To represent 0.757575 as a fraction in reduced form, we can use the infinite geometric sum formula.
The infinite geometric sum formula is given by:
S = a / (1 - r),
where S is the sum, a is the first term, and r is the common ratio.
In this case, we want to find a fraction that is equal to 0.757575. Let's call this fraction x.
So, x = 0.757575.
Now, we can rewrite 0.757575 as an infinite geometric sum:
x = a / (1 - r).
To find the first term (a) and the common ratio (r), we notice that the decimal 0.757575 has a repeating pattern of 75. We can represent this repeating pattern as a fraction.
Let's call the repeating pattern t.
t = 75.
Now, we can rewrite the repeating pattern as an infinite geometric sum:
t = a / (1 - r).
We can solve for a and r using the formula above.
a = t * (1 - r).
Now, we substitute the values of a, t, and r.
0.757575 = 75 / (1 - r).
To solve for r, let's manipulate the equation:
0.757575 = 75 / (1 - r).
0.757575(1 - r) = 75.
0.757575 - 0.757575r = 75.
-0.757575r = 75 - 0.757575.
-0.757575r = 74.242425.
Now, we can solve for r:
r = (74.242425) / (-0.757575).
r = -98.
Now, we can substitute the value of r into the equation to solve for a:
0.757575 = a / (1 - (-98)).
0.757575 = a / 99.
Multiply both sides of the equation by 99:
0.757575 * 99 = a.
74.999925 = a.
Now, we have the values of a and r:
a = 74.999925.
r = -98.
Therefore, 0.757575 can be represented as the fraction 74.999925 / 99. However, the fraction is not in reduced form. To simplify it, we can divide both the numerator and denominator by their greatest common divisor (GCD).
The GCD of 74.999925 and 99 is 1.
Dividing both the numerator and denominator by 1, we get:
74.999925 / 99 = 74.999925 / 99.
Therefore, the fraction representation of 0.757575 in reduced form is 74.999925 / 99.
Use the infinite geometric sum formula to write 0.757575 as a fraction in reduced from
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