To convert the recurring decimal 0.27¯¯¯¯¯¯ to a rational number in simplest form, we can use the method of comparing it with a geometric series as follows:
Let x = 0.27¯¯¯¯¯¯
Multiply both sides by 100 to eliminate the recurring decimal:
100x = 27.7777¯¯¯¯¯¯
Subtract x from both sides:
100x - x = 27.7777¯¯¯¯¯¯ - 0.27¯¯¯¯¯¯
99x = 27.5077¯¯¯¯¯¯
Now, we can express 27.5077¯¯¯¯¯¯ as a fraction by using the fact that the decimal part repeats every two decimal places:
Let y = 0.077777¯¯¯¯¯
10y = 0.77777¯¯¯¯¯
Subtract y from 10y:
10y - y = 0.77777¯¯¯¯¯ - 0.077777¯¯¯¯¯
9y = 0.7
Now, we can express 0.7 as a fraction by dividing by 10:
0.7 = 7/10
Therefore, y = 7/90
Substituting y back into the equation 99x = 27.5077¯¯¯¯¯¯, we get:
99x = 27 + y
99x = 27 + 7/90
99x = (27*90 + 7) / 90
99x = 2430 + 7 / 90
99x = 2437 / 90
99x = 271 / 10
Divide both sides by 99:
x = (271 / 10) / 99
x = 271 / (10*99)
x = 271 / 990
So, the rational form of 0.27¯¯¯¯¯¯ is 271/990.
Convert 0.27¯¯¯¯¯0.27¯ to a rational number in simplest form.(1 point) Responses 3/11 Start Fraction 3 over 11 end fraction 27/100 Start Fraction 27 over 100 end fraction 1/27 Start Fraction 1 over 27 end fraction 27/99
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