To find the equivalent rational number for the recurring decimal 5.454545..., we need to follow a few mathematical steps.
Let's denote the decimal as 'x'. So, x = 5.454545...
We know that the decimal part is recurring every two digits, hence we can express x as the following algebraic equation:
x = 5 + 0.454545...
We can then multiply this equation by 100 (since two digits are repeating) to shift the decimal point two places to the right. This gives us:
100x = 545.454545...
Now, subtract the initial equation (x = 5.454545...) from the one we just got:
100x - x = 545.454545... - 5.454545...
99x = 540
After rearranging the equation to find 'x', we get:
x = 540 / 99
If we divide these two numbers, we get the numerical value of x, which is approximately:
x = 5.505050505050505
So, the recurring decimal 5.454545... is approximately equivalent to the rational number 540/99 or in decimal form, 5.505050505050505.