Visualize 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? There is one whole pizza, evenly divided into two equal parts, so each boy gets one of the two equal parts. In math, we write $\tfrac{1}{2}$ to mean one out of two parts.

On Tuesday, Andy and Bobby share a pizza with their parents, with each person getting an equal amount of the whole pizza. There is one whole pizza, divided evenly into four equal parts, so each person has $\tfrac{1}{4}$ of the pizza. On Wednesday, the family invites some friends over for a pizza dinner — twelve people in all. If they share the pizza equally, each person gets $\tfrac{1}{12}$ 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 of those parts are included. The denominator $b$ cannot equal zero because division by zero is undefined.

The circle above has been divided into three parts of equal size. Each part represents $\tfrac{1}{3}$ of the circle. This kind of model is called a fraction circle — other shapes, such as rectangles, can also model fractions.

Example. Name the fraction of the shape that is shaded in each figure: (a) a circle cut into eight equal wedges, five of them shaded; (b) a $3 \times 3$ grid of nine equal squares, two of them shaded.

We ask two questions: how many equal parts are there (the denominator), and of those, how many are shaded (the numerator)?

(a) There are eight equal parts, and five are shaded, so the fraction shaded is $\tfrac{5}{8}$.

(b) There are nine equal parts, and two are shaded, so the fraction shaded is $\tfrac{2}{9}$.

Model improper fractions and mixed numbers

Suppose you had eight equal fifth-pieces of a pizza. You used five of them to make one whole pizza, and you had three fifths left over. Using fraction notation, eight fifths is written $\tfrac{8}{5}$. Since eight fifths is one whole (five fifths), plus three more fifths, we can write it as $1\tfrac{3}{5}$, read as “one and three-fifths.”

Fractions such as $\tfrac{5}{4}$, $\tfrac{3}{2}$, $\tfrac{5}{5}$, and $\tfrac{7}{3}$ are called improper fractions — the numerator is greater than or equal to the denominator, so the fraction’s value is greater than or equal to one. When the numerator is smaller than the denominator, the fraction is a proper fraction, with value less than one. Fractions such as $\tfrac{1}{2}$, $\tfrac{3}{7}$, and $\tfrac{11}{18}$ are proper fractions.

Example. Name the improper fraction modeled by two circles, each cut into thirds, with all three thirds of the first circle shaded and one of the three thirds of the second circle shaded. Then write it as a mixed number.

Each circle is divided into three pieces, so each piece is $\tfrac{1}{3}$ of a circle. There are four shaded pieces total, so there are four thirds, $\tfrac{4}{3}$. The figure also shows one whole circle plus one third, which is $1\tfrac{1}{3}$. So $\tfrac{4}{3} = 1\tfrac{1}{3}$.

Convert between improper fractions and mixed numbers

The division expression $\tfrac{11}{6}$ (which can also be written as $6 \overline{)11}$) 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. Convert $\tfrac{33}{8}$ to a mixed number.

Divide the denominator into the numerator: $\tfrac{33}{8}$ means $8 \overline{)33}$.

The quotient is $4$, the remainder is $1$, and the divisor is $8$. Write the mixed number as quotient, plus remainder over divisor:

To convert a mixed number to an improper fraction, we reverse the idea: multiply the whole number by the denominator, add the numerator, and write the result over the original denominator.

Example. Convert the mixed number $4\tfrac{2}{3}$ to an improper fraction.

Multiply the whole number by the denominator: the whole number is $4$ and the denominator is $3$, so $4 \cdot 3 = 12$. Add the numerator to the product: $12 + 2 = 14$. Write the sum over the original denominator:

Model equivalent fractions

If Andy eats $\tfrac{1}{2}$ of a pizza and Bobby eats $\tfrac{2}{4}$ of an identical pizza, have they eaten the same amount? In other words, does $\tfrac{1}{2} = \tfrac{2}{4}$?

Imagine a rectangular tile representing one whole. A second identical tile is cut into two equal halves, and a third identical tile is cut into four equal fourths. Two of the fourth-pieces line up exactly with one half-piece, so $\tfrac{2}{4} = \tfrac{1}{2}$. Likewise, three sixth-pieces line up exactly with one half-piece, so $\tfrac{3}{6} = \tfrac{1}{2}$ as well.

Find equivalent fractions

How can we use mathematics, rather than pictures, to change $\tfrac{1}{2}$ into $\tfrac{4}{8}$? Imagine cutting a pizza that’s divided into two pieces, and cutting each of those two pieces into four smaller pieces. The whole pizza is now cut into eight pieces instead of two. Mathematically:

This leads to the Equivalent Fractions Property: multiplying the numerator and denominator of a fraction by the same nonzero number does not change the fraction’s value.

Example. Find three fractions equivalent to $\tfrac{2}{5}$.

To find equivalent fractions, we multiply the numerator and denominator by the same number (but not zero). Multiplying by $2$, $3$, and $5$:

So $\tfrac{4}{10}$, $\tfrac{6}{15}$, and $\tfrac{10}{25}$ are all equivalent to $\tfrac{2}{5}$.

