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If Sam types at a rate of $x$ words per minute, calculate how many minutes, in terms of $x$ will it take him to type 500 words?
(a) $500x$
(b) $500-x$
(c) $500+x$
(d) $\dfrac{500}{x}$

Last updated date: 16th Jul 2024
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Hint: We first try to form the proportionality equation for the variables. We take an arbitrary constant. We use the given values of the variables to find the value of the constant. Finally, we put the constant’s value to find the equation.

Complete step-by-step solution:
We have been given the relation between two variables where we assume number of typed words as r and time as t.
The inversely proportional number is actually directly proportional to the inverse of the given number. The relation between r and t is inverse relation.
It’s given r varies directly as t which gives $r\propto t$.
To get rid of the proportionality we use the proportionality constant which gives $r=kt$.
Here, the number k is the proportionality constant. It’s given $r=x$ when $t=1$.
We put the values in the equation $r=kt$ to find the value of k. So, $x=k$.
Therefore, the equation becomes with the value of k as $r=xt$.
Now we simplify the equation to get the value of t for number of typed words being 500
\[500=xt \Rightarrow t=\dfrac{500}{x}\].
Therefore, the required time in minutes is \[\dfrac{500}{x}\].
The correct option is D.

Note: In a direct proportion, the ratio between matching quantities stays the same if they are divided. They form equivalent fractions. In an indirect (or inverse) proportion, as one quantity increases, the other decreases. In an inverse proportion, the product of the matching quantities stays the same.