written as fractions, the decimal numbers and are and , respectively. can you see a pattern? use this knowledge to convert 0.0625 into a fraction. then find its simplest form.
The Correct Answer and Explanation is :
To convert the decimal number 0.0625 into a fraction and find its simplest form, let’s first observe some decimal to fraction conversions to identify a pattern.
Common decimal equivalents include:
- 0.1 = 1/10
- 0.25 = 1/4
- 0.5 = 1/2
- 0.75 = 3/4
- 0.125 = 1/8
- 0.0625 = ?
From this, we see that many decimal numbers can be expressed as fractions, often with powers of 2 or 5 as denominators. Specifically, fractions that have a denominator of 10, 100, or 1000 are common in decimal representation.
Now, let’s convert 0.0625 into a fraction. The decimal 0.0625 can be read as “62.5 thousandths,” which translates to:
[
0.0625 = \frac{625}{10000}
]
Next, we can simplify this fraction. To do this, we need to find the greatest common divisor (GCD) of the numerator (625) and the denominator (10,000).
- Finding the GCD:
- The prime factorization of 625 is (5^4) (since (625 = 5 \times 5 \times 5 \times 5)).
- The prime factorization of 10,000 is (10^4 = (2 \times 5)^4 = 2^4 \times 5^4).
From the prime factorizations, we see that the GCD is (5^4 = 625).
- Dividing both the numerator and denominator by their GCD:
[
\frac{625 \div 625}{10000 \div 625} = \frac{1}{16}
]
Thus, (0.0625) simplifies to (\frac{1}{16}).
Conclusion
In summary, the decimal (0.0625) can be converted into the fraction (\frac{625}{10000}). By finding the GCD of 625 and 10,000, we simplify it to (\frac{1}{16}), the simplest form of this fraction. This process illustrates the relationship between decimals and fractions and the importance of simplification in expressing numbers in their most concise form.