Question

When the repeating decimal $10.363636$ is written in simplest functional form $\dfrac{p}{q}$, find the value of $p+q$

Answer 1

Hint: To convert the given repeating decimal value in fraction form we will start by putting the value equal to $x$. Then we will multiply 100 on both sides and subtract the new obtained equation by the old equation. Finally we will simplify the obtained equation to get the desired answer.

Complete step-by-step solution:
We have to write the decimal number given below into simplest functional form:
$10.363636$
Let us take that the above number is equal to some variable as below:
$x=10.363636$……$\left( 1 \right)$
Now multiply both sides by 100 as below:
$100\times x=10.363636\times 10$
$100x=1036.363636$….$\left( 2 \right)$
Next subtract equation (2) from equation (1) as below:
$\begin{align}
  & 100x-x=1036.3636363-10.363636 \\
 & \Rightarrow 99x=1026 \\
 & \Rightarrow x=\dfrac{1026}{99} \\
\end{align}$
Now we can simplify the above fraction by dividing numerator and denominator by 3 as follows:
$\begin{align}
  & x=\dfrac{\dfrac{1026}{3}}{\dfrac{99}{3}} \\
 & \therefore x=\dfrac{114}{11} \\
\end{align}$
So $10.363636$ is written as $\dfrac{114}{11}$ in the simplest fraction form.
Comparing $\dfrac{114}{11}$ by $\dfrac{p}{q}$ we get,
$\begin{align}
  & p=114 \\
 & q=11 \\
\end{align}$….$\left( 3 \right)$
Finally we have to find the value of:
$p+q$
Equate the value from equation (3) above we get,
$\begin{align}
  & p+q=114+11 \\
 & \therefore p+q=125 \\
\end{align}$
Hence value of $p+q$ is 125.

Note: A repeating decimal is a way of representing a number whose digits are periodic and the infinitely repeated portion is not zero. We can show that a number is rational if its decimal representation is repeating or terminating. A rational number is the one which can be represented in the form of $\dfrac{p}{q}$ where $q\ne 0$ and the number which can’t be represented this way is known as irrational numbers. Every repeating decimal number satisfies a linear equation with integer coefficients and its unique solution is a rational number.