Solving Word Problems with Equations
Marcus and Jayden scored 27 points together in a basketball game, and Marcus scored twice as many as Jayden. Could you find each score without guessing? Solving an equation is the mechanical part. The hard part, and the part that really matters, is setting it up: turning words into an equation with a variable. With a clear method and a few models, most word problems become routine.
The Four-Step Method
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Choose the variable and write what it means: "let $x$ = the number of cards Marcus has." Don't write only the letter. Say which number it stands for.
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Translate each fact in the story using that same $x$.
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Set up an equation using the fact you haven't used yet (usually a total, or two things that are equal), then solve it with inverse operations.
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Check your answer in the story, not in your equation (the equation might be wrong), and answer in a full sentence with units.
Two facts come up all the time. The next consecutive integer after $x$ is $x + 1$. And when something is left over, you subtract: if you had $x$ dollars and spent $\$12$, you have $x - 12$ left.
If you add a number to its consecutive integer (the number that comes right after it), the result is $43$. Set up an equation to find out what the number is.
If you add a number to its consecutive integer (the number that comes right after it), the result is $43$. Set up an equation to find out what the number is.
A group of friends buys a large pizza that costs $\$12$. If after paying for it they have $\$8$ left over from the money they collected, how much money did they have at…
A group of friends buys a large pizza that costs $\$12$. If after paying for it they have $\$8$ left over from the money they collected, how much money did they have at the beginning? Represent the initial amount with the letter $x$.
"Twice as Many," "More Than," "Fewer Than"
Here is how common phrases translate. Suppose a quantity is $x$.
| Words | Expression |
|---|---|
| twice as many as $x$ | $2x$ |
| three times as many as $x$ | $3x$ |
| $5$ more than $x$ | $x + 5$ |
| $5$ fewer than $x$ | $x - 5$ |
Let's solve the problem from the start of the page. Let $x$ be Jayden's points. Marcus scored twice as many, so he scored $2x$. A tape diagram shows why the total is $x + 2x$:
Marcus is playing a video game and scored twice as many points in the current level as in the previous level. If between the two levels he has a total of $150$ points,…
Marcus is playing a video game and scored twice as many points in the current level as in the previous level. If between the two levels he has a total of $150$ points, how many points did he get in each level? Call $x$ the points scored in the first level.
On a shelf in a comic book store, there are three times as many action manga as mystery ones. If there are $48$ comics in total between both genres, how many are there…
On a shelf in a comic book store, there are three times as many action manga as mystery ones. If there are $48$ comics in total between both genres, how many are there of each kind? Call $x$ the mystery comics.
For Ethan's birthday, two types of invitations were handed out: digital and paper. The number of digital invitations was three times the number of paper ones. If a total…
For Ethan's birthday, two types of invitations were handed out: digital and paper. The number of digital invitations was three times the number of paper ones. If a total of 48 invitations were sent, how many people received the paper invitation and how many received the digital format?
Sharing a Total
When several people (or things) add up to a total, let $x$ be the amount of one of them. Write the others using $x$. The equation says that all the parts together equal the total.
Choose as $x$ the amount that the others are compared to. Then the other parts come out directly as $3x$ or $x + 12$, with no fractions.
Three friends are comparing how many songs they have on their favorite playlist. Ashley has twice as many songs as Marcus, and Sarah has 15 more songs than Marcus. If…
Three friends are comparing how many songs they have on their favorite playlist. Ashley has twice as many songs as Marcus, and Sarah has 15 more songs than Marcus. If between the three of them they have a total of 155 songs, how many songs does each one have on their playlist? Set up an equation where $x$ is the number of songs Marcus has.
Emma bought a video game on sale and a new controller. The controller cost her $\$25$ less than the video game. If she paid a total of $\$89$ for both items, what is the…
Emma bought a video game on sale and a new controller. The controller cost her $\$25$ less than the video game. If she paid a total of $\$89$ for both items, what is the original price of each item without counting other discounts?
The PE teacher bought three boxes of tennis balls. In the second box, there are twice as many balls as in the first, and in the third, there are $10$ more balls than in…
The PE teacher bought three boxes of tennis balls. In the second box, there are twice as many balls as in the first, and in the third, there are $10$ more balls than in the second. If there are $85$ balls in total, how many balls does each box contain?
