Convert .454545454545......(repeating decimal) to a rational number in simplest form.

Answer 1

Let $x=0.4545454545\dots$.

To convert this repeating decimal to a fraction, we first observe the repeated part, which is "45". This consists of two digits.

To eliminate the repeating decimal, we can multiply $x$ by 100 (since there are two digits that repeat):

$$100x=45.4545454545\dots$$

Now, we can set up an equation using the original $x$:

$$x=0.4545454545\dots$$ $$100x=45.4545454545\dots$$

Next, we subtract the first equation from the second:

$$100x-x=45.4545454545\dots -0.4545454545\dots$$

This simplifies to:

$$99x=45$$

Now, solve for $x$:

$$x=\frac{45}{99}$$

Next, we simplify the fraction $\frac{45}{99}$. The greatest common divisor (GCD) of 45 and 99 can be found. The prime factorization of both numbers is:

• $45=3^2\times 5$
• $99=3^2\times 11$

The GCD is $3^2=9$.

Now, we divide both the numerator and the denominator by their GCD (9):

$$\frac{45÷9}{99÷9}=\frac{5}{11}$$

Thus, the decimal $0.4545454545\dots$ can be expressed as the fraction:

$$\boxed{{\displaystyle \frac{5}{11}}}$$