Convert 0.27 ̄ ̄ ̄ to a rational number in simplest form.
3/11
1/27
27/100
27/99
0.27 ̄ ̄ ̄ can be written as 27/99 in simplest form.
To convert the decimal 0.27̄̄̄ to a rational number in simplest form, we can use a variable.
Let x = 0.27̄̄̄
Multiply both sides of the equation by 100, since there are two recurring digits after the decimal point:
100x = 27.̄̄̄
Now, subtract x from 100x to eliminate the recurring decimal:
100x - x = 27.̄̄̄ - 0.̄̄̄27̄̄̄
Simplifying the left side:
(100 - 1)x = 27.̄̄̄
99x = 27.̄̄̄
To obtain a rational number, divide both sides of the equation by 99:
(99x)/99 = 27.̄̄̄/99
x = 27/99
Hence, 0.27̄̄̄ can be converted to the rational number 27/99 in simplest form.
To convert the repeating decimal 0.27 ̄ ̄ ̄ to a rational number, we can represent it as a fraction where the numerator is the repeating part of the decimal and the denominator is composed of the same number of nines as the number of repeating digits.
Step 1: Assign a variable to the repeating decimal.
Let x = 0.27 ̄ ̄ ̄
Step 2: Multiply both sides of the equation by 1000 (to move the decimal point three places to the right) to eliminate the repeating part.
1000x = 270.27 ̄ ̄ ̄
Step 3: Subtract the original equation from the equation obtained in step 2 to eliminate the repeating part.
1000x - x = 270.27 ̄ ̄ ̄ - 0.27 ̄ ̄ ̄
999x = 270
Step 4: Solve for x by dividing both sides of the equation by 999.
x = 270/999
To simplify the fraction 270/999, we can find the Greatest Common Divisor (GCD) of the numerator and denominator and divide both by the GCD.
Step 5: Find the GCD of 270 and 999.
The GCD of 270 and 999 is 9.
Step 6: Divide both the numerator and denominator by 9.
270/9 / 999/9 = 30/111
So, 0.27 ̄ ̄ ̄ is equal to 30/111, which is the simplest form of the rational number.