4.2 Multiply and divide fractions  (Page 4/7)

In a previous chapter, we worked with opposites and absolute values. [link] compares opposites, absolute values, and reciprocals.

Divide fractions

Why is $12÷3=4?$ We previously modeled this with counters. How many groups of $3$ counters can be made from a group of $12$ counters?

There are $4$ groups of $3$ counters. In other words, there are four $3\text{s}$ in $12.$ So, $12÷3=4.$

What about dividing fractions? Suppose we want to find the quotient: $\frac{1}{2}÷\frac{1}{6}.$ We need to figure out how many $\frac{1}{6}\text{s}$ there are in $\frac{1}{2}.$ We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in [link] . Notice, there are three $\frac{1}{6}$ tiles in $\frac{1}{2},$ so $\frac{1}{2}÷\frac{1}{6}=3.$

Model: $\frac{1}{4}÷\frac{1}{8}.$

Solution

We want to determine how many $\frac{1}{8}\text{s}$ are in $\frac{1}{4}.$ Start with one $\frac{1}{4}$ tile. Line up $\frac{1}{8}$ tiles underneath the $\frac{1}{4}$ tile.

There are two $\frac{1}{8}\text{s}$ in $\frac{1}{4}.$

So, $\frac{1}{4}÷\frac{1}{8}=2.$

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Model: $2÷\frac{1}{4}.$

Solution

We are trying to determine how many $\frac{1}{4}\text{s}$ there are in $2.$ We can model this as shown.

Because there are eight $\frac{1}{4}\text{s}$ in $2,2÷\frac{1}{4}=8.$

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Let’s use money to model $2÷\frac{1}{4}$ in another way. We often read $\frac{1}{4}$ as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in [link] . So we can think of $2÷\frac{1}{4}$ as, “How many quarters are there in two dollars?” One dollar is $4$ quarters, so $2$ dollars would be $8$ quarters. So again, $2÷\frac{1}{4}=8.$

Using fraction tiles, we showed that $\frac{1}{2}÷\frac{1}{6}=3.$ Notice that $\frac{1}{2}·\frac{6}{1}=3$ also. How are $\frac{1}{6}$ and $\frac{6}{1}$ related? They are reciprocals. This leads us to the procedure for fraction division.