Express the repeating decimal as a fraction.
0.156156156...
To express the repeating decimal 0.156156156... as a fraction, we need to identify the repeating pattern. In this case, we see that the digits 156 repeat.
Step 1: Let's call the repeating decimal x.
Step 2: Multiply x by 1000 to shift all the repeating digits to the left of the decimal point:
1000x = 156.156156...
Step 3: Subtract equation (2) from equation (1) to eliminate the repeating digits after the decimal point:
1000x - x = 156.156156... - 0.156156...
Simplifying, we have:
999x = 156
Step 4: Divide both sides of the equation by 999:
x = 156/999
Therefore, the repeating decimal 0.156156156... can be expressed as the fraction 156/999.
To express the repeating decimal 0.156156156... as a fraction, we can follow these steps:
Step 1: Let x = 0.156156156...
Step 2: Multiply both sides of the equation by 1000 to shift the decimal point three places to the right: 1000x = 156.156156...
Step 3: Subtract the original equation (step 1) from the altered equation (step 2) to eliminate the repeating part: 1000x - x = 156.156156... - 0.156156156...
Simplifying, we get: 999x = 156
Step 4: Divide both sides of the equation by 999 to solve for x: x = 156/999
This can be simplified further by finding the greatest common divisor (GCD) of the numerator (156) and the denominator (999), which is 3:
x = (156/3) / (999/3) = 52/333
Therefore, the repeating decimal 0.156156156... can be expressed as the fraction 52/333.