Fractions

A fraction names part of a whole (or part of a group). It is written as $\dfrac{\text{numerator}}{\text{denominator}}$.

• The denominator (bottom) tells how many equal pieces the whole is divided into.
• The numerator (top) tells how many of those pieces you have.

In $\dfrac{3}{4}$, the whole is cut into $4$ equal parts and we have $3$ of them. Two fractions are equivalent when they name the same amount, even though the numbers look different. You make an equivalent fraction by multiplying (or dividing) the top and bottom by the same nonzero number.

The same amount, cut differently.

A fraction is in lowest terms (fully simplified) when the numerator and denominator share no common factor except $1$. To simplify, divide the top and bottom by their greatest common factor (GCF).

The same amount, cut differently.

To decide which of two fractions is larger:

• Common denominators: rewrite both with the same denominator, then compare numerators.
• Cross-multiplying: for $\dfrac{a}{b}$ and $\dfrac{c}{d}$, compare $a\times d$ with $b\times c$. The fraction whose numerator is part of the larger product is larger.

The same amount, cut differently.

An improper fraction has a numerator greater than or equal to its denominator (like $\tfrac{11}{4}$). A mixed number is a whole number next to a proper fraction (like $2\tfrac{3}{4}$). They can name the same amount.

• Improper $\to$ mixed: divide numerator by denominator. The quotient is the whole number; the remainder over the denominator is the fraction.
• Mixed $\to$ improper: multiply the whole number by the denominator, add the numerator, and keep the denominator.

When two fractions have the same denominator, add or subtract the numerators and keep the denominator. Then simplify.

To add or subtract fractions with different denominators, rewrite them over a common denominator. The easiest one to use is the least common denominator (LCD), which is the least common multiple of the denominators. Then add or subtract the numerators and simplify.

Add (or subtract) the whole-number parts and the fraction parts separately. Give the fractions a common denominator first. Sometimes subtracting forces you to borrow (regroup) one whole into fraction form.

To multiply fractions, multiply the numerators and multiply the denominators. Simplify at the end (or cancel common factors first).

For mixed numbers, first change each to an improper fraction, then multiply.

To divide by a fraction, multiply by its reciprocal (flip the second fraction). The reciprocal of $\dfrac{c}{d}$ is $\dfrac{d}{c}$.

Change any mixed numbers to improper fractions before flipping.

• Read carefully and decide what operation the story describes.
• “Of” a quantity means multiply; “left over” or “how much more” often means subtract; “in all” means add; sharing equally means divide.
• Do the arithmetic, then simplify and label your answer.

A complex fraction is a fraction whose numerator, denominator, or both are themselves fractions, like $\dfrac{\ \frac{2}{3}\ }{\ \frac{4}{5}\ }$. A big fraction bar means “divide,” so a complex fraction is just a division problem in disguise:

Rewrite it as a division, multiply by the reciprocal, and simplify as usual.

The same amount, cut differently.

When you compare $\dfrac{a}{b}$ and $\dfrac{c}{d}$ (with positive $b$ and $d$), multiplying both by $bd$ does not change which is larger, since $bd>0$. This turns $\dfrac{a}{b}$ into $ad$ and $\dfrac{c}{d}$ into $bc$ --- exactly the cross-products. That is the whole justification for the cross-multiplying rule.

A surprising cousin is the mediant of $\dfrac{a}{b}$ and $\dfrac{c}{d}$, formed by adding tops and bottoms:

For positive fractions, the mediant always lands strictly between the two original fractions. (Warning: this is not how you add fractions --- it is a different, special construction.)

A unit fraction has a numerator of $1$, like $\dfrac{1}{2}$ or $\dfrac{1}{7}$. The ancient Egyptians wrote every other fraction as a sum of distinct unit fractions, for example $\dfrac{2}{3}=\dfrac{1}{2}+\dfrac{1}{6}$.

One reliable recipe is the greedy method: repeatedly subtract the largest unit fraction that is not bigger than what remains, until nothing is left. It always terminates, and the denominators grow quickly.

A continued fraction builds a number out of a whole part plus $1$ over another whole part plus $1$ over $\ldots$, stacking reciprocals downward. Every ordinary fraction can be rewritten this way by repeatedly separating the whole-number part and flipping the leftover. For example,

Reading it back from the bottom up: $2+\tfrac12=\tfrac52$, then $1\div\tfrac52=\tfrac25$, then $1+\tfrac25=\tfrac75$. Continued fractions reveal the “best” simple approximations to a number.

The harmonic mean of two positive numbers $a$ and $b$ averages their reciprocals instead of the numbers themselves:

It is the right average when the quantities are rates. If you drive one mile at $30$ mph and one mile at $60$ mph, your average speed is the harmonic mean

which is less than the plain average of $45$, because you spend more time at the slower speed.