Chains of Comparisons
Sometimes one amount is compared to a second, and the second is compared to a third. Start with the amount that doesn't depend on anyone else, call it $x$, and build the others one link at a time. Pay close attention to who each phrase is talking about.
In a video game store, they have sold three times as many copies of "Elden Ring" as of "FIFA". In addition, they have sold $15$ fewer units of "Zelda" than of "Elden…
In a video game store, they have sold three times as many copies of "Elden Ring" as of "FIFA". In addition, they have sold $15$ fewer units of "Zelda" than of "Elden Ring". If they sold $125$ video games in total among the three titles, how many copies of each were sold?
A digital content creator reviewed his monthly statistics. The comedy video has twice as many views as the technology video, and the travel video has $4000$ more views…
A digital content creator reviewed his monthly statistics. The comedy video has twice as many views as the technology video, and the travel video has $4000$ more views than the comedy video. If the three videos have a total of $24000$ views, how many views does each video have?
Three siblings, Alex, Bella, and Charlie, want to buy a gift for their parents that costs $\$105…
Three siblings, Alex, Bella, and Charlie, want to buy a gift for their parents that costs $\$105$. Alex contributes an amount of money, Bella puts in twice as much as Alex, and Charlie puts in $\$15$ less than Bella. How much money does each one contribute so that they have exactly enough for the gift?
Age Problems
In age problems, make a small table for "today" and "in some years." Time passes at the same rate for everyone, so you add the same number to every age.
Some age problems put $x$ on both sides of the equation. When that happens, subtract the smaller $x$-term from both sides so that the $x$'s end up on one side only.
Chloe is $3$ years older than her little brother. If their ages add up to $19$ years, how old is each of them? Call $x$ the age of the little brother.
Chloe is $3$ years older than her little brother. If their ages add up to $19$ years, how old is each of them? Call $x$ the age of the little brother.
Hector is 5 years older than his sister Olivia. In 8 years, the sum of their ages will be 41 years. How old is each of them currently? (Remember to add 8 years to each…
Hector is 5 years older than his sister Olivia. In 8 years, the sum of their ages will be 41 years. How old is each of them currently?
(Remember to add 8 years to each person's age to set up the future sum).
Ashley's mother is currently three times as old as she is. In $12$ years, the mother will be twice as old as Ashley. How old is each of them currently? Use a table to…
Ashley's mother is currently three times as old as she is. In $12$ years, the mother will be twice as old as Ashley. How old is each of them currently? Use a table to help you organize the data for the present and the future.
Rectangle Problems
In geometry problems, always draw the figure and label the sides using $x$. Then use the perimeter: the sum of all the side lengths. For a rectangle, $P = \ell + w + \ell + w$.
When a side is described with "more than" or "less than" (for example, "the length is 4 feet more than the width"), the length is $x + 4$ and the perimeter equation becomes $x + (x + 4) + x + (x + 4) = P$.
We want to fence a small rectangular garden plot to plant tomatoes. The length of the plot is twice the width. If we used $18$ feet of fencing to completely surround it…
We want to fence a small rectangular garden plot to plant tomatoes. The length of the plot is twice the width. If we used $18$ feet of fencing to completely surround it (perimeter), what are the width and length of the plot?
We want to design a rectangular area for an esports tournament in the school gym. The length of the area must be $8$ feet greater than the width. If we need exactly $56$…
We want to design a rectangular area for an esports tournament in the school gym. The length of the area must be $8$ feet greater than the width. If we need exactly $56$ feet of safety tape to enclose the total perimeter, what are the dimensions (length and width) of the gaming area?
A gardener wants to fence a rectangular plot with a perimeter of $120$ feet. To make better use of the land, he decided that the length should be twice the width minus…
A gardener wants to fence a rectangular plot with a perimeter of $120$ feet. To make better use of the land, he decided that the length should be twice the width minus $6$ feet. What will be the final dimensions of the plot?
Isosceles Triangle Problems
An isosceles triangle has two sides of equal length. So if one of the equal sides is $s$, the other is also $s$, and the perimeter is base $+\ s + s$. Don't forget to count the equal side twice.
We are designing a logo for an esports team in the shape of an isosceles triangle. We want the perimeter to be $30$ cm. If the base measures $10$ cm and the other two…
We are designing a logo for an esports team in the shape of an isosceles triangle. We want the perimeter to be $30$ cm. If the base measures $10$ cm and the other two sides are equal, how much must each of those sides ($x$) measure?
