Fractions and decimals

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1 Activities for this section:

Work On Wording

2 Fractions and decimals

You might have noticed that we use the same language to describe $\frac {1}{10}$ and $0.1$ . Both of them are called “one tenth”, but we have defined them differently. Let’s see how these ideas are actually the same thing.

Explain why $\frac {1}{10}$ and $0.1$ are the same value.

We will start by drawing a whole which also has the value $1$ .

To find $\frac {1}{10}$ of this whole, we need to cut it into $\answer [given]{10}$ equal pieces, based on the meaning of the denominator, and then shade $\answer [given]{1}$ piece, based on the meaning of the numerator. Let’s do that in our picture.

Using our definition of fractions, this shaded piece is $\frac {1}{10}$ of the whole. But what if we thought about this using bundling? We started with one unit, and then unbundled it into $\answer [given]{10}$ equal sections. By definition, each section has the value $\answer [given]{0.1}$ because when we unbundle, we move exactly one place value right.

In other words, this shaded region is $\frac {1}{10}$ of the unit, and its value is also $0.1$ . Since it’s the same shaded region, these values must be equal. $\frac {1}{10} = 0.1$

Similarly, we could unbundle each of the $0.1$ -sized pieces, meaning that the new, smaller pieces would each have the value $0.01$ . Unbundling each of the $10$ equal pieces into $10$ more equal pieces would mean that our whole is split into $100$ equal pieces, or that each piece is worth $\frac {1}{100}$ . In this way we can see that $\frac {1}{100} = 0.01.$ This process could continue as long as we like.

Using thinking like we just described, what are the decimal values of the fractions below?

$\frac {1}{1000} = \answer {0.001}$
$\frac {1}{10000} = \answer {0.0001}$
$\frac {1}{100000} = \answer {0.00001}$

We can use this connection between fractions and decimals to write other fractions as decimals as well.

Write $\frac {9}{20}$ as a decimal.

Since we want to use one of the facts above, we want to change this fraction into an equivalent fraction whose denominator is $10$ , $100$ , $1000$ , or some other power of $10$ . We know that $20 \times \answer [given]{5} = 100$ , so we can make an equivalent fraction with denominator $100$ in this case. We have $\frac {9}{20} = \frac {\answer [given]{45}}{100}.$ Now, we know that $\frac {1}{100} = \answer [given]{0.01}$ as a decimal, and that according to our meaning of fractions $\frac {45}{100}$ is $45$ pieces, each of size $\frac {1}{100}$ . This means that $\frac {45}{100}$ is also $45$ pieces, each of size $\answer [given]{0.01}$ as a decimal. We can think about this as $45$ individual blocks, each of value $0.01$ . These $45$ blocks could be organized into $\answer [given]{4}$ bundles and $\answer [given]{5}$ individual blocks, and so the total value of all of these blocks would be $\answer [given]{0.45}$ . In other words, $\frac {9}{20} = 0.45$ as a decimal.

The previous example shows us that if we can make an equivalent fraction whose denominator is a power of $10$ (like $10$ , $100$ , $1000$ , etc), we can write the fraction as a decimal by looking at the numerator of that fraction.

Using the strategy of the example above, what is the decimal equivalent of the fraction $\frac {5}{8}$ ?

$\answer {0.625}$

Using a strategy similar to the one above, how would you write the decimal $0.735$ as a fraction?

$\frac {\answer {735}}{1000}$

2026-07-27 22:07:17