1.42857142857 as a fraction
Introduction
Understanding and representing numbers in different forms is an essential skill in mathematics. In this article, we will explore the decimal number 1.42857142857 and express it as a fraction. By breaking down the decimal into its constituent parts and applying basic mathematical principles, we can find the fractional representation of this number. Let's delve into the world of fractions and discover the beauty of mathematical conversions.
What is a fraction?
Before we dive into converting 1.42857142857 to a fraction, let's refresh our understanding of fractions. A fraction represents a part of a whole or a ratio between two quantities. It consists of a numerator (top number) and a denominator (bottom number), separated by a horizontal line. Fractions can be proper (numerator is smaller than the denominator), improper (numerator is greater than or equal to the denominator), or mixed (a whole number combined with a proper fraction).
Converting decimals to fractions
Decimal numbers can be converted to fractions by identifying the place value of each digit. The decimal point separates the whole number part from the fractional part. To convert a decimal to a fraction, we assign the decimal value to the numerator and place the appropriate power of 10 as the denominator. By simplifying the fraction, we can often express the decimal as a fraction in its simplest form.
Understanding 1.42857142857
The decimal number 1.42857142857 can be interpreted as 1 whole unit plus the fraction 0.42857142857. To convert this decimal to a fraction, we need to determine the exact value of the fractional part. By examining the pattern of digits, we can see that it repeats after the decimal point. This recurring pattern indicates that the number is a repeating decimal.
Writing the repeating decimal as a fraction
To express a repeating decimal as a fraction, we assign variables to the repeating part and solve for its value. In the case of 1.42857142857, we can represent the repeating part as x. Multiplying both sides of the equation by a power of 10 equal to the number of digits in the repeating pattern allows us to eliminate the repeating part. By subtracting the original equation from the multiplied equation, we can solve for x. Let's go through the steps to convert 1.42857142857 to a fraction.
Step 1: Assigning variables
Let x represent the repeating part of the decimal, which is 42857142857 in this case.
Step 2: Setting up the equation
Multiply both sides of the equation by a power of 10 equal to the number of digits in the repeating pattern (in this case, 11):
1011 * x = 42857142857
Step 3: Subtracting the original equation
Subtract the original equation from the multiplied equation:
1011 * x - x = 42857142857 - 1.42857142857
Step 4: Simplifying the equation
Combine like terms on both sides of the equation:
1011 * x - x = 42857142856
1011 * x - 1 * x = 42857142856
(1011 - 1) * x = 42857142856
Step 5: Solving for x
Divide both sides of the equation by (1011 - 1) to solve for x:
x = 42857142856 / (1011 - 1)
Final fraction representation
By substituting the value of x into the equation, we can find the fractional representation of the repeating decimal 1.42857142857:
1.42857142857 = 1 + x = 1 + (42857142856 / (1011 - 1))
Simplifying the fraction
To simplify the fraction further, we can apply mathematical techniques such as finding the greatest common divisor (GCD) between the numerator and denominator. By dividing both the numerator and denominator by their GCD, we can reduce the fraction to its simplest form.
Conclusion
In conclusion, the decimal number 1.42857142857 can be expressed as a fraction by identifying its repeating pattern and solving for its value. By following the steps outlined in this article, we determined that the fraction representation of 1.42857142857 is 1 + (42857142856 / (1011 - 1)). Understanding how to convert decimals to fractions expands our mathematical toolkit and enhances our ability to work with numbers in different forms.
References
1. Math Is Fun. (n.d.). Decimal to Fraction Calculator. Retrieved from https://www.mathsisfun.com/decimal-fraction-calculator.html
2. Khan Academy. (n.d.). Repeating decimals as fractions. Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-divisibility-tests/v/converting-repeating-decimals-to-fractions-1