Multiplying and Dividing Fractions

What “a fraction of a fraction” means

When you see $\frac{1}{2} \times 6$, you already read it as half of six, which is $3$. The word of is what multiplication means here, and it does not change when the second number is also a fraction. So $\frac{1}{2} \times \frac{1}{3}$ means one half of one third: start with one third of a whole, then take half of that piece.

Picture a single whole as a square. Slice it into thirds with two vertical cuts and shade one of the three strips. That shaded strip is $\frac{1}{3}$ of the square. Now take half of just that strip by cutting it across the middle. Half of the strip is a smaller rectangle, and the question is what fraction of the whole square that small rectangle is. Extend the horizontal cut all the way across and you can count. The square is now divided into $6$ equal small rectangles, and the doubly shaded piece is exactly one of them. So one half of one third is one sixth:

Look at where the $6$ came from. The whole square was cut into thirds across its width and into halves down its height. The two sets of cuts make the number of small rectangles the denominators multiplied together, $3 \times 2 = 6$. The doubly shaded piece is the overlap of “one of the thirds” and “one of the halves.” So the number of shaded rectangles is the numerators multiplied together, $1 \times 1 = 1$. Multiply the bottoms to count the pieces in the new whole, and multiply the tops to count how many of those pieces you keep.

The rule for multiplying fractions

The same picture works when each fraction keeps more than one part. Take $\frac{2}{3} \times \frac{4}{5}$, which reads as two thirds of four fifths:

The five columns and three rows cut the square into $15$ equal rectangles, which is where the $15$ comes from. The kept block is $2$ rows crossing $4$ columns, so it covers $8$ of them, and that is the $8$. Simplify the answer whenever you can: here $8$ and $15$ share no factor above $1$, so $\frac{8}{15}$ is already in lowest terms.

Multiplication rule. For any fractions $\frac{a}{b}$ and $\frac{c}{d}$ with $b$ and $d$ not zero, $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.$

The square diagram shows the rule for fractions between $0$ and $1$, where you shade part of one whole. The very same rule, multiply the tops and multiply the bottoms, carries over to improper fractions and to whole numbers. A whole number is a fraction over $1$, so $5 \times \frac{2}{7} = \frac{5}{1} \times \frac{2}{7} = \frac{10}{7}$. You just may end up with a result larger than one whole.

Canceling before you multiply

Multiplying the tops and bottoms first and simplifying afterward always works, but the numbers can grow large and then need a big simplification. There is a shortcut that keeps the numbers small: cancel any common factor between a numerator and a denominator before multiplying.

Take $\frac{3}{4} \times \frac{8}{9}$. Before multiplying, notice that $3$ on top shares the factor $3$ with $9$ on the bottom. The $8$ on top also shares the factor $4$ with the $4$ on the bottom. Divide each pair by its common factor. The $3$ becomes $1$ and the $9$ becomes $3$ after dividing both by $3$. The $8$ becomes $2$ and the $4$ becomes $1$ after dividing both by $4$:

Multiplying the small leftovers gives $\frac{2}{3}$ directly, in lowest terms, with no large product to reduce. To see that this is the same answer the long way gives, multiply first instead: $\frac{3}{4} \times \frac{8}{9} = \frac{24}{36}$. Then $\frac{24}{36}$ simplifies by its greatest common factor $12$ to $\frac{2}{3}$. Same result, but canceling first kept the numbers to single digits.

That cancel paired the $3$ on top of one fraction with the $9$ on the bottom of the other. It is allowed because the product is one fraction, $\frac{3 \times 8}{4 \times 9}$. In that single fraction the $3$ is a numerator and the $9$ is a denominator. Simplifying a fraction means dividing its top and its bottom by a common factor, exactly the rule from the equivalent-fractions lesson. So it makes no difference which of the two starting fractions each number came from.

The reciprocal of a nonzero fraction

The reciprocal of a nonzero fraction is what you get by swapping its numerator and denominator, that is by flipping it over. The reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$, and the reciprocal of $\frac{2}{7}$ is $\frac{7}{2}$. A whole number such as $4$ is $\frac{4}{1}$, so its reciprocal is $\frac{1}{4}$.

The reason the reciprocal matters is a single clean fact: a nonzero fraction times its reciprocal is always $1$.

Two numbers whose product is $1$ are called reciprocals of each other. You can read the relation both ways: $\frac{3}{5} \times \frac{5}{3} = 1$, so each is the reciprocal of the other. That is the whole reason a reciprocal matters: multiplying by it undoes a multiplication by the original nonzero fraction.

Why dividing by a nonzero fraction means multiplying by its reciprocal

Division asks a counting question: $a \div b$ asks “how many $b$‘s fit into $a$?” Keep that meaning and apply it to fractions. The expression $\frac{3}{4} \div \frac{1}{4}$ asks “how many fourths fit into three fourths?” The picture below answers that directly.

That picture answers the question for one size of piece. Below you can change the size. The top bar is fixed at three fourths. The bottom bar is that same three fourths, with each of its parts cut into as many pieces as you choose.

The numerator on the bottom bar counts how many of the smaller pieces fit into three fourths. Cut each fourth into $2$ and the bar reads $\frac{6}{8}$, so six eighths cover the same amount. That means six one-eighth pieces fit into three fourths, and $\frac{3}{4} \div \frac{1}{8} = 6$. Notice what flipping does here: $6 = \frac{3}{4} \times \frac{8}{1}$, which is $\frac{3}{4}$ times the reciprocal of $\frac{1}{8}$. Cut into $3$ instead and the bar reads $\frac{9}{12}$, so $\frac{3}{4} \div \frac{1}{12} = 9$, matching $\frac{3}{4} \times \frac{12}{1} = \frac{36}{4} = 9$.

Now run the cut all the way to its finest and watch which direction the count moves. The pieces keep shrinking and more of them fit, which is exactly why dividing by a smaller fraction gives a larger answer.

Why does flipping the divisor work in general? Dividing undoes multiplying: the answer to $\frac{3}{4} \div \frac{2}{3}$ is whatever number, multiplied by $\frac{2}{3}$, gets back to $\frac{3}{4}$. Multiplying by $\frac{2}{3}$‘s reciprocal $\frac{3}{2}$ is exactly what cancels $\frac{2}{3}$ away, because $\frac{2}{3} \times \frac{3}{2} = 1$, the reciprocal fact from the last section. So multiplying by the reciprocal of the divisor always finds the number that division asks for.

Division rule. To divide by a nonzero fraction, multiply by its reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.$

The rule needs $c$ to be nonzero, because the reciprocal of $\frac{0}{d}$ would be $\frac{d}{0}$, and nothing can be divided by zero.

So to divide by a nonzero fraction, leave the first fraction alone, flip the second, and multiply, using the canceling shortcut if it helps. For example,

where the last step divides top and bottom by their greatest common factor $6$. A common slip is to flip the wrong fraction. Only the divisor, the fraction you are dividing by, gets turned over; the first fraction stays exactly as it is.

Common mistakes