What is 0.48 repeating as a fraction in simplest form?
The Correct Answer and Explanation is:
To convert the repeating decimal (0.48\overline{48}) (where “48” repeats indefinitely) into a fraction, we can follow a systematic approach.
Step 1: Let (x) be the repeating decimal.
Let:
[
x = 0.48484848\ldots
]
Step 2: Multiply (x) by a power of 10 to shift the decimal point.
Since “48” has two digits, we multiply (x) by (100) to shift the decimal point two places to the right:
[
100x = 48.48484848\ldots
]
Step 3: Set up an equation to eliminate the repeating part.
Now we have two equations:
- (x = 0.48484848\ldots)
- (100x = 48.48484848\ldots)
Next, we subtract the first equation from the second to eliminate the repeating decimal:
[
100x – x = 48.48484848\ldots – 0.48484848\ldots
]
This simplifies to:
[
99x = 48
]
Step 4: Solve for (x).
Now, we solve for (x):
[
x = \frac{48}{99}
]
Step 5: Simplify the fraction.
To simplify (\frac{48}{99}), we need to find the greatest common divisor (GCD) of (48) and (99).
The prime factorization of (48) is (2^4 \times 3), and for (99) it is (3^2 \times 11). The GCD is (3) since it’s the only common factor.
Now, divide both the numerator and the denominator by their GCD:
[
\frac{48 \div 3}{99 \div 3} = \frac{16}{33}
]
Conclusion
Thus, the repeating decimal (0.48\overline{48}) can be expressed as the fraction (\frac{16}{33}) in its simplest form. This fraction represents the same value as the repeating decimal, confirming that our conversion was accurate.