Time & Distance

Concept

This type of problems is basically interrelated with Speed, Distance & Time. From the basic three interrelated data any one type is hidden to find out that query applying mathematical rules. As example, a man travel from city A to city B and it took him three hours. The distance between the cities is 250 km. Here one information from the Distance, Speed & Time is absent. Here, Speed is inversely proportional to time and is directly corresponding to the distance. It states that the speed is equal to the distance covered per unit time. Again, Time is oppositely proportional to speed and is directly corresponding to the distance. It states that the time is equivalent to the distance covered per unit speed.

Tips to solve Time & Distance problem:

Distance is time covered with a certain speed. So, Distance=Speed x Time
The distance covered per unit of time is called speed. So, Speed=Distance/Time
The distance covered per unit of speed is called time. So, Time=Distance/Speed

Q. A car travels at a constant speed of 60 km/h for 2 hours, then at 80 km/h for 1 hour. What is the average speed of the car for the whole journey?

A) 66.67 km/h

The average speed of the car is given by the total distance divided by the total time. The total distance is $\displaystyle 60 \times 2 + 80 \times 1 = 200 \text{ km}$ , and the total time is $\displaystyle 2 + 1 = 3 \text{ hours}$ . Therefore, the average speed is $\displaystyle \frac{200}{3} = 66.67 \text{ km/h}$ .

B) ) 70 km/h

C) 73.33 km/h

D) 76.67 km/h

Discuss

Q. A train covers a distance of 120 km in 2 hours, then stops for 30 minutes, then covers another 80 km in 1 hour. What is the average speed of the train for the whole journey?

A) 40 km/h

B) 50 km/h

The average speed of the train is given by the total distance divided by the total time. The total distance is $\displaystyle 120 + 80 = 200 \text{ km}$ , and the total time is $\displaystyle 2 + 0.5 + 1 = 3.5 \text{ hours}$ . Therefore, the average speed is $\displaystyle \frac{200}{3.5} = 57.14 \text{ km/h}$ .

C) 60 km/h

D) 80 km/h

Discuss

Q. A cyclist travels at a speed of 15 km/h for the first half of a journey, then at a speed of 25 km/h for the second half. If the total distance is 60 km, how long does the journey take?

A) 2 hours

B) 2.5 hours

C) 3 hours

The time taken for each half of the journey is given by the distance divided by the speed. The distance for each half is $\displaystyle \frac{60}{2} = 30 \text{ km}$ , and the speeds are $\displaystyle 15 \text{ km/h}$ and $\displaystyle 25 \text{ km/h}$ respectively. Therefore, the time taken for the first half is $\displaystyle \frac{30}{15} = 2 \text{ hours}$ , and the time taken for the second half is $\displaystyle \frac{30}{25} = 1.2 \text{ hours}$ . The total time is $\displaystyle 2 + 1.2 = 3.2 \text{ hours}$

D) 4 hours

Discuss

Q. A runner completes a lap of a circular track in 40 seconds. If the radius of the track is 50 meters, what is the runner’s average speed in m/s?

A) π m/s

B) 2.5π m/s

The average speed of the runner is given by the distance divided by the time. The distance for one lap of the circular track is given by the circumference, which is ( 2\pi r ), where ( r ) is the radius. Therefore, the distance is $\displaystyle 2\pi \times 50 = 100\pi \text{ meters}$ , and the time is 40 seconds. Therefore, the average speed is $\displaystyle \frac{100\pi}{40} = 2.5\pi \text{ m/s}$ .

C) 3.5π m/s

D) 4.5π m/s

Discuss

Q. A bus leaves a station at 9:00 am and travels at a constant speed of 40 km/h. Another bus leaves the same station at 10:00 am and travels at a constant speed of 50 km/h in the same direction as the first bus. At what time will the second bus overtake the first bus?

A) 11:00 am

B) 12:00 pm

C) 1:00 pm

The second bus will overtake the first bus when they have covered the same distance from the station. Let t be the time taken by the second bus to overtake the first bus, measured from when it leaves the station. Then, the distance covered by the second bus in t hours is 50t km, and the distance covered by the first bus in t + 1 hours is 40(t + 1) km (since it left one hour earlier). Equating these distances, we get: • 50t = 40(t + 1) • Simplifying and rearranging, we get: • t = 4 Therefore, it takes four hours for the second bus to overtake the first bus, which means it will happen at 1:00 pm.

D) 2:00 pm

Discuss

Q. A plane flies from city A to city B at a speed of 500 km/h, then from city B to city C at a speed of 600 km/h. The distance between city A and city B is 1500 km, and the distance between city B and city C is 1800 km. What is the average speed of the plane for the whole trip?

A) 540 km/h

B) 545.45 km/h

C) 550 km/h

The average speed of the plane is given by the total distance divided by the total time. The total distance is $\displaystyle 1500 + 1800 = 3300 \text{ km}$ , and the total time is given by adding the times taken for each leg of the trip, which are given by dividing the distances by the speeds. Therefore, the total time is $\displaystyle \left(\frac{1500}{500}\right) + \left(\frac{1800}{600}\right) = 6 \text{ hours}$ . Therefore, the average speed is $\displaystyle \frac{3300}{6} = 550 \text{ km/h}$ .

D) 650 km/h

Discuss

Q. A car travels from point P to point Q at a constant speed of x km/h, then from point Q to point R at a constant speed of y km/h. The distance between point P and point Q is equal to the distance between point Q and point R. What is the average speed of the car for the whole journey in terms of x and y?

A) $\displaystyle \frac{x + y}{2}$ km/h

B) $\displaystyle \frac{2xy}{x + y}$ km/h.

The average speed of the car is given by the total distance divided by the total time. The total distance is twice the distance between point P and point Q, which we can call $\displaystyle d$ km. The total time is given by adding the times taken for each leg of the journey, which are given by dividing $\displaystyle d$ by $\displaystyle x$ and $\displaystyle y$ respectively. Therefore, the total time is $\displaystyle \frac{d}{x} + \frac{d}{y}$ . Therefore, using algebra, we can simplify: Average speed = $\displaystyle \frac{2d}{\left(\frac{d}{x} + \frac{d}{y}\right)}$ Average speed = $\displaystyle \frac{2dxy}{dx + dy}$ Average speed = $\displaystyle \frac{2xy}{x + y}$ Therefore, in terms of $\displaystyle x$ and $\displaystyle y$ , the average speed of the car for the whole journey is $\displaystyle \frac{2xy}{x + y}$ km/h.

C) The speed is $\displaystyle \frac{x^2 + y^2}{x + y}$ km/h.

D) None of these

Discuss

Time & Distance

Concept

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