Convert 0.27¯¯¯¯¯
to a rational number in simplest form
To convert the repeating decimal 0.27¯¯¯¯¯ to a rational number in simplest form, we can use the method of substitution. Let x = 0.27¯¯¯¯¯.
To find the value of x, we multiply both sides of the equation by 100 to eliminate the repeating decimal:
100x = 27.¯¯¯¯¯
Next, we subtract the original equation from this newly obtained equation:
100x - x = 27.¯¯¯¯¯ - 0.27¯¯¯¯¯
Simplifying:
99x = 27.72
To remove the decimal, we multiply both sides of the equation by 100:
99x * 100 = 27.72 * 100
9,900x = 2,772
Dividing both sides of the equation by 9,900:
x = 2,772/9,900
To simplify this rational number, we can divide both the numerator and denominator by their greatest common divisor, which is 36:
x = (2,772 ÷ 36) / (9,900 ÷ 36)
Simplifying:
x = 77/275
To convert 0.27¯¯¯¯¯¯ to a rational number in simplest form, follow these steps:
Step 1: Let's represent 0.27¯¯¯¯¯¯ as x.
x = 0.27¯¯¯¯¯¯
Step 2: Multiply both sides of the equation by 100 to remove the repeating decimal.
100x = 27.777777...
Step 3: Subtract the original equation from the equation obtained in step 2 to eliminate the repeating part.
100x - x = 27.777777... - 0.27¯¯¯¯¯¯
Simplifying the equation:
99x = 27.507777...
Step 4: Divide both sides of the equation by 99 to solve for x.
99x / 99 = 27.507777... / 99
Simplifying further:
x = 0.277777...
Step 5: Now, we can express x as a fraction.
Let's represent x as a fraction:
x = 0.277777...
Step 6: Multiply both sides of the equation by 10 to move the decimal point to the right.
10x = 2.777777...
Step 7: Subtract the original equation from the equation obtained in step 6 to eliminate the repeating part.
10x - x = 2.777777... - 0.277777...
Simplifying the equation:
9x = 2.5
Step 8: Divide both sides of the equation by 9 to solve for x.
9x / 9 = 2.5 / 9
Simplifying further:
x = 2.5/9
Therefore, 0.27¯¯¯¯¯¯ can be expressed as the rational number 2.5/9 in simplest form.