Question

The number of minutes needed to solve an exercise set of variation problems varies directly as the number of problems and inversely as the number of people working on the solutions. It takes 4 people 36 minutes to solve 18 problems. How many minutes will it take 6 people to solve 42 problems?
a.54 minutes
b.56 minutes
c.60 minutes
d.None of these

Answer 1

Hint: As we know that when the number of problems increases the time also gets increased as it varies directly and the number of people gets decreased as it varies inversely.

Complete step-by-step answer:
We can easily determine the relationship between the number of people, number of problems and the time taken by them and after substituting the given data we can have the time taken when the problem and number of people get increased.
Consider the data that is as the number of people increases from 4 to 6 and number of problems also increases from 18 to 42 thus, we need to find the time when the previous time was 36 minutes.
Also, we know that the minutes is directly proportional to the number of problems and the minutes is inversely proportional to the number of people.
Hence, we can have the relationship between all three as:
\[{\text{Minutes}} = k\left( {\dfrac{{{\text{Number of problems}}}}{{{\text{Number of people}}}}} \right)\]
Where \[k\] is the constant of proportionality.
Now, consider the first set that is 36 minutes, 18 problems and 4 people to do it.
We will substitute the values in the derived relationship to evaluate the value of \[k\].
Thus, we get,
\[
   \Rightarrow {\text{36}} = k\left( {\dfrac{{{\text{18}}}}{{\text{4}}}} \right) \\
   \Rightarrow 36 = k\left( {\dfrac{9}{2}} \right) \\
   \Rightarrow k = 36\left( {\dfrac{2}{9}} \right) \\
   \Rightarrow k = 4\left( 2 \right) \\
   \Rightarrow k = 8 \\
\]
Now, we will substitute the value of \[k\] in the derived relationship.
That is
\[{\text{Minutes}} = 8\left( {\dfrac{{{\text{Number of problems}}}}{{{\text{Number of people}}}}} \right)\]
Now, consider the second set, that is we need to calculate the time when the number of problems are increased to 42 and the number of people are increased to 6.
Thus, we have,
\[
  {\text{Minutes}} = 8\left( {\dfrac{{{\text{42}}}}{{\text{6}}}} \right) \\
   = 8\left( 7 \right) \\
   = 56 \\
\]
Thus, we get that the time taken to solve 42 problems is 56 minutes when 6 people are working on it.
Hence, the correct option is B.

Note: As we can recheck through the relationship that when the number of problems has been increased then the number of minutes also got increased, that is it varies directly. The constant of proportionality is necessary to find the number of minutes.