Maths: How to Convert Decimal Point to Fraction

Hi. 👋

Let's go.

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Non-Repeating Decimal

0.05 to Fraction

0.05 can be written as $5/100$

$5/100$ can be reduced by finding the GCF (Greatest Common Factor) from $5$ and $100$, which is $5$.

Then divide both numerator (the top fraction ➡️ $5$) and denominator (bottom part ➡️ $100$) by $5$.

$0.05 = 5/100 = 5/100 : (5/5) = (5 : 5)/(100 : 5) = 1/20$

So, $0.05 = 1/20$ ✅

0.65 to Fraction

0.65 can be written as $65/100$

$65/100$ can be further simplified by finding the GCF, $5$.

We divide both fraction parts by $5$.

$0.65 = 65/100 = 65/100 : (5/5)$ = $(65 : 5)/(100 : 5) = 13/20$

Therefore, $0.65$ is $13/20$ ✅

3.175 to Fraction

3.175 can be written as $3 + 0.175$

Convert the $0.175$ to fraction ➡️ $175/1000$

So we have a mixed fraction $3 175/1000$

$3.175 = 3 + 0.175$

$3 + 175/1000 = 3 + (175/1000 : (25/25))$

$3 + (175:25)/(1000:25) = 3 7/40$

$3.175 = 3 7/40$ ✅

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Repeating Decimal

0.111111... to Fraction

We have one-pattern repeating sequence ➡️ $1$.

💡 This uses a bit of algebra as the fraction conversion.

Let's take $n = 0.111111...$

Shift the $n$ value one digit to the right — multiply by $10$.

$10n = 1.111111...$

Now, if we subtract $10n$ and $n$, the result will be an integer.

$10n - n = 1.1111... - 0.1111...$

$9n = 1$

$n = 1/9$

Thus, $0.11111111.... = 1/9$ ✅

1.23123123123... to Fraction

We have three-pattern repeating sequence ➡️ $123$.

💡 Before we do the prior method, we need to observe the pattern before shifting to either right or left.

ℹ️ These examples only emphasise the right shift.

Let's take $n = 1.23123123123...$

We shift the decimal point 3 digits to the right — multiply by $1000$.

$1000n = 1231.23123123...$

Again, similar to above, if we subtract $1000n$ and $n$, it will produce an integer.

$1000n - n = 1231.23123123... - 1.23123123...$

$999n = 1230$

$n = 1230/999$

Apparently $1230$ and $999$ have $3$ as the GCF.

Divide both by $3$ to simplify the fraction.

$n = 1230/999 : (3/3) = (1230:3)/(999:3) = 410/333$

We can write the improper fraction $410/333$ as mixed fraction ➡️ $1 77/333$

So, $1.23123123123... = 410/333 = 1 77/333$ ✅

15.65415654156541... to Fraction

We have five-pattern repeating sequence ➡️ $15654$.

Let $n = 15.6541565415654...$

Shift 5 digits to the right — multiply by $100000$.

Thus, $100000n = 1565415.65415654...$

$100000n - n = 1565415.65415654... - 15.65415654...$

$99999n = 1565400$

$n = 1565400/99999$

$15.65415654156541... = 1565400/99999 = 15 65415/99999 = 15 (65415:3)/(99999:3)$

$= 15 21805/33333$ (no further reduction)

$15.65415654156541... = 15 21805/33333$ ✅

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Other Patterns

Repeating decimal can be as such:

• $10000.543215432154321.....$
• $509.9999999...$
• $29.787878787...$
• $0.000234500123123123123...$
• And so on...

Of course, and?

🤔

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Technique Summary

Essentially:

1. Spot the repeating pattern — we can underline or highlight the pattern.
2. Then we do the algebra accordingly.

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