To convert the repeating decimal 3.24̅ into a fraction, where the bar indicates that ’24’ repeats indefinitely, we can follow a systematic method.
Let’s denote the repeating decimal as:
x = 3.24̅
This can be rewritten as:
x = 3.24242424…
Now, to eliminate the repeating part, we can multiply both sides of the equation by 100 (since the repeating block ’24’ has two digits):
100x = 324.242424…
Next, we also write the original equation:
x = 3.242424…
Now we have:
- 100x = 324.242424…
- x = 3.242424…
We can subtract the second equation from the first:
100x – x = 324.2424… – 3.242424…
Which simplifies to:
99x = 321
Now, solving for x gives:
x = 321/99
Next, we simplify this fraction. Both the numerator and the denominator can be divided by 3:
x = 107/33
Thus, the rational number equivalent to 3.24̅ is:
107/33