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Linear Equations & Word Problems

Linear equations/inequalities, word problems, rates, work, mixtures, distance-speed-time

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Linear Equations & Word ProblemsAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

A linear equation in one unknown has exactly one solution, unless the $x$ terms cancel. Then you get something always true, as when $2(x+1)=2x+2$ becomes $0=0$ (every number works), or never true, as when $2x+1=2x+3$ becomes $0=2$ (no number works). Example: $3x+7=22$ has the one solution $x=5$ .
Distance $=$ rate $\times$ time, and work done $=$ rate $\times$ time. Finishing a job alone in $t$ hours means a rate of $\frac1t$ of the job per hour. When workers or pipes work together, their rates add. Example: pipes that fill a tank in $4$ and $12$ hours together fill $\frac14+\frac1{12}=\frac13$ of it per hour, so they take $3$ hours.
Average speed is total distance divided by total time, never the average of the speeds. Example: drive $30$ miles at $30$ mph ( $1$ hour) and back at $15$ mph ( $2$ hours). That is $60$ miles in $3$ hours, or $20$ mph, not $22.5$ .
Mixtures. The amount of the pure substance stays the same when you mix. If $V_1$ liters at concentration $c_1$ and $V_2$ liters at $c_2$ make a mix at concentration $c$ , then $c_1V_1+c_2V_2=c(V_1+V_2)$ . Example: $2$ liters of $10\%$ acid and $3$ liters of $20\%$ acid hold $0.2+0.6=0.8$ liters of acid in $5$ liters, so the mix is $16\%$ acid.
Ages. The difference between two people's ages never changes. Example: Ann is $8$ years older than Ben, so she is twice his age when Ben is $8$ and she is $16$ .

How AMC 10 tests it

Problems 1–8: a short word problem (ages, coins, a trip in two parts) that becomes one equation like $ax+b=c$ .
Problems 5–12: work or pipe problems where the key idea is to add rates, not times.
Round-trip speed problems where the average of the two speeds is one of the wrong answer choices.
Problems asking for something other than the natural unknown: $x+y$ , the other person's age, a percent.

Standard approaches

Give a letter to the unknown you are asked for, and turn each sentence into one equation.
Turn everything into a rate per unit of time. For trips, call the distance $1$ or a handy common multiple.
Pick a friendly total to avoid fractions: the least common multiple of the times, or $100$ for percents. Example: if Al does a job in $4$ hours and Bo in $6$ , call the job $12$ units. Their rates are $3$ and $2$ units per hour, so together the job takes $\frac{12}5$ hours.
If the equation gets messy, test the answer choices against the story.

Worked example

A pump fills an empty tank in $6$ hours, and a drain empties a full tank in $12$ hours. The tank starts empty with the pump running and the drain closed. After $t$ hours the drain is opened, and the tank becomes exactly full $5$ hours after that. What is $t$ ?

(A) $3$ (B) $\frac{10}{3}$ (C) $\frac{7}{2}$ (D) $4$ (E) $\frac{9}{2}$

Call a full tank $1$ . The pump adds $\frac16$ of a tank per hour and the drain removes $\frac1{12}$ . With both open, the tank gains $\frac16-\frac1{12}=\frac2{12}-\frac1{12}=\frac1{12}$ of a tank per hour.

During the first $t$ hours only the pump runs, so it adds $\frac t6$ . During the next $5$ hours both are open, adding $5\cdot\frac1{12}=\frac5{12}$ . Together these make one full tank:
$\frac t6+\frac5{12}=1\quad\Longrightarrow\quad \frac t6=\frac7{12}\quad\Longrightarrow\quad t=\frac{42}{12}=\frac72 .$
Check: in $\frac72$ hours the pump fills $\frac7{12}$ , and $\frac7{12}+\frac5{12}=1$ .

The answer is $\boxed{\textbf{(C)}\ \frac72}$ .

Pitfalls

Averaging the two speeds instead of dividing total distance by total time.
Adding times instead of rates. Example: if one person takes $3$ hours alone and another takes $4$ , together they take $\frac{12}7$ hours, not $7$ , which is longer than either one alone.
Solving for $x$ but answering a different quantity from the one asked.
Forgetting a unit conversion, such as minutes to hours. Example: $20$ minutes is $\frac13$ hour, not $0.2$ hour.

