Express the repeating decimal 0.513 (the 13 is repeating so the decimal is 0.5131313131313...) as a fraction in lowest terms using the infinite geometric series method.
0.5131313131313...
= .5 + .013131313...
= .5 + .013 + .00013 + .0000013 + ...
= .5 + an infinite geometric series
for the GS, a= .013 , r = .01
sum ∞ = a/(1-r) = .013/(1-.01) = (13/1000) / (99/100)
= 13/990
so sum∞ = 1/2 + 13/990 = 254/495
easier way:
for numerator, write down all digits to the end of the first repeat, from that subtract the digits that don't repeat : 513 - 5 = 508
for denominator, write down a 9 for each repeating digit, followed by a 0 for each non-repeating digit : 990
so ..
.5131313 = 508/990 = 254/495
To express the repeating decimal 0.513 as a fraction, we can use the infinite geometric series method.
Step 1: Let's represent the repeating decimal as x:
x = 0.5131313131313...
Step 2: Multiply both sides of the equation by 1000 to move the decimal point:
1000x = 513.13131313...
Step 3: Subtract the original equation from the previous equation to eliminate the repeating part:
1000x - x = 513.13131313... - 0.5131313131...
999x = 513
Step 4: Solve for x by dividing both sides of the equation by 999:
x = 513/999
Step 5: Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. In this case, there is no common factor other than 1, so we have:
x = 513/999
Therefore, the repeating decimal 0.513 can be expressed as the fraction 513/999 in lowest terms using the infinite geometric series method.
To express the repeating decimal 0.513 as a fraction using the infinite geometric series method, we can set up an equation and solve for the fraction.
Step 1: Identify the repeating pattern
Looking at the repeating decimal 0.513131313..., we can see that the repeating pattern is 13.
Step 2: Write the decimal as a sum of two parts
Let's express the decimal as the sum of the non-repeating part and the repeating part.
0.513 = 0.510 + 0.003
The non-repeating part is 0.510 (which is obtained by dropping the repeating part) and the repeating part is 0.003.
Step 3: Set up the equation
The formula for the sum of an infinite geometric series is:
S = a / (1 - r)
Where:
S represents the sum of the series,
a represents the first term of the series,
and r represents the common ratio between the terms.
We need to write the repeating part as an infinite geometric series. Since the repeating pattern is 13, we have:
a = 0.013
r = 0.01 (we divide the repeating pattern by 100, as there are two digits in the repeating pattern)
Now, substituting these values into the formula:
S = 0.003 / (1 - 0.01)
Step 4: Simplify the equation
To simplify the equation, we can calculate the value of (1 - 0.01) and then divide 0.003 by the resulting value.
1 - 0.01 = 0.99
S = 0.003 / 0.99
Step 5: Calculate the final fraction
To express the fraction in lowest terms, we need to reduce it. In this case, we can simplify by dividing both the numerator and denominator by 0.001:
S = (0.003 / 0.001) / (0.99 / 0.001)
S = 3 / 990
Therefore, the repeating decimal 0.513 can be expressed as the fraction 3/990 in lowest terms using the infinite geometric series method.