Question

In the month of January, the railway police caught $4000$ ticketless travelers. In February, the number rose by $5\% $. However, due to constant vigilance by the police and the railway staff, the number reduced by $5\% $ and in April it further reduced by $10\% $. The total number of ticketless travelers caught in the month of April was
A) $3125$
B) $3255$
C) $3575$
D) $3591$

Answer 1

Hint:In the above question we are given the number of ticketless travelers in month of January and we are given the $\% $ increase in ticketless travelers in February, so by using formula-$\% $ change $ = 100 \times \dfrac{{{\text{change}}}}{{{\text{original value}}}}$.We will find the total number of ticketless travelers in February. Now again using $\% $ change $ = 100 \times \dfrac{{{\text{change}}}}{{{\text{original value}}}}$ for number of ticketless travelers in month of February and $\% $ decrease in number of ticketless travelers in march, we will find the total number of ticketless travelers in march. Similarly, we find the required value, the total number of ticketless travelers in April using the $\% $ decrease in the month of April and the total number of ticketless travelers in march.

Complete step-by-step answer:
We are given that-
Total number of ticketless travelers in January is $4000$ ……………….(1)
In February increase in number of ticketless travelers is by $5\% $ ……………….(2)
In march decrease in number of ticketless travelers is by $5\% $ ……………….(3)
In April decrease in number of ticketless travelers is by $10\% $ ……………….(4)
And we have to find the number of ticketless travelers in the month of April.
So, in order to solve the question we will proceed by finding the number of ticketless travelers in February, March and April one by one.
Therefore, now let’s consider (2),
Here we have
In February the increase in the number of ticketless travelers is by $5\% $.
Let change or increase in the month of February be ${c_1}$.
We know any $\% $ change is equal to-
 $\% $ change $ = 100 \times \dfrac{{{\text{change}}}}{{{\text{original value}}}}$ ……………….(5)
Now using (5), where $\% {\text{ increase = 5\% }}$,
$5 = 100 \times \dfrac{{{c_1}}}{{{\text{original value}}}}$
We know here original value is total number of ticketless travelers in January, using (1)
$5 = 100 \times \dfrac{{{c_1}}}{{4000}}$
Now solving this we get
${c_1} = 200$ ……………….(6)
Now the increase in the number of ticketless travelers is $200$ in the month of February. So, the total number of ticketless travelers in month of February is
$ = {\text{number of ticketless travelers in january + increase in number of ticketless travelers in february}}$
Now substituting values from (1), we get
${\text{ = 4000 + }}{{\text{c}}_1}$
Now substituting values from (6), we get
So, total number of ticketless travelers in month of February $ = 4000 + 200 = 4200$ ……………….(7)
Now let’s consider (3)
In march decrease in number of ticketless travelers is by $5\% $
Now here the decrease is corresponding to the month of February.
Now using (7), we get
So original value or initial value for march month is $4200$ ……………….(8)
Let the change or decrease in the number of ticketless travelers in the month of march be ${c_2}$.
Now again using (5), $\% $ change $ = 100 \times \dfrac{{{\text{change}}}}{{{\text{original value}}}}$ and the above values, where $\% {\text{ decrease = 5}}$
 ${\text{5 = 100}} \times \dfrac{{{c_2}}}{{{\text{original value}}}}$
Now, using (8), we get,
${\text{5 = 100}} \times \dfrac{{{c_2}}}{{4200}}$
Now solving this further, we get
${c_2} = 210$ ……………….(9)
Now the decrease in the number of ticketless travelers is by $210$ in the month of march. So, the total number of ticketless travelers in month of march is
$ = {\text{Number of ticketless travelers in february - decrease in number of ticketless travelers in march}}$
Now substituting values from (7), we get
${\text{ = 4200 - }}{{\text{c}}_2}$
Now substituting values from (9), we get
So, total number of ticketless travelers in month of march \[ = 4200 - 210 = 3990\] ……………….(10)
Now let’s consider (4)
In April decrease in number of ticketless travelers is by $10\% $
Now here the decrease is corresponding to the month of march.
Now using (10), we get
So original value or initial value for April month is $3990$ ……………….(11)
Let the change or decrease in the number of ticketless travelers in the month of march be ${c_3}$.
Now again using (5), $\% $ change $ = 100 \times \dfrac{{{\text{change}}}}{{{\text{original value}}}}$ and the above values, where $\% {\text{ decrease = 10}}$
${\text{10 = 100}} \times \dfrac{{{c_3}}}{{{\text{original value}}}}$
Now, using (11), we get,
${\text{10 = 100}} \times \dfrac{{{c_3}}}{{3990}}$
Now solving this further, we get
${c_3} = 399$ ……………….(12)
Now the decrease in the number of ticketless travelers is $399$ in April. So, the total number of ticketless travelers in month of April is
$ = {\text{Number of ticketless travelers in march - decrease in number of ticketless travelers in april}}$
Now substituting values from (11), we get
${\text{ = 3990 - }}{{\text{c}}_3}$
Now substituting values from (12), we get
So, total number of ticketless travelers in month of April \[ = 3990 - 399 = 3591\]

So, the correct answer is “Option D”.

Note:In the above question students generally make mistakes in taking original value or initial value.
They usually take the initial value to be $4000$ in each case, which is completely wrong because the initial value for each month is corresponding to its preceding month. For example, the initial value for the month of February is the total number of ticketless travels in January.
And the initial value for the month of march is the total number of ticketless travelers in February.
Similarly, the initial value for the month of April is the total number of ticketless travelers in march.
Also, to solve the above question we must know the formula:$\% $ change $ = 100 \times \dfrac{{{\text{change}}}}{{{\text{original value}}}}$.