The Correct Answer and Explanation is:
Correct Answer: 491/495
Explanation
The problem asks to convert the repeating decimal 0.991 to a fraction in its simplest form. The dots over the second ‘9’ and the ‘1’ indicate that these two digits form the repeating block. Therefore, the number is 0.9919191…
To convert this repeating decimal to a fraction, we can use an algebraic method.
- Set up the equation:
Let x be equal to our decimal number.
x = 0.9919191… - Multiply to isolate the repeating part:
Our goal is to create two equations where the decimal part after the decimal point is identical, so we can subtract them to eliminate the repeating tail.
First, we multiply x by 10 to move the decimal point past the non-repeating digit (‘9’).
10x = 9.919191… (Equation 1) - Multiply to shift the repeating block:
Next, we multiply the original equation for x by a power of 10 that moves the decimal point past the first full repeating block. Since there is one non-repeating digit and a two-digit repeating block, we multiply by 1000 (10³).
1000x = 991.919191… (Equation 2) - Subtract the equations:
Now we subtract Equation 1 from Equation 2. This step cancels out the infinitely repeating decimal part.
1000x = 991.919191…
– 10x = 9.919191…
——————–
990x = 982 - Solve for x:
To find the value of x as a fraction, we divide both sides by 990.
x = 982 / 990 - Simplify the fraction:
The final step is to simplify the fraction to its lowest terms. We can see that both the numerator (982) and the denominator (990) are even numbers, so they share a common factor of 2.
982 ÷ 2 = 491
990 ÷ 2 = 495
This gives us the fraction 491/495. The number 491 is prime, so the fraction cannot be simplified further.
Thus, the decimal 0.991 converted to a fraction in its simplest form is 491/495.
