0.135 Repeating as a Fraction
The decimal number 0.135 with a repeating pattern is a fascinating topic in mathematics. To understand its conversion to a fraction, we need to delve into the world of recurring decimals and fractions.
What is a Repeating Decimal?
A repeating decimal is a decimal number that has a sequence of digits that repeats indefinitely. In the case of 0.135, the digits "135" repeat infinitely. This type of decimal is also known as a periodic decimal or a non-terminating, repeating decimal.
Converting 0.135 Repeating to a Fraction
To convert 0.135 repeating to a fraction, we can use the following steps:
- Let x = 0.135 (the repeating decimal number)
- Multiply x by 1000 (since the repeating pattern has 3 digits)
1000x = 135.135
- Subtract x from both sides to eliminate the repeating pattern
999x = 135
- Divide both sides by 999 to get the final result
x = 135/999
Simplifying the Fraction
The resulting fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD is 27.
x = (135 ÷ 27) / (999 ÷ 27)
x = 5/37
Final Answer
The repeating decimal 0.135 is equal to the fraction 5/37.
In conclusion, converting a repeating decimal to a fraction involves multiplying, subtracting, and dividing to eliminate the repeating pattern. This process allows us to express the decimal number in a simplified fraction form, which can be useful in various mathematical applications.
