Question

Convert 0.72.... to a rational number in simplest form.

Answer 1

To convert the repeating decimal \(0.72\ldots\) (which can be represented as \(0.727272...\)) into a rational number, we can use the following method:

Let \(x = 0.727272...\).

Next, we can multiply both sides of this equation by 100, which is a power of 10 that shifts the decimal point two places to the right:

\[
100x = 72.727272...
\]

Now, we can see that the decimal part \(0.727272...\) on the right-hand side is the same as our original equation for \(x\):

\[
100x = 72 + x
\]

Now we can isolate \(x\) by subtracting \(x\) from both sides:

\[
100x - x = 72
\]
\[
99x = 72
\]

Now, we solve for \(x\) by dividing both sides by 99:

\[
x = \frac{72}{99}
\]

Next, we simplify \(\frac{72}{99}\). We can find the greatest common divisor (GCD) of 72 and 99.

The prime factorization of 72 is:
\[
72 = 2^3 \times 3^2
\]

The prime factorization of 99 is:
\[
99 = 3^2 \times 11
\]

The GCD of 72 and 99 is \(3^2 = 9\).

Now we divide both the numerator and the denominator by their GCD:

\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\]

Thus, the simplest form of \(0.727272...\) as a rational number is:

\[
\boxed{\frac{8}{11}}
\]