.5625 As A Fraction: Simplifying Made Easy!

gptAI

5 min read

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11-15- 2024

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To convert the decimal number 0.5625 into a fraction, we can simplify this task step by step. Understanding how to simplify fractions is essential for clarity in mathematics. 🌟

Understanding Decimal to Fraction Conversion

When converting a decimal to a fraction, the first step is to recognize the place value of the decimal. The decimal 0.5625 is in the fourth decimal place, which means it can be expressed as a fraction over 10,000. Thus, we write it as:

[ 0.5625 = \frac{5625}{10000} ]

Finding the Greatest Common Divisor (GCD)

Next, to simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

Using Prime Factorization

To find the GCD, we can break down the numbers into their prime factors.

• The prime factorization of 5625:
  • 5625 ÷ 5 = 1125
  • 1125 ÷ 5 = 225
  • 225 ÷ 5 = 45
  • 45 ÷ 5 = 9
  • 9 ÷ 3 = 3
  • 3 ÷ 3 = 1

So, the factorization is:
[ 5625 = 5^4 \times 3^2 ]

• The prime factorization of 10000:
  • 10000 ÷ 10 = 1000
  • 1000 ÷ 10 = 100
  • 100 ÷ 10 = 10
  • 10 ÷ 10 = 1

So, the factorization is:
[ 10000 = 10^4 = (2 \times 5)^4 = 2^4 \times 5^4 ]

Determining the GCD

Now we can identify the common factors:

• For 5625: ( 5^4 \times 3^2 )
• For 10000: ( 2^4 \times 5^4 )

The GCD is calculated from the lowest powers of the common prime factors:

• The common factor is ( 5^4 ) only.
  Thus, GCD = 625.

Simplifying the Fraction

Now, we can simplify the fraction ( \frac{5625}{10000} ) using the GCD we found:

[ \frac{5625 \div 625}{10000 \div 625} = \frac{9}{16} ]

Thus, the simplified form of the fraction for 0.5625 is:

[ \textbf{0.5625 as a fraction is } \frac{9}{16} ]

Why Simplifying Fractions is Important

Simplifying fractions allows for easier calculations and helps in better understanding relationships in mathematics. Simplified fractions are clearer and more concise.

Benefits of Using Simplified Fractions

1. Clarity: Simplified fractions are easier to read and understand. 📖
2. Ease of Computation: Smaller numbers make for quicker calculations, particularly in addition and subtraction. ⚙️
3. Reduced Errors: Fewer digits reduce the likelihood of mistakes in calculations. ✅

Practice Problems

To reinforce this knowledge, try converting these decimals to fractions and simplifying them:

Solutions

• 0.75 becomes ( \frac{3}{4} ) after simplification.
• 0.125 simplifies to ( \frac{1}{8} ).
• 0.3 remains ( \frac{3}{10} ) as it cannot be simplified further.
• 0.8 simplifies to ( \frac{4}{5} ).

Conclusion

Converting decimals to fractions might seem daunting at first, but by breaking it down into manageable steps, it becomes a straightforward process. Learning how to simplify fractions is an invaluable skill, enhancing not only your understanding of fractions but also your overall math proficiency. Remember, with practice, you’ll find that simplifying fractions is a breeze! 🌈