Traps that recur

Assuming that once Michael catches the truck he stays ahead, or missing the two meetings where the gap only touches zero at the end of a stop or a drive. (2008 AMC 10B #25)
Forgetting to subtract the lunch break from every day's working time, or converting 2:12 PM and 7:12 PM to hours incorrectly (372 and 672 minutes after 8:00). (2012 AMC 10A #19)
Comparing the second mean to the original mean instead of to the enlarged list's mean (it should be m + 2 - 1 = m + 1 with n + 2 numbers). (2002 AMC 10B #25)
Writing c + m = 8p and m/4 + c/6 = 8, then stalling with two equations in three unknowns instead of using that c and m are positive. (2000 AMC 10 #22)

1.2025 AMC 10A #1
Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at $1:30$ , traveling due north at a steady $8$ miles per hour. Betsy leaves on her bicycle from the same point at $2:30$ , traveling due east at a steady $12$ miles per hour. At what time will they be exactly the same distance from their common starting point?
1. A) $3:30$
2. B) $3:45$
3. C) $4:00$
4. D) $4:15$
5. E) $4:30$
2.2024 AMC 10A #2
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T=aL+bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?
1. A) $240$
2. B) $246$
3. C) $252$
4. D) $258$
5. E) $264$
3.2023 AMC 10A #2
The weight of $\frac{1}{3}$ of a large pizza together with $3 \frac{1}{2}$ cups of orange slices is the same weight of $\frac{3}{4}$ of a large pizza together with $\frac{1}{2}$ cups of orange slices. A cup of orange slices weigh $\frac{1}{4}$ of a pound. What is the weight, in pounds, of a large pizza?
1. A) $1\frac{4}{5}$
2. B) $2$
3. C) $2\frac{2}{5}$
4. D) $3$
5. E) $3\frac{3}{5}$
4.2023 AMC 10B #5
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$ . Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$ . How many numbers are written on the blackboard?
1. A) $10$
2. B) $5$
3. C) $6$
4. D) $8$
5. E) $9$
5.2023 AMC 10A #1
Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$ . Alicia bikes towards $B$ at $18$ miles per hour. Leaving at the same time, Beth bikes toward $A$ at $12$ miles per hour. How many miles from City $A$ will they be when they meet?
1. A) $20$
2. B) $24$
3. C) $25$
4. D) $26$
5. E) $27$
6.2022 AMC 10A #3
The sum of three numbers is $96.$ The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
7.2021 AMC 10B #4
At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
1. A) $23$
2. B) $32$
3. C) $37$
4. D) $41$
5. E) $64$
8.2021 AMC Fall 10A #2
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
1. A) $16$
2. B) $17$
3. C) $18$
4. D) $19$
5. E) $20$
9.2021 AMC 10A #2
Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have?
1. A) $600$
2. B) $650$
3. C) $1950$
4. D) $2000$
5. E) $2050$
10.2021 AMC Fall 10A #4
Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$ -mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A?
1. A) $2 \frac{3}{4}$
2. B) $3 \frac{3}{4}$
3. C) $4 \frac{1}{2}$
4. D) $5 \frac{1}{2}$
5. E) $6 \frac{3}{4}$
11.2020 AMC 10A #4
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
1. A) $20$
2. B) $22$
3. C) $24$
4. D) $25$
5. E) $26$
12.2020 AMC 10A #1
What value of $x$ satisfies $x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?$
1. A) ${-}\frac{2}{3}$
2. B) $\frac{7}{36}$
3. C) $\frac{7}{12}$
4. D) $\frac{2}{3}$
5. E) $\frac{5}{6}$
13.2019 AMC 10A #3
Ana and Bonita are born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$
1. A) $3$
2. B) $5$
3. C) $9$
4. D) $12$
5. E) $15$
14.2018 AMC 10B #2
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes?
1. A) $64$
2. B) $65$
3. C) $66$
4. D) $67$
5. E) $68$
15.2017 AMC 10B #5
Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating $10$ pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?
1. A) $10$
2. B) $20$
3. C) $30$
4. D) $40$
5. E) $50$
16.2017 AMC 10A #2
Pablo buys popsicles for his friends. The store sells single popsicles for $\$1$ each, $3$ -popsicle boxes for $\$2$ each, and $5$ -popsicle boxes for $\$3$ . What is the greatest number of popsicles that Pablo can buy with $\$8$ ?
1. A) $8$
2. B) $11$
3. C) $12$
4. D) $13$
5. E) $15$
17.2017 AMC 10B #2
Sofia ran $5$ laps around the $400$ -meter track at her school. For each lap, she ran the first $100$ meters at an average speed of $4$ meters per second and the remaining $300$ meters at an average speed of $5$ meters per second. How much time did Sofia take running the $5$ laps?
1. A) $\text{5 minutes and 35 seconds}$
2. B) $\text{6 minutes and 40 seconds}$
3. C) $\text{7 minutes and 5 seconds}$
4. D) $\text{7 minutes and 25 seconds}$
5. E) $\text{8 minutes and 10 seconds}$
18.2015 AMC 10A #2
A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box?
1. A) $3$
2. B) $5$
3. C) $7$
4. D) $9$
5. E) $11$
19.2015 AMC 10A #6
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?
1. A) $\frac{5}{4}$
2. B) $\frac{3}{2}$
3. C) $\frac{9}{5}$
4. D) $2$
5. E) $\frac{5}{2}$
20.2015 AMC 10B #3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$ , and one of the numbers is $28.$ What is the other number?
1. A) $8$
2. B) $11$
3. C) $14$
4. D) $15$
5. E) $18$

