Divide fractions using models

Division of fractions is understood as finding how many groups of one fractional size fit into another quantity, or determining the size of each group when a quantity is partitioned into a given number of fractional groups. Number lines, area models, and tape diagrams connect this meaning to the relationship between division and multiplication and explain why dividing by a fraction is equivalent to multiplying by its reciprocal; the focus is positive fractions, whole numbers, and mixed numbers, not negative or algebraically generalized cases.

Detailed Explanation: Divide fractions using models

Division by a fraction asks:

How many groups of that fraction fit into the amount?

Example

Find:

2\frac14 \div \frac34

1. Represent the total amount

Write $2\frac14$ in fourths:

2\frac14=\frac94

A tape diagram for $\frac94$ looks like 9 pieces, each $\frac14$ :

\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert

2. Make groups of $\frac34$

Each group must contain 3 pieces of size $\frac14$ :

\underbrace{\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert }_{\frac34} \quad \underbrace{\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert }_{\frac34} \quad \underbrace{\left \vert \frac14\right \vert \frac14\left \vert \frac14\right \vert }_{\frac34}

There are 3 groups of $\frac34$ .

Therefore,

2\frac14\div\frac34=3

The model also explains the multiplication rule:

\frac94\div\frac34 = \frac94\times\frac43 = 3

We multiply by $\frac43$ , the reciprocal of $\frac34$ , because three groups of $\frac34$ make one group of $\frac94$ :

\frac34\times 3=\frac94

Learn by doing: Divide fractions using models

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Fraction Division - Pizza Concept Intro - Picture and Number to Picture

Earned ?