Distance, Rate and Time Problems

0:00 Welcome to this explainer. Look, if you're getting ready for the ASVAB and feeling, well, maybe just a little uneasy about the math sections, I want you to take a deep breath right now, because today we're going completely calculator free. I'm going to show you exactly how you can confidently solve literally every single motion problem on the ASVAB entirely by hand, just by using a few straightforward logical steps. Here's our quick roadmap for today. We've got the core motion formula, two vehicle motion rules, round trips and stops. Short cuts and worked examples, common motion traps, and finally a chance to check yourself. All right, let's dive right into section one, the core motion formula. Okay, so here is our absolute core relationship, D equals R times T, distance equals rate times time, and honestly, that's it. Assuming a steady or average speed over a trip, this one simple equation is the only key you need. It's like a three-piece puzzle. If a question gives you any two of these pieces, you can always unlock the missing third piece. You just divide distance by time to get the rate, or you can divide distance by rate to get the time.

1:05 Now, the biggest pitfall you're going to encounter here is unit consistency. The units of your rate completely decide the units for your time and distance. Take a look at this example. Say a train runs at 72 kilometers per hour for 25 minutes. If you just grab those numbers and multiply 72 by 25 directly, you get 1,800 kilometers. Think about that for a second. That is an impossibly fast 25-minute train ride. No way. Instead, you have to convert those 25 minutes into hours first by dividing by 60. Then you multiply 72 by the fraction 25 over 60, and boom, you get the correct answer, 30 kilometers. And you can do all of this math without a calculator for sure. Let's say you need to find the time by dividing distance by speed. You've got a 275 mile drive at 50 miles per hour. Well, 275 divided by 50 is 5.5 hours. To make sense of that decimal in the real world, simply multiply the 0.5 hours by 60. That gives you exactly 30 minutes. So your total trip lasts 5 hours and 30 minutes. Pretty easy, right? Two vehicle motion rules.

2:11 So what happens when two objects are approaching each other, or maybe separating in opposite directions? Well, they're actually working together to shrink or grow the gap between them. So you simply add their speeds. If two buses are 380 kilometers apart, and driving toward each other at 90 kilometers per hour and 100 kilometers per hour, they are actively closing that gap at a combined 190 kilometers each hour. Now, contrast this with a chase scenario. If a faster object is chasing a slower one on the exact same route, the gap only shrinks by the difference in their speeds. In this case, you subtract. A car going 57 miles per hour, closing in on a bus, doing 48 miles per hour, shrinks the gap by exactly 9 miles per hour. But what if they don't start at the same time? Stay calm, we just take it step by step. Imagine two towns, 260 miles apart. Car A leaves at 40 miles per hour, and exactly one hour later, Car B leaves the other town at 70 miles per hour. Deal with Car A's solo hour first. In that first hour, it travels 40 miles, shrinking our total gap down to 220 miles.

3:18 Now they're both driving toward each other. Add their speeds. 40 plus 70 is 110 miles per hour. 220 divided by 110 equals two hours. So they meet two hours after Car B actually starts driving. Let's jump into section three. Round trips and stops. You've got to be aware of the equal distance average speed trap. It's a classic. If you go 30 miles per hour one way and 60 miles per hour on the way back, your first instinct is probably to average those speeds to 45, right? Actually, that is completely wrong. Average speed is always total distance divided by total time. Because you travel slower on one leg of the trip, you naturally spend more time at that slower speed, which completely pulls the true average down. For those specific speeds, the true average is actually 40 miles per hour, leaning heavily toward that slower pace. And how do we handle a built in rest stop by mentally isolating your distance from your time? Let's break it down. Say a driver goes 104 miles in two hours. Then they rest for 30 minutes. Now remember, this adds time to the clock but absolutely zero distance.

4:23 After the rest, they go 90 miles in 1.5 hours. Just add your total miles. 104 plus 90 is 194. Then add your total hours. 2 plus 0.5 for the rest plus 1.5 is 4 hours. 194 divided by 4 gives you a simple, clean, 48.5 miles per hour overall average. Alright, section 4, shortcuts and worked examples. When chaos hits on a two object problem, just draw a simple motion table on your scratch paper. Literally just make a grid. Traveler A and Traveler B as your rows and rate time distance as your columns. Doing this instantly organizes the chaos. You'll clearly see how the rows relate to each other. Either they cover equal distances for a catch-up scenario or their distances add up to the total gap for a meeting scenario. Even better, you can use the relative speed shortcut we discussed. Let's look at worked example 2, exactly as it's written in the lesson. A jogger running 6 miles per hour is 0.6 miles ahead of a friend running 8 miles per hour. Instead of making two complicated equations, treat the gap itself as one object. The friend gains 2 miles per hour on the slower jogger.

5:30 The catch-up time is simply the gap of 0.6 divided by that relative speed of 2, which equals 0.3 hours. Multiply that by 60 and you get 18 minutes. Look at that. A two jogger problem turned into a single calculator-free division step. So what if the head start is in time not distance? Check out worked example 3. A truck leaves at 45 miles per hour. 40 minutes later, a car follows at 60 miles per hour. First things first, turn that 40 minute head start into a physical lead. 45 miles per hour multiplied by the fraction 40 over 60 hours gives the truck a 30 mile lead. From there, it's just a standard chase. The car gains 15 miles per hour. 30 miles divided by 15 miles per hour equals exactly 2 hours to catch-up. Let's move to section 5. Common motion traps. Here are the 5 big traps test writers love to use. Hand it to you on a silver platter. Number 1, averaging the speeds directly instead of using total distance over total time. 2, forgetting that a rest stop adds time but no distance. 3, using the wrong relative speed, like accidentally adding speeds in a chase instead of subtracting them.

6:38 4, mixed units, like simply forgetting to change minutes to hours. And 5, ignoring a time head start and dividing before converting it to a physical distance lead. Honestly, just being aware of these 5 specific traps is literally half the battle won. Finally, section 6. Check yourself. Let's see what you've learned. Let's start with question 1. A bus travels at 64 kilometers per hour for 2.5 hours. How far does it travel? Is it A, 160 kilometers, B, 25.6 kilometers, C, 128 kilometers, or D, 66.5 kilometers? I want you to pause this explainer right now. Grab a pencil and solve this by hand using our core formula before we reveal the answer. Go ahead. I'll wait. Okay, are you ready? Using distance equals rate times time, we just do direct multiplication here. 64 multiplied by 2.5. Well, we know half of 64 is 32, so 64 times 2 is 128, plus that extra 32 gives us 160. The answer is A, 160 kilometers.

7:39 If you nail that, you've absolutely mastered the core formula. Let's step it up to a two vehicle problem for question 5. Two trains are 495 kilometers apart on parallel tracks and head toward each other. One at 75 kilometers per hour and the other at 90 kilometers per hour. How long until they meet? Is it A, 6.6 hours, B, 5.5 hours, C, 3 hours, or D, 33 hours? Once again, hit pause right now, check your relative speed rules, do the mental math, and let's see what you get. Let's break it down together. Because they are approaching each other, they are working together to close that gap. So we add the speeds. 75 plus 90 equals a relative speed of 165 kilometers per hour. To find the time, we take the distance of 495 and divide it by 165. Well, 165 times 3 is exactly 495, so the answer is 3 hours, which is option C. When you master these predictable, logical rules, you really transform a scary math test into a simple puzzle.

8:46 You are now officially unstoppable on these motion problems. So my final question for you is, now that you don't need a calculator to track moving trains, what other ASVAB math fears are you ready to conquer next? Keep practicing, trust your structure, and I'll see you in the next explainer.