Arithmetic

Time, Speed & Distance

Medium

The wheels of bicycles $A$ and $B$ have radii $30$ cm and $40$ cm, respectively. While traveling a certain distance, each wheel of $A$ required $5000$ more revolutions than each wheel of $B$ . If bicycle $B$ traveled this distance in $45$ minutes, then its speed, in km per hour, was

$16\pi$

$14\pi$

$18\pi$

$12\pi$

✅ Correct Option: 1

We have two bicycles with different wheel sizes traveling the same distance. The key insight is that larger wheels cover more distance per revolution, so they need fewer revolutions to travel the same distance.

Given Information:

Bicycle A: wheel radius = 30 cm

Bicycle B: wheel radius = 40 cm

Wheel A makes 5000 more revolutions than wheel B for the same distance

Bicycle B takes 45 minutes to complete this distance

When a wheel makes one complete revolution, it travels a distance equal to its circumference.

For bicycle A: Distance per revolution = $2\pi \times 30 = 60\pi$ cm

For bicycle B: Distance per revolution = $2\pi \times 40 = 80\pi$ cm

Since B's wheels are larger, each revolution covers more ground, so B needs fewer total revolutions.

Let's say bicycle B makes $n$ revolutions to travel the distance.

Then bicycle A makes $(n + 5000)$ revolutions to travel the same distance.

Since both bicycles travel the same total distance:

Distance by B = $n \times 80\pi$ cm

Distance by A = $(n + 5000) \times 60\pi$ cm

Setting them equal:

$n \times 80\pi = (n + 5000) \times 60\pi$

Dividing both sides by $\pi$ :

$80n = 60n + 300000$

$20n = 300000$

Therefore: $n = 15000$

Bicycle B makes 15000 revolutions, while bicycle A makes 20000 revolutions.

Total distance = $15000 \times 80\pi = 1200000\pi$ cm

Time taken by B = 45 minutes = $\tfrac{45}{60}$ hours = $\tfrac{3}{4}$ hours

Speed = Distance ÷ Time

Speed = $1200000\pi \div \tfrac{3}{4} = 1200000\pi \times \tfrac{4}{3} = 1600000\pi$ cm/hour

To convert cm/hour to km/hour:

1 km = 100000 cm

So we divide by 100000

Speed = $1600000\pi \div 100000 = 16\pi$ km/hour

Bicycle B's speed is $16\pi$ km/hour.

When comparing circular motion problems, we always remember that larger wheels cover more distance per revolution. This relationship helps us set up equations based on the fact that different-sized wheels traveling the same distance will have different revolution counts.

CAT 2019 Slot 2

Two ants $A$ and $B$ start from a point $P$ on a circle at the same time, with $A$ moving clock-wise and $B$ moving anti-clockwise. They meet for the first time at $10:00$ am when $A$ has covered $60\%$ of the track. If $A$ returns to $P$ at $10:12$ am, then $B$ returns to $P$ at

CAT 2024 Slot 2

A bus starts at 9 am and follows a fixed route every day. One day, it traveled at a constant speed of 60 km per hour and reached its destination 3.5 hours later than its scheduled arrival time. Next day, it traveled two-thirds of its route in one-third of its total scheduled travel time, and the remaining part of the route at 40 km per hour to reach just on time. The scheduled arrival time of the bus is

2025 Slot 1

Shruti travels a distance of 224 km in four parts for a total travel time of 3 hours. Her speeds in these four parts follow an arithmetic progression, and the corresponding time taken to cover these four parts follow another arithmetic progression. If she travels at a speed of 960 meters per minute for 30 minutes to cover the first part, then the distance, in meters, she travels in the fourth part is

CAT 2022 Slot 3

Moody takes $30$ seconds to finish riding an escalator if he walks on it at his normal speed in the same direction. He takes $20$ seconds to finish riding the escalator if he walks at twice his normal speed in the same direction. If Moody decides to stand still on the escalator, then the time, in seconds, needed to finish riding the escalator is

2025 Slot 3

Rahul starts on his journey at 5 pm at a constant speed so that he reaches his destination at 11 pm the same day. However, on his way, he stops for 20 minutes, and after that, increases his speed by 3 km per hour to reach on time. If he had stopped for 10 minutes more, he would have had to increase his speed by 5 km per hour to reach on time. His initial speed, in km per hour, was

CAT 2019 Slot 2

John jogs on track $A$ at $6$ kmph and Mary jogs on track $B$ at $7.5$ kmph. The total length of tracks $A$ and $B$ is $325$ metres. While John makes $9$ rounds of track $A$ , Mary makes $5$ rounds of track $B$ . In how many seconds will Mary make one round of track $A$ ?

The wheels of bicycles AAA and BBB have radii 303030 cm and 404040 cm, respectively. While traveling a certain distance, each wheel of AAA required 500050005000 more revolutions than each wheel of BBB. If bicycle BBB traveled this distance in 454545 minutes, then its speed, in km per hour, was

Solution

Related questions:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)

The wheels of bicycles $A$ and $B$ have radii $30$ cm and $40$ cm, respectively. While traveling a certain distance, each wheel of $A$ required $5000$ more revolutions than each wheel of $B$ . If bicycle $B$ traveled this distance in $45$ minutes, then its speed, in km per hour, was