes with the following pieces: \(5\) halves.

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Name the improper fraction modeled. Then write the improper fraction as a mixed number.

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Each circle is divided into three pieces, so each piece is \(\frac{1}{3}\) of the circle. There are four pieces shaded, so there are four thirds or \(\frac{4}{3}.\) The figure shows that we also have one whole circle and one third, which is \(1\frac{1}{3}.\) So, \(\frac{4}{3}=1\frac{1}{3}.\)
Name the improper fraction. Then write it as a mixed number.

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\(\frac{5}{3}=1\frac{2}{3}\)
Name the improper fraction. Then write it as a mixed number.

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\(\frac{13}{8}=1\frac{5}{8}\)
Draw a figure to model \(\frac{11}{8}.\)

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The denominator of the improper fraction is \(8.\) Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have \(11\) eighths. We must shade three of the eight parts of another circle.
So, \(\frac{11}{8}=1\frac{3}{8}.\)
Draw a figure to model \(\frac{7}{6}.\)

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Draw a figure to model \(\frac{6}{5}.\)

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Use a model to rewrite the improper fraction \(\frac{11}{6}\) as a mixed number.

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We start with \(11\) sixths \((\frac{11}{6}).\) We know that six sixths makes one whole.
\[\frac{6}{6}=1\]
That leaves us with five more sixths, which is \(\frac{5}{6}\ (11\ \text{sixths minus}\ 6\ \text{sixths is}\ 5\ \text{sixths}).\)
So, \(\frac{11}{6}=1\frac{5}{6}.\)
Use a model to rewrite the improper fraction as a mixed number: \(\frac{9}{7}.\)

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\(1\frac{2}{7}\)
Use a model to rewrite the improper fraction as a mixed number: \(\frac{7}{4}.\)

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\(1\frac{3}{4}\)
Use a model to rewrite the mixed number \(1\frac{4}{5}\) as an improper fraction.

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The mixed number \(1\frac{4}{5}\) means one whole plus four fifths. The denominator is \(5,\) so the whole is \(\frac{5}{5}.\) Together five fifths and four fifths equals nine fifths.
So, \(1\frac{4}{5}=\frac{9}{5}.\)
Use a model to rewrite the mixed number as an improper fraction: \(1\frac{3}{8}.\)

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\(\frac{11}{8}\)
Use a model to rewrite the mixed number as an improper fraction: \(1\frac{5}{6}.\)

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\(\frac{11}{6}\)

Convert \(\frac{11}{6}\) to a mixed number.

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	\(\frac{11}{6}\)
Divide the denominator into the numerator.	Remember \(\frac{11}{6}\) means \(11\div 6\).

Identify the quotient, remainder and divisor.
Write the mixed number as \(\text{quotient}\ \frac{\text{remainder}}{\text{divisor}}\).	\(1\frac{5}{6}\)
So, \(\frac{11}{6}=1\frac{5}{6}\)

Convert the improper fraction to a mixed number: \(\frac{13}{7}.\)

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\(1\frac{6}{7}.\)
Convert the improper fraction to a mixed number: \(\frac{14}{9}.\)

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\(1\frac{5}{9}\)

Convert the improper fraction \(\frac{33}{8}\) to a mixed number.

Αποκάλυψέ την.

	\(\frac{33}{8}\)
Divide the denominator into the numerator.	Remember, \(\frac{33}{8}\) means \(833\).
Identify the quotient, remainder, and divisor.
Write the mixed number as quotient \(\frac{\text{remainder}}{\text{divisor}}\).	\(4\frac{1}{8}\)
	So, \(\frac{33}{8}=4\frac{1}{8}\)

Convert the improper fraction to a mixed number: \(\frac{23}{7}.\)

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\(3\frac{2}{7}\)
Convert the improper fraction to a mixed number: \(\frac{48}{11}.\)

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

Convert the mixed number \(4\frac{2}{3}\) to an improper fraction.

Αποκάλυψέ την.

	\(4\frac{2}{3}\)
Multiply the whole number by the denominator.
The whole number is 4 and the denominator is 3.
Simplify.
Add the numerator to the product.
The numerator of the mixed number is 2.
Simplify.
Write the final sum over the original denominator.
The denominator is 3.	\(\frac{14}{3}\)

Convert the mixed number to an improper fraction: \(3\frac{5}{7}.\)

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\(\frac{26}{7}\)
Convert the mixed number to an improper fraction: \(2\frac{7}{8}.\)

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\(\frac{23}{8}\)

Convert the mixed number \(10\frac{2}{7}\) to an improper fraction.

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	\(10\frac{2}{7}\)
Multiply the whole number by the denominator.
The whole number is 10 and the denominator is 7.
Simplify.
Add the numerator to the product.
The numerator of the mixed number is 2.
Simplify.
Write the final sum over the original denominator.
The denominator is 7.	\(\frac{72}{7}\)

Convert the mixed number to an improper fraction: \(4\frac{6}{11}.\)

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\(\frac{50}{11}\)
Convert the mixed number to an improper fraction: \(11\frac{1}{3}.\)

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\(\frac{34}{3}\)
Use fraction tiles to find equivalent fractions. Show your result with a figure.
1. ⓐ How many eighths equal one-half?
2. ⓑ How many tenths equal one-half?
3. ⓒ How many twelfths equal one-half?
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ⓐ It takes four \(\frac{1}{8}\) tiles to exactly cover the \(\frac{1}{2}\) tile, so \(\frac{4}{8}=\frac{1}{2}.\)
ⓑ It takes five \(\frac{1}{10}\) tiles to exactly cover the \(\frac{1}{2}\) tile, so \(\frac{5}{10}=\frac{1}{2}.\)
ⓒ It takes six \(\frac{1}{12}\) tiles to exactly cover the \(\frac{1}{2}\) tile, so \(\frac{6}{12}=\frac{1}{2}.\)
Suppose you had tiles marked \(\frac{1}{20}.\) How many of them would it take to equal \(\frac{1}{2}?\) Are you thinking ten tiles? If you are, you’re right, because \(\frac{10}{20}=\frac{1}{2}.\)
We have shown that \(\frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{4}{8},\frac{5}{10},\frac{6}{12},\) and \(\frac{10}{20}\) are all equivalent fractions.
Use fraction tiles to find equivalent fractions: How many eighths equal one-fourth?

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2

Symbols used here

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\approx

approximately equal
Equal to the precision shown, not exactly.

\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: Visualize Fractions

Understand the meaning of fractions
Model improper fractions and mixed numbers
Convert between improper fractions and mixed numbers
Model equivalent fractions
Find equivalent fractions
Locate fractions and mixed numbers on the number line
Order fractions and mixed numbers
Divide the denominator into the numerator.

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.

Δοκίμασε μόνος σου.

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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