# Consider the equation $\dfrac{k}{x} = 12$ where $k$ is any number between $20$ and $65$ and $x$ is a positive integer. What are the possible values of $x$?

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Hint: We will first write the given equation $\dfrac{k}{x} = 12$ as $k = 12x$. As given in the question, $k$ is any number between $20$ and $65$. So, using this and substituting $k = 12x$, we will write $20 < 12x < 65$. Then we will simplify it. Given, $x$ is a positive integer, so we will further simplify it using the definition of positive integer to find the possible values of $x$.

The given equation is $\dfrac{k}{x} = 12$.
On cross multiplication, we can write the given equation as $k = 12x$.
Now, as given in the question, $k$ is any number between $20$ and $65$. So, we can write,
$\Rightarrow 20 < k < 65$
On putting $k = 12x$, we get
$\Rightarrow 20 < 12x < 65$
Now, as we know, the inequality remains the same on dividing by a positive number. So, on dividing by $12$, we get
$\Rightarrow \dfrac{{20}}{{12}} < x < \dfrac{{65}}{{12}}$
Cancelling the common terms, if any, from the numerator and the denominator, we get
$\Rightarrow \dfrac{5}{3} < x < \dfrac{{65}}{{12}}$
On calculating, we get
$\Rightarrow 1.67 < x < 5.42$
But, given in the question that $x$ is a positive integer. As we know that the first integer that will come just after $1.67$ is $2$ and the first integer that will come just before $5.42$ is $5$.
So, we can rewrite it as,
$\Rightarrow 2 \leqslant x \leqslant 5$
Therefore, the possible values of $x$ are $2$, $3$, $4$ and $5$.

Note: An integer can never be a fraction, a decimal, or a percent. A positive integer are those numbers that are prefixed with a plus sign $( + )$. But, most of the time positive numbers are represented simply as numbers without the plus sign. Also, zero is a neutral integer because it can neither be a positive nor a negative integer i.e., zero has no $+ ve$ sign or $- ve$ sign.