The ability to convert numbers between different formats is a fundamental skill in mathematics and has widespread applications across various fields. From engineering and computer science to finance and everyday life, understanding how to manipulate numerical representations is crucial. This article focuses on the concept of "85.5 convert," exploring how to transform the decimal number 85.5 into other numerical systems, primarily focusing on fractions, percentages, and binary representation. We'll break down each conversion step-by-step, ensuring clarity and understanding for all readers.
1. Understanding Decimal Numbers:
Before we begin conversions, it's important to understand the foundation: the decimal system. The decimal system (also known as base-10) uses ten digits (0-9) to represent numbers. Each digit's position represents a power of 10. For example, in the number 85.5:
8 represents 8 x 10<sup>1</sup> = 80
5 represents 5 x 10<sup>0</sup> = 5
5 (after the decimal point) represents 5 x 10<sup>-1</sup> = 0.5
Adding these values together (80 + 5 + 0.5) gives us the decimal number 85.5.
2. Converting 85.5 to a Fraction:
Converting a decimal to a fraction involves identifying the decimal's place value and expressing it as a fraction with a denominator that's a power of 10.
Step 1: Write the decimal as a fraction with a denominator of 1: 85.5/1
Step 2: Multiply both the numerator and denominator by a power of 10 to eliminate the decimal point. Since there's one digit after the decimal point, we multiply by 10: (85.5 x 10) / (1 x 10) = 855/10
Step 3: Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 855 and 10 is 5: 855 ÷ 5 = 171 and 10 ÷ 5 = 2
Step 4: The simplified fraction is 171/2.
Therefore, 85.5 is equivalent to the fraction 171/2.
3. Converting 85.5 to a Percentage:
A percentage represents a fraction of 100. To convert a decimal to a percentage, multiply the decimal by 100 and add a percentage sign (%).
Step 1: Multiply 85.5 by 100: 85.5 x 100 = 8550
Step 2: Add the percentage sign: 8550%
Therefore, 85.5 is equivalent to 8550%. It's important to note that percentages above 100% represent values greater than the whole.
4. Converting 85.5 to Binary (Base-2):
The binary system uses only two digits, 0 and 1. Converting a decimal to binary involves repeatedly dividing by 2 and recording the remainders.
Step 1: Integer Part (85): Repeatedly divide 85 by 2, recording the remainders:
Reading the remainders from bottom to top gives the binary representation of the integer part: 1010101
Step 2: Fractional Part (0.5): Repeatedly multiply the fractional part (0.5) by 2. If the result is greater than or equal to 1, record 1; otherwise, record 0.
0.5 x 2 = 1 (integer part is 1, fractional part is 0)
The binary representation of the fractional part is .1
Step 3: Combine the integer and fractional parts: The binary representation of 85.5 is 1010101.1
5. Summary:
This article demonstrated how to convert the decimal number 85.5 into different numerical representations: fractions, percentages, and binary. Each conversion process involves a series of logical steps based on the underlying principles of each number system. Understanding these conversions is essential for various mathematical and computational applications.
FAQs:
1. Can I convert 85.5 to other bases (e.g., base-8, base-16)? Yes, similar methods involving repeated division (for the integer part) and multiplication (for the fractional part) can be used for converting to any base.
2. What if the fraction resulting from the decimal-to-fraction conversion is a repeating decimal? Repeating decimals can be expressed as fractions using algebraic methods. For example, 0.333... is equivalent to 1/3.
3. Is there a quicker method for converting decimals to binary? While the repeated division/multiplication method is straightforward, there are more advanced techniques, such as using pre-calculated binary equivalents of powers of 10, for faster conversion.
4. Why is converting between number systems important? Different number systems are optimized for different tasks. For example, binary is ideal for computers, while decimals are more intuitive for everyday use. Conversion allows for interoperability between these systems.
5. What are some real-world applications of these conversions? Conversions are used in data representation (computers), financial calculations (percentages), engineering design (fractions), and many other fields requiring precise numerical manipulation.
Note: Conversion is based on the latest values and formulas.
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