Printable problems

Linear Equations & Word Problems

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

Print without answers Back to the topic

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)

1.2017 AMC 10B #7
Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
1. A) $2.0$
2. B) $2.2$
3. C) $2.8$
4. D) $3.4$
5. E) $4.4$
2.2016 AMC 10A #8
Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
1. A) $20$
2. B) $30$
3. C) $35$
4. D) $40$
5. E) $45$
3.2016 AMC 10B #13
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?
1. A) $25$
2. B) $40$
3. C) $64$
4. D) $100$
5. E) $160$
4.2016 AMC 10B #7
The ratio of the measures of two acute angles is $5:4$ , and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
1. A) $75$
2. B) $90$
3. C) $135$
4. D) $150$
5. E) $270$
5.2016 AMC 10B #11
Sola decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Sola’s garden?
1. A) $256$
2. B) $336$
3. C) $384$
4. D) $448$
5. E) $512$
6.2015 AMC 10A #8
Two years ago Pete was three times as old as his cousin Claire. 2 years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ?
1. A) $2$
2. B) $4$
3. C) $5$
4. D) $6$
5. E) $8$
7.2014 AMC 10B #8
A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes?
1. A) $\frac{b}{1080t}$
2. B) $\frac{30t}{b}$
3. C) $\frac{30b}{t}$
4. D) $\frac{10t}{b}$
5. E) $\frac{10b}{t}$
8.2014 AMC 10B #5
Camden constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$ , and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?
1. A) $26$
2. B) $28$
3. C) $30$
4. D) $32$
5. E) $34$
9.2013 AMC 10A #4
A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
1. A) $35$
2. B) $40$
3. C) $45$
4. D) $50$
5. E) $55$
10.2013 AMC 10B #10
A basketball team's players were successful on 50% of their two-point shots and 40% of their three-point shots, which resulted in 54 points. They attempted 50% more two-point shots than three-point shots. How many three-point shots did they attempt?
1. A) $10$
2. B) $15$
3. C) $20$
4. D) $25$
5. E) $30$
11.2012 AMC 10B #13
It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?
1. A) $36$
2. B) $40$
3. C) $42$
4. D) $48$
5. E) $52$
12.2011 AMC 10B #6
On Halloween Casper ate $1/3$ of his candies and then gave $2$ candies to his brother. The next day he ate $1/3$ of his remaining candies and then gave $4$ candies to his sister. On the third day he ate his final $8$ candies. How many candies did Casper have at the beginning?
1. A) $30$
2. B) $39$
3. C) $48$
4. D) $57$
5. E) $66$
13.2011 AMC 10A #12
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?
1. A) $13$
2. B) $14$
3. C) $15$
4. D) $16$
5. E) $17$
14.2010 AMC 10A #8
Tony works $2$ hours a day and is paid $\$0.50$ per hour for each full year of his age. During a six month period Tony worked $50$ days and earned $\$630$ . How old was Tony at the end of the six month period?
1. A) $9$
2. B) $11$
3. C) $12$
4. D) $13$
5. E) $14$
15.2010 AMC 10A #13
Angelina drove at an average rate of $80$ km/h and then stopped $20$ minutes for gas. After the stop, she drove at an average rate of $100$ km/h. Altogether she drove $250$ km in a total trip time of $3$ hours including the stop. Which equation could be used to solve for the time $t$ in hours that she drove before her stop?
1. A) $80t + 100\left(\frac{8}{3} -t\right) = 250$
2. B) $80t = 250$
3. C) $100t = 250$
4. D) $90t = 250$
5. E) $80\left(\frac{8}{3} -t\right) + 100t = 250$
16.2010 AMC 10B #14
The average of the numbers $1, 2, 3,\cdots, 98, 99,$ and $x$ is $100x$ . What is $x$ ?
1. A) $\dfrac{49}{101}$
2. B) $\dfrac{50}{101}$
3. C) $\dfrac{1}{2}$
4. D) $\dfrac{51}{101}$
5. E) $\dfrac{50}{99}$
17.2010 AMC 10B #10
Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?
1. A) $18$
2. B) $21$
3. C) $24$
4. D) $27$
5. E) $30$
18.2009 AMC 10B #15
When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$ , what is the total weight in kilograms when the bucket is full of water?
1. A) $\frac23a + \frac13b$
2. B) $\frac32a - \frac12b$
3. C) $\frac32a + b$
4. D) $\frac32a + 2b$
5. E) $3a - 2b$
19.2008 AMC 10A #11
While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing toward the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?
1. A) $2$
2. B) $4$
3. C) $6$
4. D) $8$
5. E) $10$
20.2008 AMC 10A #13
Doug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $t$ ?
1. A) $\left(\frac{1}{5}+\frac{1}{7}\right)\left(t+1\right)=1$
2. B) $\left(\frac{1}{5}+\frac{1}{7}\right)t+1=1$
3. C) $\left(\frac{1}{5}+\frac{1}{7}\right)t=1$
4. D) $\left(\frac{1}{5}+\frac{1}{7}\right)\left(t-1\right)=1$
5. E) $\left(5+7\right)t=1$

