What Does a Fraction Actually Mean?
A fraction bar is literally a division sign!
34 is identical to 3 ÷ 4
Whenever a math teacher or homework sheet asks you to "convert a fraction to a decimal", all they are asking you to do is complete the division.
Method 1: The Universal Long Division Method
This method works on in the universe, including proper fractions, improper fractions, and tricky repeating decimals.
Rule of the House:
NumeratorDenominator ⟹ Denominator \enclose{longdiv}{Numerator.00}
Worked Example 1: Terminating Decimal (58)
Convert the fraction into a decimal.
Step 1: Set up the division
Place inside the division house and on the outside:
8 \enclose{longdiv}{5}
Step 2: Add decimal point and trailing zeros
Because 8 does not go into 5, place a decimal point after 5 and add three zeros (5.000). Put the decimal point straight above on the quotient line:
8 \enclose{longdiv}{5.000}
Step 3: Divide step-by-step
- 8 goes into 50 (8 × 6 = 48).
50 - 48 = 2$. Bring down the next 0 to make **20**.
2. **8 into 20:** 8 goes into 20 **2 times** ($8 × 2 = 16$).
$20 - 16 = 4$. Bring down the next 0 to make **40**.
3. **8 into 40:** 8 goes into 40 **exactly 5 times** ($8 × 5 = 40$).
$40 - 40 = 0$.
The remainder is 0, so the decimal terminates!
\frac{5}{8} = 0.625
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## Worked Example 2: Repeating Decimal ($712$)
Some fractions never reach a remainder of zero. Instead, their digits loop forever!
> **Problem:**
> Convert **$712$** into a decimal.
### Step 1: Set up the long division ($7.000 ÷ 12$)
12 \enclose{longdiv}{7.0000}
### Step 2: Divide until you spot the pattern
1. **12 into 70:** Goes **5 times** ($12 × 5 = 60$).
$70 - 60 = 10$. Bring down 0 to make **100**.
2. **12 into 100:** Goes **8 times** ($12 × 8 = 96$).
$100 - 96 = 4$. Bring down 0 to make **40**.
3. **12 into 40:** Goes **3 times** ($12 × 3 = 36$).
$40 - 36 = 4$. Bring down 0 to make **40**.
4. **12 into 40 again:** Notice the remainder **4** is repeating forever!
### Step 3: Write with Bar Notation (Vinculum)
Only the digit **3** repeats forever (not the 5 or the 8). Place the repeating bar line **only** over the 3:
\frac{7}{12} = 0.58333... = 0.58\bar{3}
> [!WARNING]
> **Common Mistake on Repeating Bars:**
> Do not put the bar over the whole decimal! $0.\overline{583}$ means $0.583583583...$, which is wrong! Write $0.58\bar{3}$ so only the 3 is marked as repeating.
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## Method 2: The "Power of 10" Shortcut (No Division Needed!)
If the denominator can easily multiply into **10, 100, or 1,000**, you can skip long division completely:
\frac{3}{5} \times \frac{2}{2} = \frac{6}{10} = 0.6
\frac{9}{25} \times \frac{4}{4} = \frac{36}{100} = 0.36
\frac{7}{20} \times \frac{5}{5} = \frac{35}{100} = 0.35
$
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## Common Benchmark Fraction-to-Decimal Reference
| Fraction | Decimal | Repeating or Terminating? |
|---|---|---|
| $12$ | **0.5** | Terminating |
| $13$ | **$0.\bar{3}$** | Repeating |
| $14$ | **0.25** | Terminating |
| $15$ | **0.2** | Terminating |
| $18$ | **0.125** | Terminating |
| $23$ | **$0.\bar{6}$** | Repeating |
| $34$ | **0.75** | Terminating |