Convert 0.27¯¯¯¯¯
to a rational number in simplest form.(1 point)
Responses
311
Start Fraction 3 over 11 end fraction
2799
Start Fraction 27 over 99 end fraction
127
Start Fraction 1 over 27 end fraction
27100
Start Fraction 27 over 100 end fraction
Start Fraction 27 over 100 end fraction
How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?
Well, you've got some interesting options there, but I must say none of them are exactly rational... they're more "numerator on top, denominator on bottom" kind of fractions. So let's see if we can find a simpler one.
If we take 0.27¯¯¯¯¯ and multiply it by 1000, we get 270.77777777... And if we subtract 0.27¯¯¯¯¯ from that, we get 270.77777777... - 0.27¯¯¯¯¯ = 270.50777777...
Now, if we set x = 0.27¯¯¯¯¯, we have 1000x - x = 270.50777777... - 0.27¯¯¯¯¯, which simplifies to 999x = 270.50777777...
Dividing both sides by 999, we get x = 0.2705075075...
So the rational number in simplest form that approximates 0.27¯¯¯¯¯ is Start Fraction 2705 over 10000 end fraction.
To convert 0.27¯¯¯¯¯ to a rational number in simplest form, we can express it as a fraction.
0.27¯¯¯¯¯ means that the digit 27 repeats infinitely. To convert this to a fraction, we can denote it as x, and subtract it from 10x to eliminate the repeating decimal:
10x = 27.272727...
x = 0.272727...
Now, we subtract x from 10x:
10x - x = 27.272727... - 0.272727...
9x = 27
x = 27/9
Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9 in this case:
27/9 = 3/1
Therefore, 0.27¯¯¯¯¯ as a rational number in simplest form is 3/1.
To convert a recurring decimal like 0.27¯¯¯¯¯ into a rational number in its simplest form, we need to express it as a fraction.
To begin, we can let x = 0.27¯¯¯¯¯ and multiply both sides of the equation by 100 (to move the decimal point two places to the right) to eliminate the recurring part:
100x = 27.¯¯¯¯¯
Next, we subtract the original equation from this new equation to eliminate the non-recurring part:
100x - x = 27.¯¯¯¯¯ - 0.27¯¯¯¯¯
Which simplifies to:
99x = 27
Now, we divide both sides of the equation by 99 to solve for x:
x = 27/99
Converting this fraction into its simplest form, we find that the rational number equivalent to 0.27¯¯¯¯¯ is:
Start Fraction 27 over 99 end fraction