Rates

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RATES:

"Rates" - A common topic (quant/math aptitude section) to almost all competitive/admission tests! Broadly, it constitutes of 2 sub-topics: Distance-Speed-Time, Work-Time.

Distance-Speed-Time:

Though limited to just 2-3 formulae, the question structures can vary a lot. The formulae are very basic ones, which one is expected to have learnt during school days:

  • Distance = Speed * Time,
  • Average Speed = Total Distance / Total Time,
  • Relative Speed = Sum (or) Difference of the speeds based on the directions of travel.

Let’s understand usage of relevant formulae using some examples:

Example 1:

If a superfast train travels at a speed of 120km/hr for 3.5 hours, how far would it travel? Similarly, If a superfast train travels at a speed of 120km/hr, how long would it take to cover 420 km? Or, If a superfast train could travel a distance of 420 km in 3.5 hours, what was the speed at which travelled.

Solution:

Solving the above questions requires the usage of the formula listed above "Distance = Speed * Time".

Example 2:

If a car starts its journey from Ahmedabad travelling at a speed of 60 km/hr and reaches Mumbai (480 km from Ahmedabad) and then starts its journey to Bangalore(320 km from Mumbai) at a speed of 80 km/hr and reaches there. What was the average speed of the car’s journey from Ahmedabad to Bangalore?

Solution:

This would require the usage of the second formula listed above "Average Speed = Total Distance / Total Time" & also the first formula "Distance = Speed*Time too".

Example 3:

If a police (travelling at a speed of 80 km/hr) chases a thief (travelling at a speed of 60 km/hr) who is 300 km away from the police, how long would it take the police to catch the thief?

Solution:

This would require the usage of the third formula "Relative Speed = Sum (or) Difference of the speeds" based on the directions of the travel along with the first formula "Distance = Speed * Time".

(Note: in this example the relative speed would be the difference of the speeds of the police and the thief as one of them is chasing another)

Now, there may a variety of designs/structure of the questions based on the above three formulae. Based on the designs/structure, the question sub-topics (boats & streams, trains/lamppost) are named.

Let’s understand the structure of the question using some examples:

Example 4:

If a boat travels up-stream at a speed of 10 km/hr and down-stream at a speed of 20 km/hr, what is the speed of the boat in still water?

Solution:

The third formula of Relative speed can be applied to solve the above example.

Example 5:

If a Train A (length of 1 km) travels at a speed of 140 km/hr and a train B (length of 2 km), 70 km away from Train A  travels in the opposite direction on parallel tracks at a speed of 210 km/hr, how long would it take train A to cross Train B completely.

The lengths of the trains are additional considerations towards distance parameter, while solving such questions.

Let’s understand all of the above discussed using an example:

Example 6: 

Train A travelled from Cochin to Ooty (distance of 320 km) with a speed of 80 km/hr and then from Ooty to Bangalore (distance of 180 km) with a speed of 45 km/hr and then from Bangalore to Hyderabad (distance of 600 km) with a speed of 150 km/hr. Train B travelled from Hyderabad to Bangalore with a speed of 75 km/hr and then from Bangalore to Ooty with a speed of 20 km/hr and from Ooty to Cochin at a speed of 40 km/hr.

Let the two trains be facing each other (at a distance of 452 km) in opposite directions on parallel tracks. If they both start travelling at their average speeds of the above described journeys respectively and train A is 1 km long while train B is 2 km long. How long would it take each of them to cross other completely?

Solution:

Firstly, we calculate the Average Speed of each of the train’s journeys:

Train A:                

                Total Distance travelled = 320 + 180 + 600 = 1100 km

                Total Time taken = (320/80) + (180/45) + (600/150) = 4 + 4 + 4 = 12 hrs

                Average Speed = 1100/12 = 650/6 = 108.33 km/hr

Similarly, for Train B:

                Average Speed = 1100/(8+9+8) = 1100/25 = 44 km/hr

Now, the Relative Speed of two trains travelling in opposite directions would be the sum of their speeds = 108.33+44 = 152.33 km/hr

The distance they are apart at the beginning of the show = 452 km.

