The correct answer and explanation is:
The correct answer is 1/27.
To convert the repeating decimal 0.03703703703 into a fraction, we can follow a standard algebraic procedure. This method is effective for any repeating decimal.
First, we must identify the repeating pattern of digits. In the number 0.037037037…, the sequence “037” is the block of digits that repeats infinitely. This repeating block is sometimes called the repetend.
We begin by setting a variable, let’s use x, equal to the repeating decimal:
x = 0.037037037…
The next step is to create a second equation by multiplying the first equation by a power of 10. The power we choose corresponds to the number of digits in the repeating block. Since the repeating block “037” has three digits, we will multiply both sides of our equation by 10 to the power of 3, which is 1000.
1000 * x = 1000 * 0.037037037…
This multiplication shifts the decimal point three places to the right:
1000x = 37.037037…
Now we have a system of two equations:
- 1000x = 37.037037…
- x = 0.037037…
The key to this method is to subtract the second equation from the first. When we do this, the infinitely repeating decimal part after the decimal point cancels out completely.
1000x – x = (37.037037…) – (0.037037…)
999x = 37
This leaves us with a simple linear equation. To find the value of x, we can now solve for it by dividing both sides of the equation by 999:
x = 37 / 999
We have successfully converted the decimal to a fraction, but it is important to check if it can be simplified. To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator (37) and the denominator (999). The number 37 is a prime number, so its only factors are 1 and 37. This means we only need to test if 999 is divisible by 37.
Performing the division:
999 ÷ 37 = 27
Since 999 divides evenly by 37, the GCD of 37 and 999 is 37. We can now simplify our fraction by dividing both the top and bottom by 37:
x = 37 ÷ 37 / 999 ÷ 37
x = 1 / 27
Therefore, the repeating decimal 0.037037037… is equal to the fraction 1/27. You can confirm this by dividing 1 by 27, which will yield the original repeating decimal.
