Speed-Distance-Time
6 Problems • 2 sub-topics
Adalynn Le • 5/25/2026
Introduction
Speed distance time is, obviously, the relationship between speed, distance, and time. It is written as the expression \(s=\frac{d}{t}\). If you ever forget this, and if you follow the imperial system of measurement, all you need to remember is the typical unit for speed: miles per hour. This is effectively dividng the unit for distance, miles, by the unit for times, hours, to make a new unit. The same logic applies for the metric kilometers per hour.
Rates
In some countries, automobile fuel efficiency is measured in liters per \(100\) kilometers while other countries use miles per gallon. Suppose that \(1\) kilometer equals \(m\) miles, and \\(1\\) gallon equals \(l\) liters. Which of the following gives the fuel efficiency in liters per \(100\) kilometers for a car that gets \(x\) miles per gallon?
One of the concepts I touched on in the Introduction was the idea that speed doesn't have to always be covering physical distance over ellapsed units of time, it can also be used to model work over non-standard time (e.g. turns). Thus, speed is probably not the best word to describe it, and rather rates fits well too. Throughout this article, the terms will be used somewhat interchangeably.
Another thing to consider is that the \(s=\frac{d}{t}\) equation is not the only way to model the relationship between those three entities. On a trivial note, you can rearrange to solve for \(d=st\), or \(t=\frac{d}{s}\). However, the important constraint of this metric is that it assumes completely linear motion and constant speed. On the actual AMC 10, you will see much more complex problems that require you to combine rates and combine directions (see Speed in Different Directions). One of the most potent examples of how algebraic manipulation can be applied to Speed-Distance-Time relationships is shown on the left.
Another popular example of systems of equations is when you see a "race" or similar event where two entites are moving at different speeds. Typically, one will be offset and "overtake" the other. In order to solve these, all you have to do is use the given values and solve for the value that makes their equations equal or satisfies the inequality (typically, you are solving for the time that makes their distance equal). In the example to the right, we see the blue dot representing a dot witha "head start" and slower speed, and we see and orange dot that will catch up to it at the right speed. The way to solve this would be \(s_{blue} \times t + \textup{offset}= s_{orange} \times t\). This is essentially modiying the \(s=\frac{d}{t}\) equation to solve for \(d\), which must be equal, and for a value of \(t\) that is also equal. The offset is the distance of a head start that the blue dot has. Notice that this models only a linear and constant speed, and changing speeds would be much more complicated.
"Outrunning" System of Equations
Average Speed
One big way in which speed and rates can get complicated is the use of average speeds. Just as with any averaging, it is important to notice that the speeds are "weighted". In the case of taking arithmetic mean of speeds, the weights are organized by the time spent at each speed, since the average speed is typically over time. If it is looking for average speed over a set distance, the average speed can be dependent on distnace, but that is rare.
Average Speed
Assume somebody covers the same distance twice at two different speeds, going one way and then going back. What is their average speed in the time it takes them to go back and forth (assume the distance is 10 miles)
Speed: 55mph
Time: 5.5 hrs
Speed: 55mph
Time 5.5 hrs
Speed in Different Directions
The last facet of complexity in speed-distance-time problems is when an object is affected by two different forces in different directions. This typically involves a current, air-stream, or some other external force that opposes the general movement and direction of the object. In order to express this graphically, we use something called vectors. Vectors are drawn as arrows and describe both magnitude (distance) and direction. When two speeds are acting on a vector, it can be said that two vectors are acting upon it.
Say you have a boat moving in a river with a current going westwards \(3 m/s\). The boat'speed is set to \(4 m/s\) north, not counting the current. To model this, we would draw a vector of magnitude \(3\) pointing west from an origin, and then draw a vector of magnitude \(4\) pointing north. Here's where it gets complicated. As a general rule, in "vector addition", you add head to tail, so the new vector will start where the old one ends. The resultant vector, our final magnitude for the boat, will go from the start of the first vector to the end of the second. In this case, it makes a right triangle, but we won't always have such perfect angles. To find the magnitude of a resultant vector, you will typically have to use trigonometry unless you have a right triangle, like in this example, where you can use the pythagorean theorem. In this case, the final magnitude is \(\sqrt{3^2+4^2}=5 m/s\) with a direction of \(\textup{arctan}(\frac{4}{3} \approx 53.13^\circ\) north of west. This is the speed and direction of the boat, accounting for the current.
In real competition math, this would be much more complex, but the key idea to take away is that you can use vectors and trig to model multiple diretions. In our example, instead of using head-to-tail addition, we are using something called the "parallelogram method" where both vectors start at the origin. Although this works in this case, because we have a computer that can do the calculations, it is typically easier to visualize with head-to-tail addition.
Combined Speed (Currents and Wind)
You can model combined speed from two different directions using vectors! Typically, we add vectors from head-to-tail, but for simplicity we're finding the average here, essentially
Conclusion
Speed distance time are an important subset of word problems on the AMC 10 with many different facets. Different kinds of problems reqiure skillsets of algebraic manipulation, logic, and even trigonometry. The key to mastering these problems is to understand the underlying relationship between the three entities and to be able to apply it in different contexts, whether it be rates, average speeds, or vectors. With practice, you can become proficient at solving a wide variety of speed-distance-time problems and be well-prepared for the AMC 10 and other math competitions.
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