Fraction–decimal conversion

Understanding Fraction–Decimal Conversion

Fractions and decimals are two different ways to write the same kind of number: a part of a whole. This chapter focuses on moving between these two forms.

You will see three main directions:

• turning a fraction into a decimal,
• turning a decimal into a fraction,
• recognizing when decimals terminate (end) or repeat.

You are expected to already know what fractions and decimals are in general from other chapters; here we focus only on the conversion process.

Converting Fractions to Decimals

A fraction is of the form $\dfrac{a}{b}$, where $a$ and $b$ are integers and $b \ne 0$. To convert a fraction to a decimal, you divide the numerator by the denominator:

$$
\frac{a}{b} = a \div b
$$

You may get:

• a terminating decimal (it ends),
• or a repeating decimal (it goes on forever with a repeating pattern).

Fractions That Give Terminating Decimals

You get a terminating decimal when the denominator (after simplifying the fraction) has only the prime factors $2$ and/or $5$.

Examples:

1. $\dfrac{1}{2} = 1 \div 2 = 0.5$
2. $\dfrac{3}{4} = 3 \div 4 = 0.75$
3. $\dfrac{7}{8} = 7 \div 8 = 0.875$

Notice:

• $2 = 2$
• $4 = 2^2$
• $8 = 2^3$

They only use the prime factor $2$.

Another example:

$\dfrac{3}{25}$

The denominator is $25 = 5^2$, only factor $5$.
Divide:

• $3 \div 25 = 0.12$

So $\dfrac{3}{25} = 0.12$

Using Equivalent Fractions with 10, 100, 1000, ...

Sometimes it is easier to change the denominator to $10$, $100$, $1000$, etc., because these match the decimal place values.

Example: $\dfrac{7}{20}$

We want a denominator of $100$:
$20 \times 5 = 100$, so multiply top and bottom by $5$:

$$
\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100}
$$

Now write $\dfrac{35}{100}$ as a decimal:

• $\dfrac{35}{100} = 0.35$

So $\dfrac{7}{20} = 0.35$.

Another example: $\dfrac{9}{4}$

We can simply divide, or use a denominator of $100$.

• $9 \div 4 = 2.25$, so $\dfrac{9}{4} = 2.25$.

You can check with equivalent fractions:
$4 \times 25 = 100$, so:

$$
\frac{9}{4} = \frac{9 \times 25}{4 \times 25} = \frac{225}{100} = 2.25
$$

Fractions That Give Repeating Decimals

If the simplified denominator has any prime factor other than $2$ or $5$ (like $3$, $7$, $11$, etc.), the decimal will repeat.

Examples:

1. $\dfrac{1}{3}$

Do the division $1 \div 3$:

$1.000 \div 3 = 0.3333\ldots$

We write:
$$
\frac{1}{3} = 0.\overline{3}
$$
The bar over the $3$ means “3 repeats forever.”

2. $\dfrac{2}{3} = 0.\overline{6}$

Because $2 \div 3 = 0.6666\ldots$

3. $\dfrac{1}{6}$

$1 \div 6 = 0.1666\ldots$

Here the $6$ repeats, but the $1$ does not. We write:
$$
\frac{1}{6} = 0.1\overline{6}
$$

4. $\dfrac{5}{11}$

$5 \div 11 = 0.454545\ldots$

The pattern “45” repeats, so we write:
$$
\frac{5}{11} = 0.\overline{45}
$$

Simplify First

Always simplify the fraction before deciding whether the decimal terminates or repeats.

Example: $\dfrac{6}{15}$

First simplify:

• both $6$ and $15$ are divisible by $3$:
  $\dfrac{6}{15} = \dfrac{2}{5}$

Now convert $\dfrac{2}{5}$:

• $5$ is a power of $5$, so the decimal terminates.
• $2 \div 5 = 0.4$

So $\dfrac{6}{15} = 0.4$.

If you incorrectly check $15$ (which has factor $3$) before simplifying, you might wrongly expect a repeating decimal.

Converting Decimals to Fractions

Now we go the other way: starting from a decimal and writing it as a fraction. We treat terminating and repeating decimals separately.

Terminating Decimals to Fractions

A terminating decimal stops after a certain number of decimal places.

Steps:

1. Write the decimal without the decimal point as the numerator.
2. Use a denominator of $10$, $100$, $1000$, etc., depending on how many decimal places there are.
3. Simplify the fraction if possible.

Examples:

1. $0.5$

• One decimal place $\Rightarrow$ denominator $10$.
• Write as $\dfrac{5}{10}$.
• Simplify: $\dfrac{5}{10} = \dfrac{1}{2}$.

