Understanding Fractions

Equal parts of a whole

Start with one whole thing and cut it into parts that are all the same size. Cut a chocolate bar into four equal pieces, and each piece is one fourth of the bar. Take three of those pieces, and you have three fourths of the bar. We write these as

A fraction is written as two numbers separated by a bar. The number below the bar, the denominator, tells you how many equal parts the whole was cut into. The number above the bar, the numerator, tells you how many of those parts you are counting. So $\frac{3}{4}$ means “the whole was cut into $4$ equal parts, and we are taking $3$ of them.”

The whole does not have to be a bar. Cut a round pizza into $4$ equal slices and take $3$ of them. You are holding the same fraction you just shaded on the bar. Only two things matter: how many equal parts the whole was cut into, and how many of them you took. The shape of the whole is not one of them.

The two numbers do very different jobs, and keeping them straight is most of the battle. The denominator names the size of each part: fourths are bigger than fifths, which are bigger than sixths. The reason is that the more parts you cut a fixed whole into, the smaller each part has to be. The numerator just counts how many of those parts you have. Think of “fourths” as a unit, the way “feet” or “dollars” is a unit. So $\frac{3}{4}$ is three of the unit called a fourth, exactly as $3$ feet is three of the unit called a foot. One handy memory hook: the denominator is down, and it names the kind of piece you are counting.

A quick word on names. The denominator $2$ gives halves, $3$ gives thirds, and $4$ gives fourths (or quarters). From $5$ onward we use the ordinary ordinal word: fifths, sixths, sevenths, and so on. The numerator is always read as a plain count: $\frac{2}{5}$ is “two fifths,” and $\frac{4}{7}$ is “four sevenths.”

Unit fractions: the size of one part

The simplest fractions have a numerator of $1$: a single equal part, such as $\frac{1}{4}$ or $\frac{1}{8}$. A fraction with numerator $1$ is called a unit fraction, and it is the building block of every other fraction. Three fourths is just three copies of one fourth stacked together:

The denominator by itself fixes the size of one part. And here is the idea that trips up the most students: a bigger denominator makes each part smaller. Cut one whole into $8$ equal pieces instead of $4$. Then each piece has to be smaller, because the same whole is now shared among more of them.

So $\frac{1}{8}$ is less than $\frac{1}{4}$, even though $8$ is bigger than $4$. Reading the denominator backwards, thinking that more pieces must mean more, is the single most common fraction mistake.

The bar means divide

The bar in a fraction is a division sign. The fraction $\frac{3}{4}$ is another way of writing $3 \div 4$:

The denominator $b$ must not be zero here, because dividing by zero is not allowed.

Division of whole numbers does not always give a whole number. As long as you are not dividing by zero, such a division always gives a fraction. The problem $3 \div 4$ has no answer among the whole numbers, yet it has the perfectly good answer $\frac{3}{4}$. And when the division does come out even, the fraction simply is that whole number:

So a fraction is a single number, the result of one division, not two numbers sitting near each other.

Fractions live on the number line

Because a fraction is a single number, it has a home on the number line, sitting between the whole numbers you already know. Finding that home uses the two parts of the fraction in turn. To place $\frac{3}{4}$, look at the segment from $0$ to $1$. The denominator $4$ tells you to split that segment into $4$ equal steps, which creates the marks $\frac{1}{4}$, $\frac{2}{4}$, $\frac{3}{4}$, and finally $\frac{4}{4} = 1$. The numerator $3$ tells you to count $3$ steps from $0$, which stops you at the mark labelled $\frac{3}{4}$.

This is the same story as the shaded bar, stood on its side: the segment from $0$ to $1$ is the whole. The denominator says how many equal steps to cut that segment into, and the numerator says how far to count. A fraction whose numerator is smaller than its denominator lands before the $1$ mark. That happens because you have taken fewer steps than it takes to reach a whole. The closer the numerator gets to the denominator, the closer the fraction sits to $1$. And when the numerator equals the denominator, you have taken every step and arrived exactly at $1$.

Proper and improper fractions

Nothing forces the numerator to be smaller than the denominator. You can count out more parts than it takes to make one whole, simply by using more than one whole. If you have two chocolate bars and cut each into thirds, you have $6$ thirds available, and you could take $5$ of them. That is the fraction $\frac{5}{3}$: five parts, each one a third of a bar. It is more than one whole bar but less than two.

Fractions split into two kinds by this comparison:

• A proper fraction has a numerator smaller than its denominator, so its value is less than $1$. Examples: $\frac{1}{2}$, $\frac{3}{4}$, $\frac{7}{8}$.
• An improper fraction has a numerator greater than or equal to its denominator, so its value is at least $1$. Examples: $\frac{5}{3}$, $\frac{4}{4}$, $\frac{9}{2}$.

The word “improper” is unfortunate, because there is nothing wrong with such a fraction. It is simply a fraction whose value has reached or passed a whole. On the number line $\frac{5}{3}$ sits past $1$. Counting in thirds, three thirds bring you to $1$, and two more thirds carry you to $\frac{5}{3}$, which lands between $1$ and $2$.

Why a fraction over itself is one

Any fraction whose numerator equals its denominator is equal to $1$.

A close cousin is the fraction with denominator $1$. Take $\frac{5}{1}$ and read it as parts of a whole. The whole was cut into just $1$ part, so that one part is the whole itself. Taking $5$ of those parts gives $5$ wholes. The division reading agrees, since $5 \div 1 = 5$. Nothing in that depended on the number $5$: for any whole number $n$, $\frac{n}{1} = n \div 1 = n$. So every whole number is secretly a fraction:

Why the denominator can never be zero

The denominator counts the parts the whole is cut into, and you cannot cut something into zero parts and still have parts to take. The division reading shows the problem even more sharply. Take $\frac{6}{0}$, which would mean $6 \div 0$, and ask what number times $0$ gives $6$. Nothing times $0$ gives $6$, because anything times $0$ is $0$. There is no answer, so $\frac{6}{0}$ names no number at all. The same happens for $\frac{a}{0}$ whenever $a$ is not zero: division asks what times $0$ gives back $a$, and nothing does. The fraction $\frac{0}{0}$ fails the other way: every number times $0$ gives $0$, so no single answer can be chosen. We say a fraction with denominator $0$ is undefined, and you must never write one. The numerator, by contrast, is allowed to be $0$. The fraction $\frac{0}{4}$ means you took none of the four parts, which is a perfectly good amount, namely $0$.

Comparing fractions

Often you only need to know which of two fractions is larger, without finding their exact values. Two situations let you compare at a glance, and both follow directly from the meaning of the numerator and the denominator.

Same denominator: compare the numerators. Comparing $\frac{3}{8}$ and $\frac{5}{8}$, both count eighths, so the one with more eighths wins:

because $5$ eighths is more than $3$ eighths. The comparison works the same way as $5$ apples against $3$ apples, once you know the apples are the same size. When two fractions are cut into the same size of part, the one with more parts is larger.

Same numerator: compare the denominators, but backwards. Compare $\frac{2}{3}$ and $\frac{2}{5}$. Both take $2$ parts, but thirds are larger than fifths, so $2$ thirds outweighs $2$ fifths:

Suppose two fractions take the same nonzero number of parts, but the parts are different sizes. Then the one with the smaller denominator is larger, because a smaller denominator means each part is bigger. When the fractions share neither the numerator nor the denominator, neither shortcut applies directly. The next lesson, on equivalent fractions, gives the general method of rewriting both fractions so that they do share a denominator.

Common mistakes