Linear Equations & Word Problems — answersAMC 10 Step by Step

Name: Date: October 28, 2026

Keep this page separate from the problems.

Answers

#	Answer	The idea it turns on
1	E	Only each rider's distance from home matters, and that is speed times riding time. Remember Betsy has ridden one hour less than Andy.
2	B	Both trails take 69 minutes, so subtract one equation from the other. That gives a simple link between a and b: a is 1000 times b.
3	A	Take away what the two sides share. What is left says 5/12 of a pizza weighs the same as 3 cups of orange slices, which is 3/4 pound.
4	A	Tripling every number triples the total, so the original total is 15. Adding 3 to every number adds 3 once for each number, and that extra is 30.
5	E	Each hour the two riders get 18 + 12 = 30 miles closer, so they meet after 1.5 hours. Then find how far Alicia rode.
6	E	Both other numbers are described using the third one, so call the third number c. Then the first is 6c and the second is c + 40.
7	B	The 23 blue–blue pairs use 46 blue students. The other 11 blue students each sit with a yellow, which leaves 64 yellows to pair with each other: 32 pairs.
8	E	Try both sides. Only cutting the 4-inch side gives 3 by 6 = 18, so the other side is the 6-inch side, and 4 by 5 = 20.
9	C	Lara's school is one share and Portia's is three shares, so the 2600 students make four equal shares.
10	B	Find the time for each piece of each route in minutes. Route A takes 12 minutes. Route B takes 6.75 + 1.5 = 8.25 minutes. Subtract.
11	E	The answer is per hour, so look at one hour. She drives 60 miles, earns 30 dollars, uses 2 gallons costing 4 dollars, and keeps 26 dollars.
12	E	Work out the right side first (it is 1/12), then add 3/4 to both sides. Using twelfths for every fraction keeps the arithmetic small.
13	D	With Bonita b and Ana b squared, last year's condition is b squared minus 1 = 5(b minus 1). Cancel the shared factor b minus 1.
14	D	Each 30 minutes is half an hour. Find the miles driven in the first two half hours, subtract from 96, and double what is left.
15	D	Call the cherry count c, so blueberry is 2c. After eating 10 of each, 2c - 10 = 3(c - 10), so c = 20 and blueberry is 40.
16	D	The 5-popsicle box is the cheapest per popsicle. Buy two of them for $6, then spend the last $2 on a 3-popsicle box.
17	C	Find the time for each part of a lap: 100 meters takes 25 seconds and 300 meters takes 60 seconds, so one lap is 85 seconds.
18	D	Pretend all 25 tiles are triangles: that is 75 edges. Each triangle swapped for a square adds one edge, so the 9 extra edges mean 9 squares.
19	B	The sum equals 5 times the difference. Gathering terms gives 4 times the larger equals 6 times the smaller, so the ratio is 3 to 2.
20	A	If 28 is written twice, the other number is 44/3, not a whole number. So 28 is written three times and the other is 8.

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.