Time and Distance - 2

In what time will a train, 125 m long, moving at a speed of 45 kilometres per hour, cross a bridge 265 metres long ?

Solution: To cross a bridge, a train has to travel a distance equal to the sum of its own length and the length of the bridge. So here the train will have to travel (125 m + 265 m) = 390 m.

Now, the speed of the train is 45 Km per hour.
That is, the train covers 45 Km in 1 hour (= 60 minutes).

1 Km = 1000 metres

So, the train covers (45 x 1000) metres in 60 minutes.

therefore, the train covers 1 metre in 60 / (45 x 1000) minutes.

So the train covers 390 metres in 60 x 390 / (45 x 1000) = 52 /100 minutes.

1 minute = 60 seconds.

By multiplying minutes by 60 we gets seconds.
So 52/100 minutes = 52 x 60 / 100 seconds = 312/ 10 seconds = 31.2 seconds.

Answer: The train crosses the bridge in 31.2 seconds.

What is Arithmetic

Arithmetic is the oldest and most elementary branch of mathematics. It is used by everyone to carry on with their day-to-day operations. Higher up it is used for scientific and business calculations. Arithmetic is to mathematics what letter is to a word. Arithmetic is the basic foundation of mathematics.

The two basic arithmetic operations are addition (+) and subtraction (-).  In its simplest form, addition combines two numbers into one. Subtraction finds the difference between two numbers. The difference can be either, positive, negative or zero.

From addition we can derive another operation called multiplication (× or *), which is nothing but repeated addition of the same number to another.

From subtraction we can derive another operation called division (÷ or /), which is nothing but repeated subtraction of the same number from another.

Arithmetic clears the logical thinking ability of our brain and makes it more and more analytical in its thinking prowess. Logic is concerned with the principles of correct reasoning, and it is the study of arithmetic which vastly improves our reasoning ability. 

A student, in his/her infancy, begins his/her study of mathematics with positive numbers and arithmetic. Arithmetic deals with natural numbers, integers, fractions, and decimals, which the student learns in course of time. In later stages a student learns how to solve basic problems using the various tools provided by arithmetic and logic. 

Arithmetic - A few important concepts about motion

A moving train starts crossing a post, a tree, a pillar, a standing man or any object of negligible length/width when the front part of the train meets the object.

The crossing ends when the end or rear part of the train just leaves the post or pillar or whatever.

A moving train starts crossing a bridge, a platform or any object, whose length/width can be determined, when the front part of the train just meets the beginning of the bridge or platform.

 The crossing ends when the end or rear part of the train just leaves the end of the bridge or platform or whatever object of determinable length.

Thus, to cross a post, a pillar, a tree or a standing man, a train will have to travel a distance equal to its own length.

Again, to cross a platform, a bridge or any object whose length/width can be determined, a train will have to cover a distance equal to the sum of its own length and the length of the bridge or platform. 

Problem: A train travelling at a speed of 72 km per hour passes a man standing on a platform in 18 seconds. The train passes the platform in 30 seconds. Find the length of the train and that of the platform.

Solution: According to the rule, in 18 seconds the train covers a distance equal to its own length as it takes 18 seconds to pass the standing man.
In 1 hour or 60 minutes the train covers 72 km.
Therefore, in 1 minute the train covers 72/60 km.
So, in 1 second the train covers 72/60 ÷ 60 km  = 72/60 x 1/60 km
So, in 18 seconds the train covers 72/60 x 1/60 x 18 km = 0.36 km = (0.36  x 1000) metres = 360 metres.

Therefore, the length of the train is 360 metres.

Again, in 18 seconds the train travels 360 metres.
So, in 1 second the train travels 360/18 metres.
Therefore, in 30 seconds the train travels 360/18 x 30 metres = 600 metres.

According to the rule, in 30 seconds the train travels a distance equal to the sum of its own length and that of the platform. This sum is 600 metres.

Now to get the length of the platform we have to subtract the length of the train from the sum. 
Therefore, the length of the platform is (600 - 360) metres = 240 metres.

Unitary Method with Time and Distance - 1

The problem: A bus covers a distance of 660 kilometres (km) in 5 hours 30 minutes. Find the speed of the bus per hour.

Solution: 1 hour = 60 minutes
Therefore, 5 hours 30 minutes = 5 x 60 minutes + 30 minutes = 330 minutes
We have to find out how far the bus can travel in 60 minutes.
 
Now, the bus takes 330 minutes to travel 660 km. (the prime statement)

So the bus takes '1 minute' to travel 660/330 km

Therefore the bus takes '60 minutes' to travel 660/330 x 60 km = 120 km.

Answer: The speed of the bus per hour is 120 km.

In this problem we have used Unitary Method. It is a very, very important tool in arithmetic. In this method we construct the first statement, referred to in the solution as the prime statement, in a way so that the quantity or parameter that we have to find out is kept at the end of the statement.  The other parameter (or parameters) is kept at the front of the prime statement.

This construction of the prime (beginning) statement is very important. It guides us throughout the whole solution until we arrive at the answer. Here it is the speed of the bus expressed in km. So we have kept km at the end of our prime statement, and time at the front. 

In the next statement, we have to convert the first parameter, in this problem the time, into unity or 1. Hence the name Unitary Method. If time becomes 1 minute, then the distance covered by the bus will also get reduced. Hence we have divided 660 by 330. Here is the logical play of our brain. (There may be problems where, instead of division, we may have to perform multiplication.) 

In the next statement we simply replace the unity by the desired value, here "1 minute" by 60 minutes" to get the desired distance in km, and accordingly we have multiplied (as 60 minutes' distance will be greater than 1 minutes' distance) 660/330 by 60. 

Here we are dealing with just two parameters, namely time taken and distance covered. There may be more than two parameters, which will be discussed in a later post. The rule of Unitary Method remains all the same there also.

Try out the following problem yourself:

A train covers a distance of 500 km in 2 hours 5 minutes. Find the speed of the train per hour.  
Answer: Speed of the train is 240 km per hour. 

Please feel free to ask any question through the comment box.