What is 0.42857142857 in fraction form without putting it over 1, 10, 100 etc.?
The correct answer and explanation is:
The decimal 0.42857142857 is a repeating decimal that closely resembles the repeating decimal representation of the fraction 3/7.
Correct answer:
0.42857142857 ≈ 3/7
Explanation (300 words):
The decimal 0.42857142857 looks like the repeating decimal sequence 0.428571… which is well-known as the decimal representation of the fraction 3/7.
Here’s how to verify this:
- Recognizing the repeating pattern:
The decimal digits “428571” repeat in the decimal expansion of 3/7. Specifically, the fraction 3/7 equals approximately: 3÷7=0.428571428571…3 ÷ 7 = 0.428571428571… This is a repeating decimal with the repeating cycle “428571” that continues infinitely. - Why does 3/7 produce this repeating decimal?
When dividing 3 by 7, the decimal expansion repeats because 7 is a prime number that does not divide evenly into powers of 10. This causes the division to result in a repeating decimal cycle of length 6 digits, which is the length of the repeating block “428571”. - Not writing it as over 1, 10, 100 etc.:
Sometimes decimals can be converted to fractions by expressing the decimal as a ratio with a power of 10 as the denominator (e.g., 0.5 = 5/10). However, for repeating decimals like 0.428571…, this approach is impractical because the decimal never terminates. Instead, you identify the fraction by recognizing the repeating cycle and recalling known fractions with repeating decimals. Alternatively, algebraic methods can be used to convert the repeating decimal to a fraction without referencing powers of 10 directly. - Algebraic approach (optional):
Let x=0.428571428571…x = 0.428571428571…. Since the repeating part is 6 digits long, multiply both sides by 106=1,000,00010^6 = 1,000,000: 1,000,000x=428,571.428571…1,000,000x = 428,571.428571… Subtract the original xx from this: 1,000,000x−x=428,571.428571…−0.428571…1,000,000x – x = 428,571.428571… – 0.428571… 999,999x=428,571999,999x = 428,571 x=428,571999,999x = \frac{428,571}{999,999} Simplifying this fraction results in 37\frac{3}{7}.
In conclusion, 0.42857142857 as a fraction is exactly 3/7—a classic example of a repeating decimal that converts neatly to a simple fraction without needing to be expressed over powers of ten.