Write this repeating decimal as a ratio of relatively prime integers:
0.324324324...
let x=0.324324324...
1000x=324.324324324324
999x=1000x-x=324
x=324/999=12/37
what is 324/999 in it lowest term? please
To express the repeating decimal 0.324324324... as a ratio of relatively prime integers, we can use a simple technique.
Step 1: Let x = 0.324324324...
Step 2: Multiply both sides of the equation by 1000 to move the decimal point three places to the right:
1000x = 324.324324324...
Step 3: Subtract the original equation from the multiplied equation:
1000x - x = 324.324324324... - 0.324324324...
This simplifies to:
999x = 324
Step 4: Solve for x by dividing both sides of the equation by 999:
x = 324/999
Since 324 and 999 share no common factors other than 1, the ratio of the repeating decimal 0.324324324... can be expressed as the relatively prime integers 324/999.
To convert the repeating decimal 0.324324324... into a ratio of relatively prime integers, we need to analyze the repeating pattern. Let's call the repeating part "x" and set up an equation to find its value.
First, we multiply the repeating decimal by a power of 10 to shift the decimal point to eliminate the repeating part:
1000x = 324.324324...
Next, since we shifted the decimal point three places to the right, we subtract the original number from the shifted number to eliminate the non-repeating part:
1000x - x = 324.324324... - 0.324324...
999x = 324
Dividing both sides of the equation by 999, we get:
x = 324/999
Now, we simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 324 and 999 is 9:
x = (324/9) / (999/9)
x = 36/111
Therefore, the repeating decimal 0.324324324... is equal to the ratio 36/111, which is a ratio of relatively prime integers since there is no common factor other than 1 between the numerator and denominator.