What is the fraction for the following decimal expansion?
Converting a decimal expansion to a fraction is a fundamental skill in mathematics, often required in various academic and practical scenarios. Whether you are a student struggling with a homework assignment or a professional dealing with financial calculations, understanding how to convert decimals to fractions is essential. In this article, we will explore the process of converting decimal expansions to fractions, providing you with a step-by-step guide to make the conversion process easier and more efficient.
The first step in converting a decimal expansion to a fraction is to identify the type of decimal you are dealing with. There are two main types of decimal expansions: terminating decimals and repeating decimals.
Terminating Decimals
A terminating decimal is a decimal that ends after a finite number of digits. For example, 0.25 and 1.5 are terminating decimals. To convert a terminating decimal to a fraction, follow these steps:
1. Write the decimal as a whole number by multiplying it by a power of 10 that has as many zeros as there are digits in the decimal part. For example, to convert 0.25 to a fraction, multiply it by 100 (10^2), resulting in 25.
2. Write the resulting whole number as the numerator of the fraction.
3. Write the power of 10 used in step 1 as the denominator of the fraction.
Using these steps, we can convert 0.25 to a fraction:
0.25 100 = 25
Numerator: 25
Denominator: 100
The fraction equivalent of 0.25 is 25/100, which can be simplified to 1/4.
Repeating Decimals
A repeating decimal is a decimal that has a sequence of digits that repeats indefinitely. For example, 0.333… (where the 3 repeats indefinitely) and 0.142857142857… (where the sequence 142857 repeats indefinitely) are repeating decimals. To convert a repeating decimal to a fraction, follow these steps:
1. Let x be the repeating decimal.
2. Multiply x by a power of 10 that is one more than the number of digits in the repeating sequence. For example, to convert 0.333… to a fraction, multiply it by 10 (since the repeating sequence has one digit).
3. Subtract the original repeating decimal from the result of step 2. This will eliminate the repeating part of the decimal.
4. Write the result as a fraction, with the original repeating decimal as the numerator and the power of 10 used in step 2 as the denominator.
Using these steps, we can convert 0.333… to a fraction:
x = 0.333…
10x = 3.333…
10x – x = 3
Numerator: 3
Denominator: 10
The fraction equivalent of 0.333… is 3/10.
In conclusion, converting decimal expansions to fractions is a valuable skill that can be applied in various situations. By following the steps outlined in this article, you can easily convert terminating and repeating decimals to their fraction equivalents. Practice and understanding the underlying concepts will make the process more intuitive and efficient.
