MATHS - 2.TIME AND DISTANCE

Once, we were going on a walk from our house to a park about 4 km. away. Our grand son accompanied us on his cycle. As he goes faster on his cycle , I just wanted to give him a little more exercise to his body and a small problem to think about. I told him, we walk at about 4 km.per hr speed , where as you may go at about 8 km.,per hour on your cycle. We are both starting at the same time and the park is about 4 km. away. So ,what you do is go on your cycle ,reach the park, turn back ,meet us wherever we are on the way , turn back again to go to the park continuing like this till we reach the park , every time turning back at the park to meet us wherever we are and then again turning back to go to the park. He did like wise enjoying the experience of meeting us to say that he has already made so many rounds etc.. So finally when we reached the park , I asked him O.K. you did a good job , but can you tell me how long a distance you travelled in this fashion. He went scilent for some time and finally replied , I feel it should be 8 km., though I cant tell you how I got it or whether it is correct or not.

That is it .He is a middle school student and he replied by what we call as “Intution” The surprising fact is that it is the correct answer. When prodded further , finally , he came out with the theory that on his cycle ,he goes at 8 km.per hour , where as , we are walking at 4 km.per hour and hence he should have covered double the distance of our walk.

Well that is the advantage of keeping things simple. Though , he could still not explain it to the satisfaction of examiners ,but the only missing link in his explanation is the fact that he cycled for same time as we were walking whatever direction he was travelling . Now imagine what could have happened, if the same problem is given for a major entrance examination with some high sounding words and the students over awed by the level of the examination start thinking big using time and distance formulae , formulating a series ,trying to find their sum ,all by seeing some imaginary hidden devils in the problem as it is given by a reknowned institute of learning. Wondering how ? Let us see how ..

Speed of walk = 4 kmph

Speed of cycle = 8 kmph

Time of start = 0 hrs say.

Distance to park = 4 km.

Jouney No.1 for cycle :

Time taken to reach park by cycle = 4/8 = 0.5 hrs .

Relative speed between cycle and walk (in same direction)=8 - 4=4 kmph.

Hence distance of seperation after journey No.1 = 0.5 hr x 4 kmph = 2 km.

Total distance travelled by cycle up to Journey No.1 = 8 x 0.5 = 4 km.

Journey No. 2 for cycle :

Relative speed between cycle and walk (in opposite direction) =8 + 4=12 kmph

Time required to meet = 2 / 12 = 0.166666..

Distance travelled by cycle = 0.16666x 8 = 1.3333 km…..

Total distance travelled by cycle up to Journey No.2 = 4 +1.33333..= 5.3333..

Like this we can continue to a series whose sum will finally turn out to be 8 km. as we deduced earlier by keeping things simple or rather what they deserve to be !!

We may recall what the dashing opening batsman in cricket , Sehwag said about his play ….” I will not think of the bowler’s reputation..I will see the ball ..if it can be hit. it will be hit “

That is the point So our tip is to look at the problem for what it deserves and not given by whom ?

Few lines on the aspect of intution brought in at the beginning of this problem. Some may say it is linked with IQ ..”GOD” given etc. May be, but then what do we mean by saying some one is intelligent? Is he good at every thing in this world, No. .one may be good at maths. another may be good in English etc. Now what do we mean by good in maths or English ? How this can be brought out in a way that is understandable. Well, one way is to see “ given a page or so of relevant material in a particular topic in which we say the person is intelligent , he should be able to read ,understand ,explain and apply it in a shorter time than others”. Then we can say that he is good in that topic. O.K., Leaving out the “God” given part from it, cant we improve our performance in this test by taking interest in the topic of our choice and putting extra hours of hard work to make up for the difference in comprehension. The answer is that it is largely possible particularly when we have time at our disposal and a strong determination to improve coupled with interest ofcourse. So let us take inspiration from the hare and the tortoise story and move on .