Question

One day the ratio of the number of first-class and second-class passengers who travelled were in the ratio $1:30$ The ratio of the first- and second-class fares is $3:1$ The total amount collected from the passengers that day was $Rs66000$ . Find the amount collected from the first-class passengers (in Rs):
$\begin{align}
  & (A)Rs3000 \\
 & (B)Rs6000 \\
 & (C)Rs9000 \\
 & (D)Rs12000 \\
\end{align}$

Answer 1

Hint: Problems on ratios can be solved if we consider the ratios to be constant. We will first assume the number of first class and second-class passengers who travelled and their respective fairs using the ratios we are given. Equating the total amount collected from the passengers that day with $66000$ we will be able to get the amount collected from the first-class passengers.

Complete step-by-step answer:
The ratio of the number of first class and second-class passengers who travelled that day is $1:30$
We assume the ratio of the number of first class and second-class passengers who travelled that day to be a constant $x$
Hence, total number of first-class passengers travelled that day is $1x$ or $x$
And the total number of second-class passengers travelled that day is $30x$
The ratio of the first- and second-class fares is $3:1$
We assume the ratio of the first- and second-class fares to be a constant $y$
Hence, the fare of each first-class passenger is $3y$ and the fare of each second-class passengers is $1y$ or $y$
Hence, the total fare gathered from first-class passengers is
$x\times 3y=3xy$
Also, the total fare gathered from second-class passengers is
$30x\times y=30xy$
$\therefore $ As, $xy$ is a constant the ratio of fare gathered from first class and second-class passengers is $3:30$
We can rewrite $3:30$ as $1:10$
Again, assuming the above ratio to be a constant $k$
The total amount of fare gathered from first-class passengers is $1k$ or $k$
And the total amount of fare gathered from second-class passengers is $10k$
The total fare gathered that day from first-class and second-class passengers combined is
$k+10k$
$=11k$
We already know that total amount collected from the passengers that day was Rs. 6600
Hence,
  $11k=66000$
$\Rightarrow k=\dfrac{66000}{11}$
$\Rightarrow k=6000$
Therefore, we conclude that the total amount of fare gathered from first-class passengers that day is $Rs6000$ (B).
So, the correct answer is “Option B”.

Note: We must be careful while assuming the ratios to be a constant as one might get confused with the different constants assumed and make silly mistakes. Also, while working with ratios we must simplify them as much as possible for easy calculation.