# Video: Determining the Time and Distance between Two Points in Opposite Directions

Two cars were moving in opposite directions on a road between 𝐴 and 𝐵. The first car started at point 𝐴 and traveled at 104 km/h, and the second started at point 𝐵 and traveled at 100 km/h. Given that the points 𝐴 and 𝐵 are 153 km apart, determine the time 𝑡 and the distance 𝑑 from point 𝐴 at which the two cars met.

03:46

### Video Transcript

Two cars were moving in opposite directions on a road between 𝐴 and 𝐵. The first car started at point 𝐴 and traveled at 104 kilometers per hour, and the second started at point 𝐵 and traveled at 100 kilometers per hour. Given that the points 𝐴 and 𝐵 are 153 kilometers apart, determine the time 𝑡 and the distance 𝑑 from point 𝐴 at which the two cars met.

So the first thing we’re gonna look at in this question, is the fact that the points 𝐴 and 𝐵 are 153 kilometers apart. So therefore, what we can say is that the total distance, which is 153 kilometers, the distance between 𝐴 and 𝐵, is going to be equal to the distance traveled by the first car plus the distance traveled by the second car. And we can add the distances as the cars are moving in the opposite direction. So now what we’re gonna do is use this and then populate it with some of the information we know. Well, we’re gonna use the speed–distance–time triangle to help us.

And because so far we’ve got everything in 𝑑, then we could say that 𝑑, our distance, is equal to the speed multiplied by the time. So therefore, we get the total distance is equal to the speed of the first car multiplied by the time taken before the two cars meet plus the speed of the second car multiplied again by the same time taken before the two cars meet. So now what we want to do is we want to rearrange this so that 𝑡 is the subject so that we can find out what 𝑡 is, the time taken before the two cars meet.

So to do this, what we’re gonna do is factor the right-hand side of the expression. And to do that, we’re gonna take out 𝑡 as a factor. And when we do this, we’re gonna get the total distance is equal to 𝑡 multiplied by 𝑠 sub one plus 𝑠 sub two. So then what we can do is divide through by 𝑠 sub one plus 𝑠 sub two. So when we do that, we’ve got the total distance divided by and then we’ve got the speed of car one plus the speed of car two is equal to our time. Great, so now what we can do is calculate our time by substituting in the values we know. So what we’re gonna get is 153 over 100 plus 104 is equal to 𝑡, our time, which is gonna give us 153 over 204 equals 𝑡, our time.

So therefore, we know that the time is gonna be equal to 0.75 hours. And that’s because 153 divided by 204 is 0.75. So therefore, we know the time taken for the two cars to meet is 45 minutes. Okay, great, so that’s the first part of the question solved. So now what we want to do is we want to find out what the distance 𝑑 from point 𝐴 is at which the two cars met. So therefore, what we’re interested in is the distance of the first car’s travel because the first car is traveling from point 𝐴.

Well, we already saw earlier that we can calculate the distance that the first car travels because this is equal to the speed of the first car multiplied by the time taken for the cars to meet. So its gonna be equal to 104 multiplied by 0.75, which is gonna be equal to 78 kilometers. Well, we could’ve worked that out using a calculator, or we could also look at it with a written method cause if you’ve got 104 multiplied by 0.75, it’s the same as 104 multiplied by three-quarters.

Well, then what we can do is divide both 104 and the denominator of four by four, which is gonna give us 26 multiplied by three over one, which will be equal to 78. So therefore, we can say that we completed the question. And we can say that the two cars met after 45 minutes of travel. And the distance 𝑑 from point 𝐴 which the two cars met was 78 kilometers.