How to Multiply and Divide Fractions: A Complete Guide for 4th and 5th Graders

What you already know about fractions

A fraction like $\dfrac{2}{3}$ has two parts:

• The numerator (the top number) tells you how many pieces you have.
• The denominator (the bottom number) tells you how many equal pieces make up a whole.

So $\dfrac{2}{3}$ means "2 of the 3 equal parts of one whole."

This guide focuses on the two operations that show up over and over in 4th and 5th grade: multiplying fractions and dividing fractions. The good news is each one only needs one rule.

Multiplying fractions: multiply straight across

Rule. To multiply two fractions, multiply the numerators and multiply the denominators:

That's it. No common denominators, no flipping, no extra setup. After multiplying, simplify if you can.

Example 1. Compute $\dfrac{2}{3} \times \dfrac{3}{4}$.

We simplified $\dfrac{6}{12}$ to $\dfrac{1}{2}$ by dividing the top and bottom by their greatest common factor, $6$.

Example 2. Compute $\dfrac{4}{5} \times \dfrac{5}{8}$.

Tip: simplify before you multiply. You can cancel any factor common to a numerator and a denominator before multiplying. In Example 2, the $5$ on top cancels with the $5$ on bottom, leaving $\dfrac{4}{8} = \dfrac{1}{2}$. Smaller numbers are easier to multiply and easier to simplify.

Multiplying a whole number by a fraction

A whole number is just a fraction with denominator $1$. So

That improper fraction also equals $3 \dfrac{1}{3}$ as a mixed number.

Dividing fractions: keep, change, flip

The trick most students remember for the rest of their lives is "Keep, Change, Flip" (sometimes shortened to KCF):

1. Keep the first fraction the same.
2. Change the division sign to multiplication.
3. Flip the second fraction (swap its numerator and denominator).

Then you have an ordinary multiplication problem.

The flipped fraction is called the reciprocal. The reciprocal of $\dfrac{c}{d}$ is $\dfrac{d}{c}$.

Why it works. Dividing by $\dfrac{c}{d}$ is the same as asking "how many groups of $\dfrac{c}{d}$ fit?", which is the same as multiplying by its reciprocal $\dfrac{d}{c}$.

Example 3. Compute $\dfrac{1}{2} \div \dfrac{1}{4}$.

Apply Keep, Change, Flip:

That makes sense: how many quarters fit inside a half? Two of them.

Example 4. Compute $\dfrac{3}{4} \div \dfrac{1}{2}$.

Dividing a fraction by a whole number

Treat the whole number as a fraction with denominator $1$, then KCF.

Example 5. Compute $\dfrac{2}{5} \div 4$.

If you split $\dfrac{2}{5}$ into $4$ equal parts, each part is $\dfrac{1}{10}$ of the whole.

Mixed numbers: convert first

A mixed number like $1\dfrac{1}{2}$ is a whole number plus a fraction. To multiply or divide a mixed number, convert it to an improper fraction first.

To convert $1\dfrac{1}{2}$:

• Multiply the whole number by the denominator: $1 \times 2 = 2$.
• Add the numerator: $2 + 1 = 3$.
• Put that over the original denominator: $\dfrac{3}{2}$.

So $1\dfrac{1}{2} = \dfrac{3}{2}$.

Example 6. Compute $1\dfrac{1}{2} \times \dfrac{2}{3}$.

Quick sanity check

Whenever you finish a fraction problem, look at the answer and ask:

• Did I simplify? Try to spot any common factor of the top and bottom.
• Does the size make sense? Multiplying two fractions less than $1$ gives an answer smaller than either input. Dividing by a fraction less than $1$ gives an answer larger than the original.

Common mistakes

• Adding instead of multiplying. Multiplying fractions does not require a common denominator. Don't try to make $\dfrac{2}{3}$ and $\dfrac{3}{4}$ have the same bottom — just multiply across.
• Forgetting to flip. The "Flip" step in KCF is non-negotiable. $\dfrac{1}{2} \div \dfrac{1}{4}$ is not $\dfrac{1}{8}$ — that would be multiplying.
• Flipping the wrong fraction. Only the second fraction (the divisor) gets flipped.
• Skipping the simplify step. $\dfrac{6}{12}$ is technically correct but $\dfrac{1}{2}$ is the form your teacher expects.