Convert 11.42424242... to a rational expression in the form of a over b where b ≠ 0.
11 42/99 = 11 14/33 = 377/33
11 42/99
To convert the decimal number 11.42424242... to a rational expression, we can determine the repeating pattern.
Let's call the repeating pattern "x." Then we can express 11.42424242... as:
11.42x
Now, to remove the decimal point, we can multiply this expression by a power of 10 that shifts the repeating pattern to the left of the decimal point:
100x
Since x represents the repeating pattern, we can subtract the original number (11.42424242...) from the shifted number (11.42x) to eliminate the repeating part:
11.42x - 11.42424242... = 0.01x
Now we have the equation:
0.01x = 11.42x - 11.42424242...
Next, we multiply both sides of the equation by a power of 10 that eliminates the decimal point:
10 * 0.01x = 10 * (11.42x - 11.42424242...)
0.1x = 114.2x - 114.2424242...
Now, let's isolate the variable by subtracting 114.2x from both sides:
0.1x - 114.2x = -114.2424242...
-114.1x = -114.2424242...
Finally, to express the rational expression in the form of a fraction, we can divide both sides of the equation by -114.1:
x = (-114.2424242...) / (-114.1)
Simplifying the fraction gives us the rational expression in the form of a over b:
x = 114.2424242... / 114.1
Therefore, 11.42424242... can be expressed as a rational expression in the form of 114.2424242... / 114.1.
To convert the recurring decimal 11.42424242... to a rational expression in the form of a over b, where b ≠ 0, we can follow these steps:
Step 1: Assign a variable to the recurring part
Let's call the recurring part of the decimal x. In this case, x = 0.42424242...
Step 2: Multiply the decimal by a power of 10 to eliminate the recurring part
To remove the recurring part, we can multiply both sides of the equation x = 0.42424242... by 100. This gives us 100x = 42.42424242...
Step 3: Subtract the original equation from the result in Step 2
Now, we can subtract the original equation (x = 0.42424242...) from the result of Step 2 (100x = 42.42424242...). This gives us:
100x - x = 42.42424242... - 0.42424242...
Simplifying the equation, we have:
99x = 42
Step 4: Solve for x
To find the value of x, we divide both sides of the equation by 99:
x = 42 / 99
Step 5: Simplify the fraction
The fraction 42/99 can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 3:
x = 14 / 33
Therefore, the rational expression for the recurring decimal 11.42424242... is 14/33.