In this module we will be seeing a word problem based on speed, distance and time. A very interesting sum based on walking and cycling. Let us take up the question and the question is this Problem Set 2, 30^th main question, sub question no.xx.

Let us read the question, from the same place at 7 am, ‘A’ started walking in the north at the speed of 5 kms per hour. After 1 hour ‘B’ started cycling in the east at the speed of 16 km per hour. At what time they will be at a distance of 52 kms apart from each other is the question, right. At what time means the question is about time. But before getting into what is to be found, you understand when we read the problem there was something called north and east. There are directions in the sum. So let us understand directions over here. Suppose this is a ground, north and east, north and east directions were mentioned in the sum. North and east are two perpendicular directions we understand, right. Now let us read the problem, part by part, step by step.

From the same place, suppose the place is this ‘O’, from the same place at 7 am, at what time, at 7 am, ‘A’ started walking in the north. ‘A’ started walking in which direction, north, means he went upwards, at a speed of 5 kms per hour. So what is the speed of ‘A’, 5 kms per hour, right.

Then what else is said, after one hour, ‘A’ started at 7 am, after one hour, after one hour means what time, after one hour means at 8 am, after one hour means at 8 am, ‘B’ started cycling in the east with a speed of  16 kms per hour. East means which direction, this direction, ‘B’ started cycling in the east, this direction. With a speed of how much, 16 kms per hour. So what is the speed of ‘B’, 16 kms per hour, right.

So let us understand this problem with the help of an animation. At 7 am ‘A’ started walking in the north, and at 8 am ‘B’ started cycling in the east. Question asked is at what time they will be at a distance of 52 kms apart from each other. So at what time the distance between ‘A’ and ‘B’ will become 52 kms is the question asked. Is the question clear, at what time? At 7 am ‘A’ started, at 8 am ‘B’ started. At what time they will be at a distance of 52 kms apart from each other is the question asked. At what time the distance between ‘A’ and ‘B’ will become 52 kms is the question asked.

So if you can see this is a right angled triangle. So let us draw the right angled triangle AOB, right. Now the speed of A is 5 kms per hour, the speed of B is 16 kms per hour. And the distance between A and B is 52 kms, this much we have understood.

Now let us tabulate this data whatever we know. First is A, next is B, now time, what time, what is the time taken by A and what is the time taken by B, let us find out that. At what time A started walking, A started walking at 7 am, right. And what time B started cycling. B started cycling at 8 am, right, we know this. The question is about time, the time one of them has to be x, and what is the other let us see. The question is about time. A started walking at 7 am and B started cycling at 8 am we know. And we also know that the distance between A and B at some time is 52 kms, right. The question asked to you is at what time the distance between A and B will become 52 kms. Suppose at 11 am, at 11 o’clock, A and B were at a distance of 52 kms apart from each other suppose, right. Now A started at what time, 7 o’clock and he reached point A at what time 11 am. B started at what time, 8 am, and he reached spot B at 11 am only, because at 11 o’clock they are at a distance of 52 kms apart from each other. So A started at 7 am and reached spot A at 11 am. B started at 8 am and reached spot B at 11 am. That means what we understand, we understand that A travelled one hour more than B, yes or no. Whatever time B travelled A travelled one more hour. If B travelled for 5 hours then A travelled for 6 hours, one hour more. That means if B travelled for x hours, A travelled for how many hours, x+1 hour. Now since we don’t know the time taken of A and B, we will assume let the time taken by B to reach spot B be x hours. So time taken by A will automatically be 1 hour more, because he travelled 1 hour more, it will be how much x+1 hour. So time is clear.

Next, we know the speed of A, what is the speed of A, 5 kms per hour. We know the speed of B also, how much is that, 16 kms per hour. Since we know time and we know speed, we can calculate distance. Distance is nothing but speed into time, distance is nothing but speed into time. So speed of A is 5 into the time taken by A is (x+1). So 5 into (x+1) so many kms. Similarly we can find out distance travelled by B also. Distance travelled by B will be nothing but again speed of B will be 16 kms per hour into time taken by B will be nothing but x hours. So 16 into x, 16x kms. Now since we know both these distances, distance covered by A and distance covered by B and also we know distance between A and B 52 kms. So let us put these distances in the place of distances, right. So all the three distances we know, we know OA, we know OB and as well as we know AB.

Now if we see this triangle, this triangle AOB is nothing but a right angled triangle. That means in this right angled triangle how to form an equation connecting all these three. We can use Pythagoras theorem.

In right angled triangle AOB Pythagoras theorem we can write

OA² + OB² = AB². Let us put the values of OA, OB and AB.

OA² means 5 into (x+1) the whole square + OB² means 16 x the whole square is equal to AB² which means 52².

Right, we got the equation, how to solve it we will see in the next module.