 QUESTION

# Abdul travelled 300 km by train and 200 km by taxi, it took him 5 hours 30 minutes. But if he travels 260 km by train and 240 km by taxi he takes 6 minutes longer. Find the speed of the train and that of the taxi.

Hint: In this question, we will use the basic concept of speed-distance formula, which states that, speed of an object can be represented as the ratio of distance covered by the object to the time taken by the object to cover the respective distance. Mathematically, we can represent it as$\text{speed of an object =}\dfrac{\text{distance covered by the object}}{\text{time taken by the object}}$.
In this question, we have to find the speed of train and taxi, so assume speed of train be ‘x’ and speed of taxi be ’y’.
We know that, speed of an object can be calculated as, $\text{speed of an object =}\dfrac{\text{distance covered by the object}}{\text{time taken by the object}}$
As we are given with the time taken by both then vehicles to cover a total distance of 500 km, so we will write the speed formula as $\text{time taken by an object =}\dfrac{\text{distance covered by the object}}{\text{speed of the object}}$
From the question, we can see that when Abdul travels 300 km by train and 200 km by taxi, he reaches his destination in 5 hours 30 minutes. This can be written as the sum of time taken by train and time taken by is equal to 5 hours 30 minutes.
We can write it mathematically as, $\text{time taken by train + time taken by taxi = 5 hours 30 minutes}$
Now, we are using $\text{time taken by an object =}\dfrac{\text{distance covered by the object}}{\text{speed of the object}}$, therefore, we will get,
$\dfrac{\text{distance covered by the train}}{\text{speed of the train}}+\dfrac{\text{distance covered by the taxi}}{\text{speed of the taxi}}=5+\dfrac{30}{60}\text{ hours}$
We can further write it as
$\dfrac{300}{\text{ x}}+\dfrac{200}{y}=5.5\text{ hours}$
$\Rightarrow 300y+200x=5.5xy$
Now, we will multiply the above equation by 2, so we get,
$\Rightarrow 600y+400x=11xy......\left( i \right)$
Similarly, when Abdul travels 260 km by train and 240 km by taxi, he reaches his destination 6 minutes later, so we can write the equation as
$\dfrac{260}{\text{ x}}+\dfrac{240}{y}=5.6\text{ hours}$
$\Rightarrow 260y+240x=5.6xy$
Now, we will multiply the above equation by 5, so, we will get,
$\Rightarrow 1300y+1200x=28xy......\left( ii \right)$
Now, we will use elimination method to get the values of ‘x’ and ‘y’. For elimination, we will multiply (i) by 3, we will get,
$1800y+1200x=33xy$
Now, we will subtract the above equation from (ii), we will get,
$1300y+1200x-1800y-1200x=28xy-33xy$
$\Rightarrow 1300y-1800y=28xy-33xy$
$\Rightarrow -500y=-5xy$
We will divide both sides by -5y, so, we get,
$\Rightarrow x=100$
Therefore, we get the value of x equal to 100 km/hr. Now, we will put value of x in equation (i), which implies, $600y+400\left( 100 \right)=11\left( 100 \right)y$
$\Rightarrow 600y+40000=1100y$
$\Rightarrow 500y=40000$
$\Rightarrow y=80$
Therefore, we get the value of y equal to 80 km/hr.
Hence, we get that the speed of the train is 100 km/hr and the speed of the taxi is 80 km/hr.

Note: In this question, remember we can apply elimination in two ways, one by taking L.C.M of x and y and another way of elimination is that we can assuming $\dfrac{1}{x}=p$ and $\dfrac{1}{y}=q$, and after finding values of p and q , simply we can reciprocate them, to find the values of x and y.