Example. Find a fraction with a denominator of $21$ that is equivalent to $\tfrac{2}{7}$.

We need to multiply the denominator by a number that results in $21$. Since $7 \cdot 3 = 21$, we multiply both the numerator and denominator by $3$:

Locate fractions and mixed numbers on the number line

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

To locate a proper fraction like $\tfrac{1}{5}$, we know its value is less than one, so it lies between the whole numbers $0$ and $1$. We divide the segment of the number line between $0$ and $1$ into five equal parts, and plot $\tfrac{1}{5}$ at the first mark.

To locate a mixed number like $3\tfrac{1}{3}$, we first note that a mixed number is a whole number plus a proper fraction, so $3\tfrac{1}{3} > 3$ but not a whole unit greater — it lies between $3$ and $4$. We divide the number line between $3$ and $4$ into three equal parts (thirds) and plot $3\tfrac{1}{3}$ at the first mark.

To locate an improper fraction like $\tfrac{7}{4}$, it is easier to first convert it to a mixed number: $\tfrac{7}{4} = 1\tfrac{3}{4}$. This tells us it lies between $1$ and $2$, one-fourth of the way past the first mark… actually three-fourths of the way. We divide the segment between $1$ and $2$ into four equal parts and plot the point at the third mark.

Example. Locate and label $\tfrac{3}{4}$, $\tfrac{4}{3}$, $\tfrac{5}{3}$, $4\tfrac{1}{5}$, and $\tfrac{7}{2}$ on a number line.

Start with the proper fraction $\tfrac{3}{4}$: it’s between $0$ and $1$. Divide that segment into four equal parts and plot $\tfrac{3}{4}$ at the third mark.

Next, locate the mixed number $4\tfrac{1}{5}$: it’s between $4$ and $5$. Divide that segment into five equal parts, and plot the point one-fifth of the way from $4$.

Now locate the improper fractions $\tfrac{4}{3}$ and $\tfrac{5}{3}$. It’s easier to convert them to mixed numbers first: $\tfrac{4}{3} = 1\tfrac{1}{3}$ and $\tfrac{5}{3} = 1\tfrac{2}{3}$. Divide the distance between $1$ and $2$ into thirds, and plot both points.

Finally, plot $\tfrac{7}{2}$. Written as a mixed number, $\tfrac{7}{2} = 3\tfrac{1}{2}$, so it lies exactly halfway between $3$ and $4$.

Thinking of negative fractions as the opposite of positive fractions helps locate them on the number line, the same way we located the opposites of whole numbers in the integers chapter. To locate $-\tfrac{15}{8}$, first find $\tfrac{15}{8}$: as a mixed number it’s $1\tfrac{7}{8}$, so it lies between $1$ and $2$. Its opposite, $-\tfrac{15}{8}$, is the same distance from $0$ but on the other side, so it lies between $-1$ and $-2$.

Order fractions and mixed numbers

We can use inequality symbols to order fractions. Remember that $a > b$ means $a$ is to the right of $b$ on the number line; as we move left to right on a number line, the values increase.

Example. Order each pair using $<$ or $>$: (a) $-\tfrac{2}{3}$ and $-1$; (b) $-3\tfrac{1}{2}$ and $-3$; (c) $-\tfrac{3}{7}$ and $-\tfrac{3}{8}$; (d) $-2$ and $\tfrac{-16}{9}$.

(a) Plotting both on a number line, $-\tfrac{2}{3}$ is to the right of $-1$, so $-\tfrac{2}{3} > -1$.

(b) $-3\tfrac{1}{2}$ is farther left (more negative) than $-3$, so $-3\tfrac{1}{2} < -3$.

(c) Both are negative proper fractions; converting to a common denominator of $56$ shows $-\tfrac{3}{7} = -\tfrac{24}{56}$ is to the right of $-\tfrac{3}{8} = -\tfrac{21}{56}$… in fact $-\tfrac{24}{56}$ is farther left, so $-\tfrac{3}{7} < -\tfrac{3}{8}$.

(d) $\tfrac{-16}{9}$ written as a mixed number is $-1\tfrac{7}{9}$, which is to the right of $-2$, so $-2 < \tfrac{-16}{9}$.

Key terms

fraction — a number written $\tfrac{a}{b}$ with $a$ the numerator and $b \neq 0$ the denominator, representing parts of a whole. proper fraction — a fraction whose numerator is smaller than its denominator (value less than one). improper fraction — a fraction whose numerator is greater than or equal to its denominator (value at least one). mixed number — a whole number combined with a proper fraction. equivalent fractions — fractions that represent the same value. Equivalent Fractions Property — multiplying (or dividing) the numerator and denominator of a fraction by the same nonzero number leaves its value unchanged.

This section is adapted from Prealgebra 2e, Section 4.1: Visualize Fractions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the fraction-circle and fraction-tile figures as prose and one inline graphic, and the long division and number-line walkthroughs as typeset math; omitted the Be Prepared quiz, Manipulative Mathematics callouts, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.