Question

In an express train, the number of passengers travelling in A.C. sleeper class, first class and sleeper class are in ratio $1:2:3$ and the fares to each of these classes are in the ratio $5:4:2$ . If the total income from this train is Rs.54000, then the income from the A.C. sleeper class is?
(a) Rs. 11210 (approx)
(b) Rs. 13210 (approx)
(c) Rs. 14210 (approx)
(d) Rs. 16210 (approx)

Answer 1

Hint: First, we will assume the ratio of passengers in 3 different class to be x i.e. $x,2x,3x$ and ratio of fares as y i.e. $5y,4y,2y$ . Then, we will multiply first class ratio the fare of first class, multiplying second class ratio to fare of second class and multiplying third class ratio to the fare of third class. On adding all this, we will equate it with total income of train i.e. Rs.54000. Then we will multiply the value obtained with the income of AC sleeper class i.e. 5xy. We will get the answer.

Complete step by step answer:
Here, we are given the ratio of passengers travelling in 3 different classes and also the ratio of fare of the classes i.e. $1:2:3$ and $5:4:2$ respectively.
We will assume A.C. sleeper class, first class and sleeper class as $ACc=x,fc=2x,sc=3x$ respectively. Also, fare of these classes as $fAC=5y,ffc=4y,fsc=2y$ .
Now, we know that total income from this train is equal to the number of passengers fare of the class. So, from this we can write in mathematical from as
$income=\left( ACc\cdot fAC \right)+\left( fc\cdot ffc \right)+\left( sc\cdot fsc \right)$
Now, we will substitute the values in the above equation, and we will get as
$54000=\left( x\cdot 5y \right)+\left( 2x\cdot 4y \right)+\left( 3x\cdot 2y \right)$
On solving the brackets, we get
$54000=5xy+8xy+6xy$
$54000=19xy$
Now, dividing both sides by 19, we get
$\dfrac{54000}{19}=xy=2842.105$ ……………………………(1)
Now, we have to find income of only AC sleeper class which we can write it as
$ACc\cdot fAC=x\cdot 5y=5xy$
We will put value of equation (1) in the above equation and on solving we get
$5xy=5\cdot 2842.105=Rs.14210.525$
Thus, the income from the A.C. sleeper class is Rs.14210 (approx).
Hence, option (c) is correct.

Note: Remember that total income is obtained by multiplying the number of passengers and fare of the tickets. Sometimes students do addition instead of multiplying which leads to wrong answers. For example $income=\left( ACc+fAC \right)+\left( fc+ffc \right)+\left( sc+fsc \right)$ and on solving we get value as $54000=x+5x+2x+4x+3x+2x=17x$ i.e. $x=3176.4$ . Then on finding income of AC sleeper class as $5x+x=6x=6\left( 3176.4 \right)=Rs.19085.4$ . This is an incorrect answer. So, be clear with the concept and formula while applying.