The perimeter of a technology park shaped like an isosceles triangle is $110$ feet. We know that each of the equal sides measures $10$ feet more than twice the base.…
The perimeter of a technology park shaped like an isosceles triangle is $110$ feet. We know that each of the equal sides measures $10$ feet more than twice the base. Calculate the length of each of the three sides of the park by setting up an equation.
Balance Scale Problems
A balance scale is a picture of an equation. If the scale is balanced, the left pan equals the right pan. Whatever you do to one pan, you must do to the other. That's the same rule you use to solve equations.
Try it. Three identical boxes (each weighing $x$) and a 2 g weight balance a 14 g weight. Move the sliders and watch what makes the scale balance.
Some scales have boxes on both pans. Then take the same number of boxes off both sides until the boxes are only on one side.
In the science lab, we have a balanced scale. On the left pan, there are $2$ identical boxes (weighing $x$ each) and a $1$ kg weight. On the right pan, there is a $7$ kg…
In the science lab, we have a balanced scale. On the left pan, there are $2$ identical boxes (weighing $x$ each) and a $1$ kg weight. On the right pan, there is a $7$ kg weight. Write the equation that represents this balance and calculate how much each box weighs.
To balance a scale in the science lab, we placed a stone of unknown weight $x$ and a $12$ gram weight on the left pan. On the right pan, for it to be balanced, we had to…
To balance a scale in the science lab, we placed a stone of unknown weight $x$ and a $12$ gram weight on the left pan. On the right pan, for it to be balanced, we had to put four identical weights, each with the same weight $x$ as the original stone. How much does the stone weigh?
Counts and Values
Some problems give two totals: how many items there are, and how much they are worth altogether (money, points, wheels…). Count and value are different. To get value, multiply how many by the value of each one.
The trick: if there are 18 items in total and $x$ of them are one type, then the other type has $18 - x$.
With money, the value of each coin goes in the "each" column, for example $\$0.50$ for a half-dollar. The equation uses dollars, so $0.50(35 - x)$ means the value of the half-dollars when there are $35 - x$ of them.
In a shopping mall parking lot, there is a total of $32$ vehicles, including cars and motorcycles. If we count the wheels of all of them (excluding spare tires), we get…
In a shopping mall parking lot, there is a total of $32$ vehicles, including cars and motorcycles. If we count the wheels of all of them (excluding spare tires), we get a total of $94$ wheels. Calculate how many cars and how many motorcycles there are in the parking lot.
(Hint: If we call $x$ the number of cars, the number of motorcycles will be $32 - x$).
In a piggy bank, there are $\$0,50$ and $\$1$ coins. In total, there are $35$ coins that add up to a value of $\$26…
In a piggy bank, there are $\$0,50$ and $\$1$ coins. In total, there are $35$ coins that add up to a value of $\$26$. How many coins of each type are in the piggy bank? (Hint: if you call $x$ the number of $\$1$ coins, the number of $\$0,50$ coins will be $35 - x$).
Fractions of What Is Left
When a problem says "used $\tfrac{1}{3}$ on Saturday, and then $\tfrac{1}{2}$ of what was left on Sunday," the second fraction applies to a smaller amount. A tape diagram makes this clear.
Let's say a campground water tank holds $x$ gallons. Campers use $\tfrac{1}{3}$ of the water on Saturday and $\tfrac{1}{2}$ of what's left on Sunday. At the end, 40 gallons remain.
You can also work backward. The 40 gallons are half of what was left on Saturday, so 80 gallons were left. Those 80 gallons are $\tfrac{2}{3}$ of the start, so the start was $80 \div \tfrac{2}{3} = 80 \times \tfrac{3}{2} = 120$ gallons. Both ways give the same answer.
On a scientific expedition through the Amazon, the team consumes $\frac{1}{4}$ of their water supplies on the first day. On the second day they consume $\frac{2}{5}$ of…
On a scientific expedition through the Amazon, the team consumes $\frac{1}{4}$ of their water supplies on the first day. On the second day they consume $\frac{2}{5}$ of what was left in the tank. If after these two days they still have $135$ gallons, how many gallons of water did they have at the start of the expedition?
Summary
Back to the start: let $x$ be Jayden's points. Marcus has $2x$, so $x + 2x = 27$, which gives $x = 9$. Jayden scored 9 points and Marcus scored 18.