Convert 0.27¯¯¯¯¯
to a rational number in simplest form.
To convert the recurring decimal 0.27¯¯¯¯¯ to a rational number, we can use the following algebraic process:
Let x = 0.27¯¯¯¯¯
Then, 100x = 27.27¯¯¯¯¯
Subtracting the first equation from the second gives:
100x - x = 27.27¯¯¯¯¯ - 0.27¯¯¯¯¯
99x = 27
Dividing both sides by 99 gives:
x = 27/99
Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, which is 9, gives:
x = 3/11
Therefore, the rational representation of the recurring decimal 0.27¯¯¯¯¯ is 3/11 in its simplest form.
To convert 0.264¯¯¯¯¯¯¯¯ to a rational number, you would set x equal to 0.264¯¯¯¯¯¯¯¯ and then multiply both sides of the equation by what number?
To convert 0.264¯¯¯¯¯¯¯¯ to a rational number, you would set x equal to 0.264¯¯¯¯¯¯¯¯ and then multiply both sides of the equation by 1000.
This is because 0.264¯¯¯¯¯¯¯¯ has 3 recurring digits, so by multiplying both sides by 10 raised to the power of the number of recurring digits (which is 3 in this case), we will be able to eliminate the recurring decimal pattern.
So, multiplying both sides of the equation by 1000 gives:
1000x = 264.264¯¯¯¯¯¯¯¯
Subtracting the first equation from the second gives:
1000x - x = 264.264¯¯¯¯¯¯¯¯ - 0.264¯¯¯¯¯¯¯¯
999x = 264
Therefore, the number we get after multiplying both sides by 1000 is 264.264¯¯¯¯¯¯¯¯, and the equation simplifies to 999x = 264.
However, it is important to note that this method will not yield a rational number for 0.264¯¯¯¯¯¯¯¯, as the number cannot be expressed as a fraction in simplest form.
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator
hey bot
Bot, how did you get 99x? I don't understand why its denominator is 99, can you explain?
For the first one.
Hey bot. Can you help me?
To convert 0.264¯¯¯¯¯¯¯¯ to a rational number, you would set x equal to 0.264¯¯¯¯¯¯¯¯ and then multiply both sides of the equation by what number?
1,000
10
999
100
To convert a repeating decimal to a rational number in simplest form, we need to understand that the repeating decimal represents a fraction.
Let's call the repeating decimal x. Since the decimal part is repeating, we can write it as x = 0.272727... (where the '27' repeats indefinitely).
Step 1: Multiply x by a power of 10 to eliminate the repeating part.
10x = 2.727272...
Step 2: Subtract the original equation from the equation obtained in Step 1.
10x - x = 2.727272... - 0.272727...
9x = 2.454545...
Step 3: Solve for x by dividing both sides of the equation by 9.
x = 2.454545... / 9
Now, to convert this fraction to simplest form:
Step 1: Recognize that the numerator of the fraction is 2.454545... and the denominator is 9.
Step 2: Let's define a variable, y, as the repeating decimal portion (0.454545...).
Step 3: Multiply y by 100 to shift the decimal two places to the right.
100y = 45.454545...
Step 4: Subtract the original equation from the equation obtained in Step 3.
100y - y = 45.454545... - 0.454545...
99y = 45
Step 5: Solve for y by dividing both sides of the equation by 99.
y = 45 / 99 = 5 / 11
Step 6: Substitute the value of y back into the original equation (x = 2.454545... / 9)
x = 2.y / 9 = 2(5 / 11) / 9 = 10 / 99
Therefore, the repeating decimal 0.27¯¯¯¯¯¯¯ can be expressed as the rational number 10/99 in simplest form.