Convert .454545454545........ to a rational number in simplest form.
Let x = 0.454545454545...
Multiplying both sides of the equation by 100, we get:
100x = 45.454545454545...
Subtracting x from 100x, we have:
100x - x = 45.454545454545... - 0.454545454545...
Simplifying, we find:
99x = 45
Dividing both sides by 99, we get:
x = 45/99
To simplify this fraction, we can divide the numerator and denominator by their greatest common divisor, which is 9:
x = (45/9) / (99/9)
Simplifying further, we have:
x = 5/11
Therefore, 0.454545454545... is equal to 5/11 in simplest form.
To convert the repeating decimal .454545454545... to a rational number in simplest form, we can use the concept of geometric series.
Step 1: Let x = 0.454545454545...
Step 2: Multiply x by 100, as we have two digits after the decimal point: 100x = 45.45454545...
Step 3: Subtract x from 100x to eliminate the repeating part: 100x - x = 45.45454545... - 0.45454545...
Simplifying the equation, we have:
99x = 45
Step 4: Solve for x by dividing both sides of the equation by 99:
x = 45/99
Step 5: Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD):
GCD(45, 99) = 9
Dividing both 45 and 99 by 9, we get:
x = 5/11
Therefore, the repeating decimal .454545454545... is equal to the rational number 5/11 in simplest form.
To convert the recurring decimal .454545454545... to a rational number, we can use the concept of algebraic manipulation.
Let's represent the given number as x = 0.454545454545...
Multiplying both sides of the equation by 100, we get:
100x = 45.454545454545...
Next, we subtract the original equation from this new equation to eliminate the recurring part:
100x - x = 45.454545454545... - 0.454545454545...
Simplifying both sides, we have:
99x = 45
Now, we can solve for x by dividing both sides of the equation by 99:
x = 45/99
To simplify this fraction, we can find the greatest common divisor (GCD) of the numerator and denominator, which is 9. Dividing both the numerator and denominator by 9 yields:
x = 5/11
Therefore, the rational number equivalent to the recurring decimal .454545454545... is 5/11 in its simplest form.