See why dividing by a fraction is the same as multiplying by its reciprocal — then run the Keep-Change-Flip shortcut on real problems.
Dividing by a fraction can feel like a magic trick: you suddenly flip the second fraction and multiply. In this lesson we earn that trick. We start with the real question division asks — "how many of these fit?" — watch the answer appear with bars, see why flipping the divisor is the same operation, and then run Keep, Change, Flip on two worked problems.
1. Division Asks "How Many Fit?"
Before any rule, remember what division means. $3 \div \tfrac12$ is not scary — it just asks: how many halves fit inside 3?
How many $\tfrac12$-pieces fit into 3 whole bars?
Each whole bar holds 2 halves, and there are 3 bars — so $3 \div \tfrac12 = 6$. Dividing by a number smaller than 1 gives an answer bigger than what you started with, because lots of little pieces fit.
2. Why Flipping the Fraction Works
Here is the heart of it. Ask "how many $\tfrac14$'s make a whole?" A quarter is one of 4 equal pieces — so it takes 4 of them to rebuild the whole. Counting how many $\tfrac14$'s fit is the same as scaling up by 4.
the piece $\tfrac14$
the whole = $4 \times \tfrac14$
Dividing by a fraction is multiplying by its reciprocal (its flip):
$\div \tfrac14$ and $\times 4$ do the exact same thing — and $4 = \tfrac41$ is just $\tfrac14$ flipped. That is where the flip comes from.
3. The Shortcut: Keep, Change, Flip
Now we can name the three moves that turn any fraction division into a multiplication. Watch what happens to $\tfrac32 \div \tfrac14$:
KEEP the first fraction exactly as it is.
CHANGE the $\div$ sign into a $\times$.
FLIP the second fraction (swap top and bottom — its reciprocal).
4. Work It Out: $\tfrac32 \div \tfrac14$
Keep-Change-Flip turned the problem into $\tfrac32 \times \tfrac41$. To multiply fractions, multiply straight across — tops together, bottoms together — then simplify.
The answer is 6 — exactly the count we found by hand back in step 1, where 6 quarters fit into $\tfrac32$. The shortcut and the meaning agree.
5. Your Turn: A Recipe Problem
A recipe needs $\tfrac23$ cup of flour. You have 4 cups. How many batches can you make? That is "how many $\tfrac23$'s fit in 4?" — a division, $4 \div \tfrac23$.
Write 4 as $\tfrac41$, then Keep, Change, Flip.
KEEP the $\tfrac41$.
CHANGE $\div$ into $\times$.
FLIP $\tfrac23$ into $\tfrac32$.
6 batches. It checks out: each batch uses $\tfrac23$ cup, and $6 \times \tfrac23 = 4$ cups — exactly what you have.
✓︎
Lesson complete
You didn't just memorize "flip and multiply" — you saw why dividing by a fraction counts how many pieces fit, which is the same as scaling by the reciprocal.
Division asks "how many fit?" — $3 \div \tfrac12 = 6$ because 6 halves fit in 3.
Dividing by a fraction equals multiplying by its reciprocal: $\div\tfrac{a}{b} = \times\tfrac{b}{a}$.