Write the following infinite decimal expansion as a fraction. You do not need to simplify.
0.492492492...
Let's call the given infinite decimal expansion as x.
If we multiply x by 1000, we shift all the digits three places to the left. Therefore, 1000x is equal to 492.492492...
Now, let's subtract x from 1000x.
1000x - x = 492.492492... - 0.492492492...
This simplifies to:
999x = 492
Dividing both sides of the equation by 999, we get:
x = 492/999
Therefore, the given infinite decimal expansion, 0.492492492..., is equal to the fraction 492/999.
since all digits are repeating the denominator will be all 9s. So the answer is 492/999.
You are correct! I apologize for the oversight in my previous response. The given infinite decimal expansion 0.492492492... can be represented as the fraction 492/999.
Step 1: Let's start by assigning a variable to the given infinite decimal expansion. Let's call it x. So, x = 0.492492492...
Step 2: Notice that the decimal part of x is repeated in a pattern of "492". To convert this into a fraction, we need to determine its denominator.
Step 3: The pattern "492" has three digits, so we'll multiply x by a power of 10 that shifts the repeating part to the left of the decimal point.
Step 4: Multiply x by 1000 to shift the pattern "492" to the left: 1000x = 492.492492...
Step 5: Subtract x from 1000x to eliminate the repeating part: 1000x - x = 492.492492... - 0.492492492...
Simplifying the left side: 999x = 492
Step 6: Now divide both sides by 999:
x = 492/999
Therefore, the fraction equivalent of the infinite decimal expansion 0.492492492... is 492/999.
To write the infinite decimal expansion 0.492492492... as a fraction, we need to identify the pattern in the repeating decimal segment.
Let's call the repeating segment "x". From the given decimal, we can observe that:
0.492492492... = 0.492 + 0.000492 + 0.000000492 + ...
In other words, we can separate the repeating segment by dividing the decimal into three parts:
0.492 (non-repeating segment) + 0.000492 (repeating segment) + 0.000000492 (repeating segment) + ...
Now, we can write an equation to represent the value of the decimal:
x = 0.000492 + 0.000000492 + ...
Multiplying both sides of the equation by 1000 (to shift the decimal point), we get:
1000x = 492.492492 + x
Simplifying the equation, we have:
999x = 492.492492
Now, we can solve for x by dividing both sides of the equation by 999:
x = 492.492492 / 999
Therefore, the fraction representation of the given infinite decimal 0.492492492... is 492.492492 / 999.