Question

If it takes 45 minutes for 4 people to paint 5 walls. How many minutes does it take 6 people to paint 4 walls ?

Answer 1

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 answer:
We have been given the relation between three variables where we assume the number of people as $q$, total time in minutes as t and number of walls as $r$. We know the relations where time is inversely proportional with number of people and directly proportional with number of walls.The inversely proportional number is actually directly proportional to the inverse of the given number.

It’s given $t$ varies directly as $r$ which gives $t\propto r$. Also, $t$ varies inversely as $q$ which can be expressed as $t\propto \dfrac{1}{q}$.So, the combined relation is $t\propto \dfrac{r}{q}$.To get rid of the proportionality we use the proportionality constant which gives $t=k\dfrac{r}{q}$.
Here, the number $k$ is the proportionality constant.It’s given $t=45,r=5,q=4$.
We put the values in the equation $t=k\dfrac{r}{q}$ to find the value of $k$.
So, $45=k\dfrac{5}{4}$. Simplifying we get
\[45=k\dfrac{5}{4} \\
\Rightarrow k=\dfrac{4\times 45}{5}\\
\Rightarrow k =4\times 9\\
\Rightarrow k =36\]
Therefore, the equation becomes with the value of k as $t=36\dfrac{r}{q}$.Now we find the value of t for $r=4,q=6$.So,
$t=36\times \dfrac{4}{6}
\therefore t=24$

Therefore, it takes 24 minutes for 6 people to paint 4 walls.

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.