AI Tutor
If you have 3 whole pizzas and cut each pizza in half, how many slices do you get?
You get 6 slices! Even though you divided the pizzas into pieces, the number of pieces grew because each slice is smaller than a whole.
Dividing by 12\frac{1}{2} gave the exact same result as multiplying by 22. But why does flipping a fraction and multiplying always work?

The Two Steps Inside Every Fraction

A fraction like 23\frac{2}{3} tells you two things: cut each whole into 3 equal pieces, then gather them in groups of 2. A reciprocal is simply a fraction flipped upside down, where the numerator and denominator swap spots, like 23\frac{2}{3} becoming 32\frac{3}{2}.
Around 850 CE, Indian mathematician Mahavira was among the first to write down that dividing by a fraction is the same as multiplying by its reversed form.
What happens when the starting amount is also a fraction instead of a whole number?

Fraction Divided by Fraction

Imagine you have 34\frac{3}{4} of a granola bar, and a single serving is 18\frac{1}{8} of a bar. We want to find 34÷18\frac{3}{4} \div \frac{1}{8}.
Since each whole has 8 eighths, 34\frac{3}{4} of a whole has 34×8=6\frac{3}{4} \times 8 = 6 eighths.