Express this decimal as a fraction.
0.8 repeating decimal
The Correct Answer and Explanation is :
The correct answer is: (0.888…) can be expressed as the fraction (\frac{8}{9}).
To express the repeating decimal (0.8\overline{8}) (which means the digit 8 repeats indefinitely) as a fraction, we can follow a systematic approach.
Step 1: Let (x) equal the repeating decimal
Let (x = 0.888…)
Step 2: Multiply by a power of 10
To eliminate the repeating part, multiply both sides of the equation by 10 (since the repeating digit is in the tenths place):
[
10x = 8.888…
]
Step 3: Set up the equation
Now we have two equations:
- (x = 0.888…) (1)
- (10x = 8.888…) (2)
Step 4: Subtract the first equation from the second
To isolate the repeating decimal, subtract equation (1) from equation (2):
[
10x – x = 8.888… – 0.888…
]
This simplifies to:
[
9x = 8
]
Step 5: Solve for (x)
Now, divide both sides by 9 to find (x):
[
x = \frac{8}{9}
]
Conclusion
Thus, the repeating decimal (0.888…) can be expressed as the fraction (\frac{8}{9}).
Explanation
Repeating decimals can often be challenging to convert into fractions due to their infinite nature. However, the method demonstrated here utilizes algebraic manipulation to handle the repetition effectively. By defining the decimal as a variable, multiplying to shift the decimal point, and then subtracting, we can eliminate the repeating portion and solve for the original variable.
This method not only works for (0.888…) but can also be applied to any repeating decimal, whether it has one digit repeating (like (0.3\overline{3})) or more (like (0.12\overline{12})). By adjusting the multiplication factor (for instance, multiplying by 100 for two repeating digits), you can manage the decimal places appropriately.
In summary, converting repeating decimals to fractions is a useful skill in mathematics, allowing for clearer representations of numbers that might seem cumbersome in their decimal form. Understanding this process lays the groundwork for more advanced mathematical concepts and operations involving rational numbers.