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Skill: Divide fractions using models

Explanation and Free Practice Resources

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.

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Detailed Explanation: Divide fractions using models

Division by a fraction asks:

How many groups of that fraction fit into the amount?

Example

Find:

214÷342\frac14 \div \frac34

1. Represent the total amount

Write 2142\frac14 in fourths:

214=942\frac14=\frac94

A tape diagram for 94\frac94 looks like 9 pieces, each 14\frac14:

∣14∣14∣14∣14∣14∣14∣14∣14∣14∣14∣14∣14∣14∣14∣14∣14∣14∣\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 34\frac34

Each group must contain 3 pieces of size 14\frac14:

∣14∣14∣14∣⏟34∣14∣14∣14∣⏟34∣14∣14∣14∣⏟34\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 34\frac34.

Therefore,

214÷34=32\frac14\div\frac34=3

The model also explains the multiplication rule:

94÷34=94×43=3\frac94\div\frac34 = \frac94\times\frac43 = 3

We multiply by 43\frac43, the reciprocal of 34\frac34, because three groups of 34\frac34 make one group of 94\frac94:

34×3=94\frac34\times 3=\frac94

Learn by doing: Divide fractions using models

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    Fraction Division - Simple by Whole Concept Intro - Picture to Answer - Don't Simplify

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