Word Problems
7 Problems • 4 sub-topics
Adalynn Le • 5/2/2026
Table of Contents
Introduction
Word problems are the foundation for everything in mathematics-especially the AMC 10. By definition, word problems are "mathematical problem[s] expressed entirely in words typically used as an educational tool" (Meriam-Webster Dictionary). Due to their nature, word problems require you to exercise a different level of thinking and make connections between mathematics and language. The AMC 10, which especially focuses on applied and puzzling mathematics, incorporates these with other topics in order to challenge you.
Understanding Word Problems
Word problems typically do not explicity tell you the formulas nor all of the numerical values nescessary to solve, at least not outright. The most important part of solving a word problem is to identify what is being asked and find what you need to get there. Let's break it down:
Identifying what is being asked
The first step should always to be to know what the question is asking for. If you really want, sometimes you can just skip to the last sentence where the problem directly asks the question. This will allow you to focus on the core point of the problem. In some situations, this can even remove some answer choices, like in the example below:
Practice Question
Jack has a collection of dimes, quarters, and nickels. If the total value of his coins is $2.50 and he has some number of each type of coin, what is the maximum number of dimes he could have?
Without solving this question, what answer(s) can you rule out?
Finding Keywords
Chances are that you've been taught the main keywords to look for in word problems, especially for operations like addition, subtraction, multiplication, and division. Recognizing these is an instant key as to what you're solving for and how to do it. If you see one you recognize, you know exactly what operation you need to do. It's like verbally explaining an equation to a friend. For reference, here's a list of the most common basic keywords:
| Addition | Subtraction | Multiplication | Division |
|---|---|---|---|
| sum | difference | product | quotient |
| total | less | times | divided by |
| combined | decreased by | of | per |
| increased by | per unit |
Other explicit situations are indicative of certain operations. For example, if the question is asking about buying something or costs, that could technically mean all of them. That being said, if you are finding the total cost of things with multiple costs, you add. If you have multiple of the same item, then you multiply. If you are finding the cost for one specific item, it's division (that's a big one). Context will also allow you, in addition to keywords, to figure out which operations to use
These are pretty simple and elementrary examples though. The AMC 10 uses much more complex scenarios. This could involve merging operations, using series/sequences, exponents, etc. Often times, these mathematical ideas are too complex to have a few keywords, so it is first important to understand what the things are and how to use them. Arithmetic series are repeated addition. Geometric series and exponents are repeated multiplication, and so on. In the end, everything boils down to a few basic operations. Let's look at a few examples that use these sorts of operations and see if you can see a common thread in the wording.
AMC 10A 2021 Spring Problem 4
Practice Problem
The first example involves an arithmetic series. The second uses a geometric series. Keep in mind that both of these are just repeated addition and multiplication, repeatedly. Let's disect them to see what the pattern is. In the first example, we see the use of the word "accelerating", which means speeding up. This means that something is growing, so we know it's a series. We also see that the question uses intervals, which is reminiscent of a common difference, or essentially segmented periods of growth. In the second example, we see the use of the word "doubles". This indicates exponential growth because the quanity is being multiplied by the same constant over time. The word "interval" is not specifically used, but the question marks growth over time, specifically 21 hours, which is our interval of time, t. In the end, both of these problems utilize a few common themes that are indicative of growth and the use of series.
With this information, we can create another table of other values for different types of series. Try to sort it on your own
| Arithmetic Series | Geometric Series |
|---|
Sort The Following
Subject to sort
Once you understand the concepts, you can string them together to make equations and model the question.
Systems of Equations
Every question needs to be modeled numerically in order to be solved algebraicly. This step can sometimes be skipped when you are dealing with more logically/visually oriented problems. In order to model problems, we use expressions and equations. When we string multiple of these together in order to model the full problem and we use the same set of variables, we get a system of equations.
Equations vs Expressions
An expression is simply a statement, whereas an equation is a comparison. When interpreting word problems, each independent event or rule can be expressed via an expression. When two rules, events or quantities must be equal, that's when you use an equation. The key thing to remember is you cannot solve an expression, or for a variable in a expression. A variable is not bound by anything. Although expressions are technically what we use to model the problem, we use equations to model the interactions between them. You do not always need a equation, because you can also use inequalities.
Using Variables
Variables can be anything. Although common mathematical practice is to use letters, the simple definition is to just represent a number or quantity. Variables are fairly simple, so there are just a few rule to keep in mind
- Variables represent ONE value
- In an expression, variables can be anything
- In an equation, variables must represent a specific value OR will have an affect on other variables in the equation
Typically, you need as many equations in your system as variables, but there are some special cases.
Diophantine Equations
How to Solve Systems of Equations
Graphing
Use desmos (or on the AMC 10) just sketch each equation and see the common meeting place (best for when there are only 2 variables)
Subtraction
Add or subtract equations to eliminate variables. Really quick but only works in some cases.
Substitution
Isolte one variable and substitute into others. Best with linear equations
A Diophantine equation is an equation that contradicts the rule that you need to have as many equations in a system as variables by bounding the values of the variables. The most typical example of this is binding to only integers. A specific enough range or similar technique can also limit the array of solutions (e.g. only positive numbers), but typically cannot make the solution set finite. Consider the two models below
A Standard Equation
Notice how there are seemingly infinite values for both of the variables, and that's considering we limited the range too!
A Diophantine Equation
y=
With this, there are only 10 total values for each variable, still dependent on each other. That makes an impossible equation possible
You can learn more about solving Systems of Equations in our article about Algebraic Manipulation
Special Cases + Popular Applications
A common staple of the AMC 10 is to use word problems modeling speed, distance and time. These are fairly straightforward, but there are two main things to remember
- Speed is equal to distance divided by time. If you forget this, just remember miles per hour, or another speed measurement
- Units are important. The units for speed are the units of distance per the units of time. Units must match and must conform to what is asked in the equation
Now, that's not to say speed, distance, time doesn't get complex. Although the concept and equation are simple, there are quite a few branches and types of problems utilizing them. Three examples are listed below. The first is "overtaking", a simple problem structure where ' there is a race or two moving objects where one starts later/at a different distance but moves faster. You can model this with systems of equations and set the distances equal to each other to find the place where they meet. The next situation is average speed. Notice that the average speed is calculated by finding the average of the speeds with weights (or importance) relative to the time spent at that speed. This means going the same distance at a fast speed and short speed gives more weight to the short speed. Finally, we have heatwind, current, or direction questions. This happens when something's movement can be modeled as the sum of two vectors, it's intended movement and another opposing vector that can act in any non-parallel direction. Often, this can require you to do conversions between speed and distance.
"Outrunning" System of Equations
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
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
Try some practice problems!
Question 1:
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Work and Rates
Work and rates is like an addition onto speed distance time. Instead of just using distance, you find the amount of something made, done, etc. This makes it diffucult because unlike speed distance time, many work questions will combine rates, and you can't accurately visualize it.
What's tricky about work is it typically does not give you time explicitly. The most important thing is to convert into speed and find the average. You'll find that in many work problems, it is most important to use speed
Conclusion
Word problems are the base for everything in the AMC 10. With a thorough understanding of how to use and apply them, you'll be more equipped for similar AMC 10 questions
Question 1:
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