To divide by a fraction, multiply by its reciprocal, the fraction with its numerator and denominator swapped. 3/4 ÷ 1/3 asks how many times 1/3 fits into 3/4, like counting how many people can each get 1/3 of a pizza from 3/4 of a pizza. Count it out and 1/3 fits twice, plus a quarter of a third time, so the quotient is 2 1/4, which is 3/4 × 3 = 9/4. Change the two fractions in the picture below and check that the count on the bars always matches flip and multiply.
Dividing by a fraction asks how many times the divisor fits into the dividend. Since 1/3 fits into 1 three times, dividing by 1/3 is the same as multiplying by 3. In the same way, 2/3 fits into 1 one and a half times, 3/2, so dividing by 2/3 means multiplying by 3/2, which is exactly 2/3 with its numerator and denominator swapped. Flip and multiply is not a trick to memorize: it first works out how many times the divisor fits into 1, then multiplies by that.
The outlined bar at the top is the whole, 1, and the blue bar below it is shaded as far as the dividend. At the start, 3/4, that is 3 of 4 equal parts. Each orange piece under it is as long as the divisor, 1/3, and laying them end to end up to the dashed line where the blue bar stops, two whole pieces fit. A short green piece is left in the third place, and the +1/4 on it means one quarter of another piece fits. So the quotient is 2 1/4, the same as the 9/4 in the formula at the bottom.
A common mistake is to divide numerator by numerator and denominator by denominator. Doing that to 3/4 ÷ 1/3 gives 3 ÷ 1 = 3 and 4 ÷ 3, which does not even divide evenly. Another is to flip the dividend instead of the divisor, which changes the question and gives a completely different value. You always flip the fraction after the ÷ sign, and when dividing by a whole number, you write it as a fraction over 1 and flip that.
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