Arithmetic

Time, Speed & Distance

Medium

The distance from $A$ to $B$ is $60 \mathrm{~km}$ . Partha and Narayan start from $A$ at the same time and move towards $B$ . Partha takes four hours more than Narayan to reach $B$ . Moreover, Partha reaches the mid-point of $A$ and $B$ two hours before Narayan reaches $B$ . The speed of Partha, in km per hour, is

$5$

$6$

$4$

$3$

✅ Correct Option: 1

We have two people, Partha and Narayan, traveling the same 60 km route from A to B. The key information:

Partha is slower (takes 4 hours more to complete the journey)

Partha reaches the halfway point (30 km) exactly 2 hours before Narayan finishes the full journey

Let's define:

$S_p$ = Partha's speed (km/h)

$S_n$ = Narayan's speed (km/h)

Remember: Time = Distance ÷ Speed

From "Partha takes 4 hours more than Narayan":

Time for Partha to reach B = $\tfrac{60}{S_p}$

Time for Narayan to reach B = $\tfrac{60}{S_n}$

Since Partha takes 4 hours more:

$\tfrac{60}{S_p} = \tfrac{60}{S_n} + 4$ ...(1)

From "Partha reaches midpoint 2 hours before Narayan reaches B":

Time for Partha to reach midpoint = $\tfrac{30}{S_p}$

Time for Narayan to reach B = $\tfrac{60}{S_n}$

Since Partha reaches the midpoint 2 hours before Narayan reaches B:

$\tfrac{30}{S_p} = \tfrac{60}{S_n} - 2$ ...(2)

Instead of solving these equations individually, let's use subtraction to eliminate one variable:

From equation (1): $\tfrac{60}{S_p} = \tfrac{60}{S_n} + 4$

From equation (2): $\tfrac{30}{S_p} = \tfrac{60}{S_n} - 2$

Subtracting equation (2) from equation (1):

$\tfrac{60}{S_p} - \tfrac{30}{S_p} = \left(\tfrac{60}{S_n} + 4\right) - \left(\tfrac{60}{S_n} - 2\right)$

$\tfrac{30}{S_p} = \tfrac{60}{S_n} + 4 - \tfrac{60}{S_n} + 2$

Notice how the $\tfrac{60}{S_n}$ terms cancel out:

$\tfrac{30}{S_p} = 6$

Therefore: $S_p = 5$ km/h

Partha's speed is 5 km/h

CAT 2020 Slot 3

$A$ and $B$ are two railway stations $90 \mathrm{~km}$ apart. A train leaves $A$ at $9:00$ am, heading towards $B$ at a speed of $40 \mathrm{~km} / \mathrm{hr}$ . Another train leaves $B$ at $10:30$ am, heading towards $A$ at a speed of $20 \mathrm{~km} / \mathrm{hr}$ . The trains meet each other at

CAT 2020 Slot 3

Vimla starts for office every day at $9$ am and reaches exactly on time if she drives at her usual speed of $40 \mathrm{~km} / \mathrm{hr}$ . She is late by $6$ minutes if she drives at $35 \mathrm{~km} / \mathrm{hr}$ . One day, she covers two-thirds of her distance to office in onethirds of her usual time to reach office, and then stops for $8$ minutes. The speed, in km/hr, at which she should drive the remaining distance to reach office exactly on time is

CAT 2018 Slot 1

Point $P$ lies between points $A$ and $B$ such that the length of $BP$ is thrice that of $AP$ . Car $1$ starts from $A$ and moves towards $B$ . Simultaneously, car $2$ starts from $B$ and moves towards $A$ . Car $2$ reaches $P$ one hour after car $1$ reaches $P$ . If the speed of car $2$ is half that of car $1$ , then the time, in minutes, taken by car $1$ in reaching $P$ from $A$ is

CAT 2023 Slot 3

A boat takes $2$ hours to travel downstream a river from port A to port B , and $3$ hours to return to port A . Another boat takes a total of $6$ hours to travel from port $B$ to port $A$ and return to port $B$ . If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port $A$ to port $B$ is

2025 Slot 2

Rita and Sneha can row a boat at 5 km/h and 6 km/h in still water, respectively. In a river flowing with a constant velocity, Sneha takes 48 minutes more to row 14 km upstream than to row the same distance downstream. If Rita starts from a certain location in the river, and returns downstream to the same location, taking a total of 100 minutes, then the total distance, in km, Rita will cover is

CAT 2023 Slot 1

The minor angle between the hours hand and minutes hand of a clock was observed at 8:48 am. The minimum duration, in minutes, after $8.48$ am when this angle increases by $50 \%$ is

Solution

Related questions:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)