The distance they have to cover together in order to cross each other completely = 452+1+2=457km

Now, the Time taken = Total Distance to be covered / Relative Speed = 457 / 152.33333 = 3 hrs.

In this example, we have tried to illustrate the usage of all the three formulae mentioned earlier.

Note: Also, in this case, the relative speed turned out to be the sum of the speeds as the trains are travelling towards each other and not one behind another.

Generally, relative speed would be used in cases where there are 2 bodies moving either towards each other or against each other or one behind another. When they are moving towards/against each other, the relative speed is the sum of their speeds while when they are moving in the same direction one behind another, the relative speed is the difference between their speeds.

Similar approach can be used to solve questions in which boats are travelling up-stream/down-stream where the sum of the speeds of the boat and the stream would be the relative speed if down-stream and the difference of their speeds, the relative speed if up-stream.

Work-Speed/Rate-Time:

This is a similar sub-topic to Distance-Speed-Time. The formula Work = Speed * Time demonstrates the similarity. There could be more than one body (person/machine) doing certain same work. Relative Speed in this case would also be either the sum of the speeds or the difference between the speeds based on their contribution towards the work.

Let’s understand using some simple examples first,

Example 1: Gadget 1 downloads data at a speed of 18mb/sec and gadget 2 downloads at a speed of 12 mb/sec. If they both download data together, how long will it take them to download 330 mb of data?

Solution:

The formula used to solve this would be "Work=Speed * Time", where Work = the amount of data to be downloaded, the Speed (relative speed) would be the sum of their speeds since both the gadgets are working towards the completion of the work.

Example 2: Gadget 1 downloads data at a speed of 32mb/sec and gadget 2 uploads data at a speed of 14 mb/sec. How long will it take an empty hard disk of capacity 9 gb to be completely filled with data when both gadgets are working. (1gb=1024mb).

Solution:

The formula again would be the same "Work=Speed*Time", where work equals 9 gb, the speed (relative speed) would be the difference between their speeds since  one of them is uploading and the other is downloading.

However, one key differentiator between Distance-Speed-Time questions and Work-Time questions is the format in which the Speed of the bodies is mentioned.

In distance-speed-time questions, the speeds are generally given directly as distance travelled per unit time. For example, 80 km/hr, 25 miles/second. In work-rate-time questions, the speeds are to be derived as work done per unit time from the information given. For example, A does a complete job in 2 hours implies, the speed of A is ½ job/hour, B does the same job in 5 hours implies, the speed of B is 1/5 job/hour. A work-rate-time question would generally be solved by calculating the speeds in such a manner.

Example 3: A pipe fills a tank in 2 hours. Another pipe empties the tank in 4 hours. How long will it take the same tank to get filled when both the pipes are doing their job?

Solution:

The pipe filling the tank takes 2 hours to fill it completely, implying that it fills ½ of the tank in 1 hour. Similarly, the second pipe emptying the tank takes 4 hours to empty it completely, implying that it empties ¼ of the tan in 1 hour. The relative speed with which the tank is getting filled would then be the difference between the individual speeds (½ – ¼) of the tank per hour. The formula Work =Speed * Time can then be applied (1 = ¼ * T) to calculate the time (4 hours) taken to fill the tank.

Example 4: A machine manufactures 24 tablets in 10 seconds while another machine does 30 tablets in 50 seconds. How long does it take both the machines working together to manufacture 100 strips of 24 tablets each?

Solution:

The speed at which the first machine is manufacturing (24 tablets in 10 seconds) = 2.4 tablets per second

The speed at which the second machine is doing it (30 tablets in 50 seconds) = 0.6 tablets per second

If both work together they will be able to do the job at a combined speed of (2.4+0.6) 3 tablets per second

Work to be done = 100 strips of 24 tablets each = 100*24 = 2400 tablets.

Time taken = Work/Speed = 2400/3 = 800 seconds = 13 minutes 20 seconds.