Question

\[ x = \frac{100}{27.50} \approx 3.63636 \] as a real fraction

Answer 1

To express the decimal \( 3.63636 \) as a fraction, we start by analyzing the repeating part of the decimal. The decimal can be written as \( 3.63636 \ldots \), where the "36" repeats indefinitely.

Let's define the repeating decimal:

\[
x = 3.63636 \ldots
\]

Next, we can separate the whole number part from the fractional part:

\[
x = 3 + 0.63636 \ldots
\]

Now, let's focus on the fractional part \( 0.63636 \ldots \). We can set this part as \( y \):

\[
y = 0.63636 \ldots
\]

To eliminate the repeating decimal, multiply \( y \) by \( 100 \) (since the repeating part has two digits):

\[
100y = 63.63636 \ldots
\]

Now we have two equations:

1. \( y = 0.63636 \ldots \)
2. \( 100y = 63.63636 \ldots \)

Next, we subtract the first equation from the second:

\[
100y - y = 63.63636 \ldots - 0.63636 \ldots
\]

This simplifies to:

\[
99y = 63
\]

Now, we can solve for \( y \):

\[
y = \frac{63}{99}
\]

We can simplify this fraction by finding the greatest common divisor (GCD) of 63 and 99, which is 9:

\[
y = \frac{63 \div 9}{99 \div 9} = \frac{7}{11}
\]

Now we can substitute \( y \) back into the equation for \( x \):

\[
x = 3 + y = 3 + \frac{7}{11}
\]

To combine into a single fraction, convert \( 3 \) to a fraction with a denominator of \( 11 \):

\[
3 = \frac{33}{11}
\]

Thus,

\[
x = \frac{33}{11} + \frac{7}{11} = \frac{33 + 7}{11} = \frac{40}{11}
\]

Therefore, the real fraction representation of \( 3.63636 \ldots \) is:

\[
\boxed{\frac{40}{11}}
\]