Math: How to Convert Decimal Point to Fraction

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 $3 175/1000$

$3.175 = 3 + 0.175$

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

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

$3.175 = 3 7/40$ ✅

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 emphasize 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)

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

Repeating decimal can be as such:

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

Of course, and?

🤔

...

Verily, this is no revelation, for such truths are known to scholars and fools alike. Yet, the true art lieth not in mere knowing, but in discerning when and how to wield these conversions with wisdom most profound.

Technique Summary

The important things to remember when converting repeating decimal to fraction:

Spot the repeating pattern 👀 — we can underline or highlight the pattern ✏️
Then we do the algebra accordingly 🧐

Related Explanation and Tool

These are related links about GCF or LCM:

An explanation and JavaScript demo + snippet about finding GCF and LCM using continuous division technique here on Monkey Raptor.
GCFLCM.C: Greatest Common Factor and Least Common Multiple Calculator on Port Raptor.

Last modified on 2025-04-13T04:22:56-07:00

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