express the given repeating decimal
0.159159159 as a fraction
n = 0.159159159...
three numbers 159 repeat
so multiply by 10^3 = 1000
1000n =159.159159159 now subtract n
n = .159159159
-------------------------------
999n = 159.000000....
so
n = 159/999
or
159*10^-3 + 159*10^-6 +159*10^-9 ....
each term is 10^-3 times previous
sum of infinite geometric series with first term g and ratio r
S = g/(1-r)
here g = .159
and r = .001
so
S = .159/(1-.001) = .159/.999 =159/999
To express the repeating decimal 0.159159159 as a fraction, we need to set up an equation to solve for the fraction.
Let's represent the repeating decimal as the variable x:
x = 0.159159159...
To get rid of the repeating part, we can multiply both sides of the equation by a power of 10 that will shift the decimal point to the right of the repeating part.
Multiplying by 1000 (since there are three digits in the repeating part):
1000x = 159.159159...
Now, we can subtract the original equation from the equation we obtained after multiplying:
1000x - x = 159.159159... - 0.159159159...
Simplifying both sides:
999x = 159
Dividing both sides by 999:
x = 159/999
Therefore, the fraction representation of the repeating decimal 0.159159159 is 159/999.
To express the repeating decimal 0.159159159 as a fraction, we need to first determine the pattern of the repeating part.
Step 1: Assign a variable to the repeating part. Let x represent the repeating part 159.
Step 2: Multiply both sides by a power of ten that will shift the decimal point to the right of the repeating part. Since there are three digits in the repeating part, we will multiply by 1000.
1000x = 159.159159...
Step 3: Subtract the original equation from the equation obtained in Step 2 to eliminate the repeating part:
1000x - x = 159.159159... - 0.159159159...
999x = 159
Step 4: Solve for x by dividing both sides of the equation by 999:
x = 159/999
Step 5: Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator, and dividing both by the GCD:
x = 53/333
Therefore, the repeating decimal 0.159159159 can be expressed as a fraction 53/333.