Write 0.27 as a fraction or mixed number in simplest form
The Correct Answer and Explanation is :
To convert the decimal 0.27 into a fraction in simplest form, we start by recognizing that 0.27 can be expressed as a fraction over 1:
[
0.27 = \frac{0.27}{1}
]
Next, we can eliminate the decimal by multiplying both the numerator and denominator by 100, which shifts the decimal point two places to the right:
[
0.27 \times 100 = 27 \quad \text{and} \quad 1 \times 100 = 100
]
Thus, we have:
[
0.27 = \frac{27}{100}
]
Now, we need to check if this fraction can be simplified. A fraction is in simplest form when the numerator and denominator have no common factors other than 1. We can do this by finding the greatest common divisor (GCD) of 27 and 100.
Finding the GCD:
- Factors of 27: The prime factorization of 27 is (3^3) (since (27 = 3 \times 3 \times 3)).
- Factors of 100: The prime factorization of 100 is (2^2 \times 5^2) (since (100 = 2 \times 2 \times 5 \times 5)).
Since 27 has the prime factors of 3, and 100 has the prime factors of 2 and 5, there are no common prime factors between them. Therefore, the GCD of 27 and 100 is 1.
Since the GCD is 1, the fraction (\frac{27}{100}) is already in simplest form. Thus, we conclude that:
[
0.27 = \frac{27}{100}
]
As a mixed number, 0.27 does not exceed 1, so it cannot be expressed as a mixed number with a whole part. Therefore, the simplest fractional representation of 0.27 is:
[
\frac{27}{100}
]
In summary, converting a decimal to a fraction involves eliminating the decimal by using powers of 10, simplifying the resulting fraction by finding the GCD, and confirming that it is in its simplest form.