Divide Fractions and Mixed Numbers

Dividing fractions sounds tricky, but there is a one-step trick that makes it just as easy as multiplying. The big idea: dividing by a fraction is the same as multiplying by its reciprocal — the fraction flipped upside down. Once you flip the second fraction, you simply multiply across like always.

Picture It First — How Many Fit?

• Division asks: how many of the divisor fit inside the dividend?
• Example: 1 / (1/4) asks how many 1/4-pieces fit in 1 whole — the answer is 4.
• 6 / (1/2) asks how many halves fit in 6 wholes — there are 12 halves, so the quotient is 12.
• When you divide by a fraction smaller than 1, the quotient is larger than the dividend.

The Keep-Change-Flip Rule

• KEEP the first fraction (the dividend) the same.
• CHANGE the divide sign to a multiply sign.
• FLIP the second fraction (the divisor) to its reciprocal — top and bottom swap.
• Then multiply across: tops together, bottoms together. Simplify if you can.

Worked Example — Whole Divided by a Fraction

• Solve 2 / (1/3). Write 2 as 2/1.
• Keep 2/1, change / to x, flip 1/3 to 3/1.
• Now multiply: 2/1 x 3/1 = (2 x 3) / (1 x 1) = 6/1 = 6.
• Check by picturing: each whole holds 3 thirds, and 2 wholes hold 2 x 3 = 6 thirds.

Fraction Divided by Fraction

• Solve 3/4 / (1/2). Keep 3/4. Change / to x. Flip 1/2 to 2/1.
• 3/4 x 2/1 = (3 x 2) / (4 x 1) = 6/4 = 1 1/2.
• Solve 5/6 / (2/3). Keep 5/6. Flip 2/3 to 3/2.
• 5/6 x 3/2 = 15/12 = 1 3/12 = 1 1/4 after simplifying.

Dividing Mixed Numbers

• First, change every mixed number into an improper fraction.
• 1 1/2 becomes 3/2 (because 2 x 1 + 1 = 3 over 2).
• 2 1/4 becomes 9/4 (because 4 x 2 + 1 = 9 over 4).
• Then keep-change-flip and multiply across, simplifying the answer.