So $0.5 = \dfrac{1}{2}$.

2. $0.75$

• Two decimal places $\Rightarrow$ denominator $100$.
• Write as $\dfrac{75}{100}$.
• Simplify: divide top and bottom by $25$:
  $\dfrac{75}{100} = \dfrac{3}{4}$.

So $0.75 = \dfrac{3}{4}$.

3. $2.34$

This is a mixed number in fraction form:

• Two decimal places, so think of $2.34$ as $\dfrac{234}{100}$.

Now simplify:

• Divide top and bottom by $2$:
  $\dfrac{234}{100} = \dfrac{117}{50}$.

So $2.34 = \dfrac{117}{50}$.

4. $0.008$

• Three decimal places, so denominator $1000$.
• $0.008 = \dfrac{8}{1000}$.
• Simplify: divide top and bottom by $8$:
  $\dfrac{8}{1000} = \dfrac{1}{125}$.

So $0.008 = \dfrac{1}{125}$.

Repeating Decimals to Fractions (Simple Patterns)

There is a general algebraic method for any repeating decimal. Here we focus on the simplest and most common cases: repeating blocks that start immediately after the decimal point.

Case 1: A single digit repeats

Example: $0.\overline{3}$

Let $x = 0.\overline{3}$.

Then

• $10x = 3.\overline{3}$

Subtract the first equation from the second:

• $10x - x = 3.\overline{3} - 0.\overline{3}$
• $9x = 3$

So
$$
x = \frac{3}{9} = \frac{1}{3}
$$

So $0.\overline{3} = \dfrac{1}{3}$.

Another example: $0.\overline{7}$

Let $x = 0.\overline{7}$.

Then

• $10x = 7.\overline{7}$

Subtract:

• $10x - x = 7.\overline{7} - 0.\overline{7}$
• $9x = 7$

So
$$
x = \frac{7}{9}
$$

So $0.\overline{7} = \dfrac{7}{9}$.

In general, for one repeating digit $a$:
$$
0.\overline{a} = \frac{a}{9}
$$

Case 2: A block of digits repeats

Example: $0.\overline{45} = 0.454545\ldots$

Let $x = 0.\overline{45}$.

The repeating block has 2 digits, so multiply by $100$ (two zeros):

• $100x = 45.\overline{45}$

Subtract the original equation:

• $100x - x = 45.\overline{45} - 0.\overline{45}$
• $99x = 45$

So
$$
x = \frac{45}{99}
$$

Now simplify (divide by $9$):
$$
x = \frac{5}{11}
$$

So $0.\overline{45} = \dfrac{5}{11}$.

Another example: $0.\overline{16}$

Let $x = 0.\overline{16}$.

Two repeating digits:

• $100x = 16.\overline{16}$

Subtract:

• $100x - x = 16.\overline{16} - 0.\overline{16}$
• $99x = 16$

So
$$
x = \frac{16}{99}
$$

So $0.\overline{16} = \dfrac{16}{99}$.

In general, if the repeating block has $n$ digits, we use $10^n$:

• If $x = 0.\overline{\text{(block)}}$, then
  ^n x - x = \text{(block)}$,
  giving
  $x = \dfrac{\text{(block)}}{10^n - 1}$.

Examples:

• $0.\overline{3} = \dfrac{3}{9}$
• $0.\overline{72} = \dfrac{72}{99}$
• $0.\overline{526} = \dfrac{526}{999}$

You can then simplify the fraction if possible.

Types of Decimals from Fractions

Once a fraction is in simplest form:

• If the denominator has only prime factors $2$ and $5$, the decimal terminates.
• Example: $\dfrac{7}{40}$; $40 = 2^3 \cdot 5$ so it terminates.
• If the denominator has any other prime factor (like $3$, $7$, $11$, etc.), the decimal repeats.
• Example: $\dfrac{5}{12}$; $12 = 2^2 \cdot 3$, the factor $3$ causes a repeating decimal.

You do not need to fully compute the decimal to know if it ends or repeats; checking the simplified denominator is enough.

Practical Tips and Common Pitfalls

• Always simplify fractions before deciding if their decimals terminate or repeat.
• Be careful with mixed numbers: convert them to improper fractions or to decimals carefully.
• When turning decimals into fractions:
• Count decimal places to choose the correct power of $10$.
• Then simplify.
• When you see a repeating decimal in a problem, look for a simple fraction form (like ninths, elevenths, ninety-ninths).

Working fluently with both forms lets you choose whichever is more convenient for calculations in later topics, such as percentages and ratios.