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	C	Both halves have the same length d, and the walking distance is exactly d. Biking time d/17 plus walking time d/5 equals 44 minutes, written in hours.
2	C	Work backwards from 0 coins. To undo a crossing, add back the 40 coins and then cut the amount in half. Three undo steps give 20, then 30, then 35.
3	D	With q sets of quadruplets there are 4q sets of triplets and 12q sets of twins, so the babies number 4q + 12q + 24q = 40q = 1000.
4	C	Call the angles 5x and 4x. The smaller angle has the bigger complement, so 90 - 4x = 2(90 - 5x), which gives x = 15.
5	B	Corners are shared, so n posts per short side and 2n per long side give 6n - 4 = 20. Each side is (posts - 1) gaps of 4 yards.
6	B	Name the ages from two years ago so Pete is 3 times Claire. The four-years-ago fact then gives Claire 6, so today they are 20 and 8.
7	E	Two conversions are needed: 3 minutes is 180 seconds, and every 3 feet is one yard. Find feet per second first, then scale.
8	A	Call the panes 2x wide and 5x tall. Across there are 4 panes and 5 borders, down there are 2 panes and 3 borders; the two totals must match.
9	C	A win means the team scored double the other team, so its score must be even. That makes the even scores the wins and the odd scores the one-run losses.
10	C	Let x be the three-point tries; then there are 1.5x two-point tries. Points are 1.5x + 1.2x = 2.7x, and 2.7x = 54 gives x = 20.
11	B	Speeds add, not times. Pretend the escalator has 120 steps: Clea walks 2 steps per second, and with the escalator she moves 5, so the escalator alone moves 3.
12	A	Undo the story backwards from the end. Add back what he gave away, then remember what is left after eating a third is two thirds of what he had.
13	A	Call the number of two-point shots y. They give 2y points, three-pointers also give 2y points, and free throws give y + 1 points, so 5y + 1 = 61.
14	D	Each day Tony earns exactly his age in dollars. The daily average 630 / 50 = 12.6 is not whole, so he turned 13 during the period.
15	A	Take the 20-minute stop out of the 3 hours first: she drove for 8/3 hours. Then distance = 80 times the first part plus 100 times the rest.
16	B	There are 100 numbers, not 99. Their sum is 4950 + x, and that sum must equal 100 times 100x, so 9999x = 4950.
17	C	Change the speeds to miles per minute: 1/2 in the sun and 1/3 in the rain. Then one equation with the rain minutes as the unknown finishes it.
18	E	Going from half full to two-thirds full adds one-sixth of the water, and that one-sixth weighs a - b. Add the last one-third (twice that) to the two-thirds weighing.
19	D	The trip to shore takes 15 minutes, so the boat may gain at most 30 ÷ 15 = 2 gallons each minute. LeRoy must bail the other 8.
20	D	Together they paint 1/5 + 1/7 of the room each hour, but they only paint for t minus 1 hours, because one hour of the total is lunch.

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.