Find the rational fraction in lowest terms represented by the periodic decimal. show work 0.44444 . . .
The Correct Answer and Explanation is:
Correct Answer:
The rational fraction in lowest terms for 0.44444… is 4/9.
Step-by-Step Solution:
Let the repeating decimal be represented by x:
x = 0.44444…
Since the decimal repeats after one digit, multiply both sides by 10 to shift the decimal point:
10x = 4.44444…
Now subtract the first equation from the second:
10x = 4.44444…
x = 0.44444…
Subtracting gives:
10x – x = 4.44444… – 0.44444…
9x = 4
Solve for x:
x = 4 รท 9
Thus, the rational fraction is 4/9, which is already in its lowest terms.
Explanation:
Repeating decimals are numbers that have a digit or a group of digits that repeat infinitely. The decimal 0.44444… is an example of a repeating decimal with a single digit repeating, which is 4.
Every repeating decimal can be written as a rational fraction, which means it can be expressed as a ratio of two integers. The method used here involves algebraic manipulation. By setting the repeating decimal equal to a variable (x) and then multiplying both sides by a power of 10 that aligns the repeating digits, we create two equations that help eliminate the repeating part.
The subtraction step simplifies the decimal into a basic arithmetic equation, making it easy to isolate x and solve for its value as a fraction. In this case, the process shows that 0.44444… equals 4/9, and since 4 and 9 share no common factors besides 1, the fraction is already in lowest terms.
This technique works for any repeating decimal, whether the repeating part has one digit or multiple digits. Converting repeating decimals to fractions is important in understanding the relationship between decimals and rational numbers, reinforcing the idea that all repeating decimals are rational by definition.
