Question

A tractor is moving with a speed of 20 km/hr, x km ahead of a truck moving with a speed of 35 km/hr. If it takes 20 minutes for a truck to overtake the tractor, then what is x equal to?
(A) 5 km
(B) 6 km
(C) 7 km
(D) 8 km

Answer 1

Hint: First of all convert the time 20 minutes into hour using the relation, 60 minutes = 1 hour. Now, use the formula \[\text{Distance=Speed}\times \text{Time}\] , and calculate the distance covered by the tractor and the truck in \[\dfrac{1}{3}\] hour. As the tractor is x kilometers ahead from the truck and to overtake the tractor, the truck has to reach the distance as the tractor reaches the distance after \[\dfrac{1}{3}\] hour . So, \[OB=CD\] . Here, A and C is the starting point of the tractor and truck respectively. Point B and D and is the endpoint where the truck overtakes the tractor. The distance \[OA=x\] . Now, solve it further and get the value of x.

Complete step-by-step answer:
According to the question, it is given that a tractor is moving with a speed of 20 km/hr, x km ahead of a truck moving with a speed of 35 km/hr. If it takes 20 minutes for a truck to overtake the tractor.
The speed of the tractor = 20 km/hr ……………………………..(1)
The speed of the truck = 35 km/hr ………………………………(2)
The time taken by the truck to overtake tractor = 20 minutes …………………….……..(3)
We know the relation between hour and minutes, 60 minutes = 1 hour.
\[\Rightarrow \text{60}\,\text{minutes=1 hour}\]
\[\Rightarrow 1\text{ minutes=}\dfrac{1}{60}\text{ hour}\] …………………………………(4)
Now, from equation (3) and equation (4), we get
The time taken by the truck to overtake tractor = 20 minutes = \[20\times \dfrac{1}{60}\] hour = \[\dfrac{1}{3}\] hour ……………………….(5)
We know the formula, \[\text{Distance=Speed}\times \text{Time}\] ………………………..(6)
From equation (1), we have the speed of the tractor.
Using the formula shown in equation (6), we can get the distance covered by the tractor in \[\dfrac{1}{3}\] hour.
The distance covered by the tractor in \[\dfrac{1}{3}\] hour, AB = \[20km/hr\times \dfrac{1}{3}hr=\dfrac{20}{3}km\].
The tractor is already x distance ahead from the point of the start of the truck.
So, the total distance from point O, OB = \[\left( x+\dfrac{20}{3} \right)km\]………………………………(7)
From equation (2), we have the speed of the truck.
Using the formula shown in equation (6), we can get the distance covered by the truck in \[\dfrac{1}{3}\] hour.
The distance covered by the truck in \[\dfrac{1}{3}\] hour, CD = \[35km/hr\times \dfrac{1}{3}hr=\dfrac{35}{3}km\] ………………………………(8)
To overtake the tractor, the truck has to reach at the distance as the tractor reaches the distance after \[\dfrac{1}{3}\] hour .

From the figure, we can say that the distance, \[OB=CD\] ………………………………..(9)
Now, from equation (7), equation (8), and equation (9), we get
\[\begin{align}
  & \Rightarrow OB=CD \\
 & \Rightarrow OA+AB=CD \\
 & \Rightarrow x+\dfrac{20}{3}=\dfrac{35}{3} \\
 & \Rightarrow x=\dfrac{35}{3}-\dfrac{20}{3} \\
 & \Rightarrow x=\dfrac{35-20}{3} \\
 & \Rightarrow x=\dfrac{15}{3} \\
 & \Rightarrow x=5 \\
\end{align}\]
Therefore, the value of x is 5 km.
Hence, option (A) is the correct one.

Note: In this question, one might calculate the distance covered by the tractor by multiplying the speed and the time directly. Like, \[\text{Distance=20km/hr}\times 20\text{minutes}\] . This is wrong because the unit of time is in minutes while the unit of the speed is in km/hr. So, we have to convert the time in an hour using the relation, 60 minutes = 1 hour.