Write the decimal −0.43 (the 3 is repeating) as a fraction in simplest form.
The Correct Answer and Explanation is:
To convert the repeating decimal (-0.43\overline{3}) (where the “3” repeats indefinitely) into a fraction, we can follow a systematic method. Let’s denote the repeating decimal as (x):
[
x = -0.43333\ldots
]
To eliminate the repeating part, we can multiply (x) by 10 (which shifts the decimal point one place to the right):
[
10x = -4.3333\ldots
]
Now, we have two equations:
- (x = -0.4333\ldots)
- (10x = -4.3333\ldots)
Next, we can subtract the first equation from the second:
[
10x – x = -4.3333\ldots – (-0.4333\ldots)
]
This simplifies to:
[
9x = -4.3333\ldots + 0.4333\ldots
]
The right side simplifies as follows:
[
-4.3333\ldots + 0.4333\ldots = -4.3333 + 0.4333 = -4
]
Now we have:
[
9x = -4
]
To solve for (x), divide both sides by 9:
[
x = -\frac{4}{9}
]
Thus, the decimal (-0.43\overline{3}) can be expressed as the fraction (-\frac{4}{9}).
Explanation
The process of converting a repeating decimal into a fraction involves recognizing the repeating part and using algebraic manipulation. By assigning the decimal to a variable, multiplying by a power of ten to shift the decimal point, and then setting up equations to eliminate the repeating part, we can isolate the variable and solve for it.
In this case, we took advantage of the fact that multiplying by 10 allowed us to align the repeating digits. Subtracting the original equation from the multiplied equation allowed us to focus solely on the non-repeating parts. The final steps involved straightforward algebra to arrive at a simple fraction.
The resulting fraction (-\frac{4}{9}) is in simplest form because the numerator and denominator share no common factors other than 1. Hence, converting repeating decimals into fractions not only helps in understanding their value but also facilitates operations involving fractions in mathematics.