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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)

1.2007 AMC 10B #12
Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$ ?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
2.2007 AMC 10B #6
The $2007 \text{ AMC }10$ will be scored by awarding $6$ points for each correct response, $0$ points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the $25$ problems, Sarah has decided to attempt the first $22$ and leave only the last $3$ unanswered. How many of the first $22$ problems must she solve correctly in order to score at least $100$ points?
1. A) $13$
2. B) $14$
3. C) $15$
4. D) $16$
5. E) $17$
3.2005 AMC 10A #7
Josh and Mike live $13$ miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $8$
4.2004 AMC 10A #6
Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
1. A) $22$
2. B) $23$
3. C) $24$
4. D) $25$
5. E) $26$
5.2004 AMC 10B #15
Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth?
1. A)$1.15
2. B)$1.20
3. C)$1.25
4. D)$1.30
5. E)$1.35
6.2003 AMC 10B #5
Moe uses a mower to cut his rectangular $90$ -foot by $150$ -foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn?
1. A) $0.75$
2. B) $0.8$
3. C) $1.35$
4. D) $1.5$
5. E) $3$
7.2003 AMC 10B #12
Al, Betty, and Clare split $\$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year, they have a total of $\$1500$ dollars. Betty and Clare have both doubled their money, whereas Al has managed to lose $\$100$ dollars. What was Al's original portion?
1. A)$250
2. B)$350
3. C)$400
4. D)$450
5. E)$500
8.2002 AMC 10B #12
For which of the following values of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$ ?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
9.2002 AMC 10A #12
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
1. A) $45$
2. B) $48$
3. C) $50$
4. D) $55$
5. E) $58$
10.2021 AMC Fall 10A #11
Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?
1. A) $70$
2. B) $84$
3. C) $98$
4. D) $105$
5. E) $126$
11.2014 AMC 10A #15
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home?
1. A) $140$
2. B) $175$
3. C) $210$
4. D) $245$
5. E) $280$
12.2012 AMC 10A #16
Three runners start running simultaneously from the same point on a $500$ -meter circular track. They each run clockwise around the course maintaining constant speeds of $4.4$ , $4.8$ , and $5.0$ meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
1. A) $1,000$
2. B) $1,250$
3. C) $2,500$
4. D) $5,000$
5. E) $10,000$
13.2011 AMC 10A #15
Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?
1. A) $140$
2. B) $240$
3. C) $440$
4. D) $640$
5. E) $840$
14.2010 AMC 10B #11
A shopper plans to purchase an item that has a listed price greater than $\$100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\$30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds $\$100$ .
Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or Coupon C. What is $y - x$ ?
1. A) $50$
2. B) $60$
3. C) $75$
4. D) $80$
5. E) $100$
15.2010 AMC 10B #15
On a $50$ -question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$ . What is the maximum number of questions that Jesse could have answered correctly?
1. A) $25$
2. B) $27$
3. C) $29$
4. D) $31$
5. E) $33$
16.2009 AMC 10A #20
Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?
1. A) $20$
2. B) $30$
3. C) $55$
4. D) $65$
5. E) $80$
17.2008 AMC 10B #18
Bricklayer Brenda would take nine hours to build a chimney alone, and Bricklayer Brandon would take $10$ hours to build it alone. When they work together, they talk a lot, and their combined output decreases by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chimney?
1. A) $500$
2. B) $900$
3. C) $950$
4. D) $1000$
5. E) $1900$
18.2008 AMC 10A #15
Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?
1. A) $120$
2. B) $130$
3. C) $140$
4. D) $150$
5. E) $160$
19.2007 AMC 10A #13
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
1. A) $\frac 23$
2. B) $\frac 34$
3. C) $\frac 45$
4. D) $\frac 56$
5. E) $\frac 78$
20.2006 AMC 10A #15
Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $\text{250 m/min}$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $\text{300 m/min}$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the start do they pass each other?
1. A) $29$
2. B) $42$
3. C) $45$
4. D) $47$
5. E) $50$

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	D	Going back N years, Tom loses N years but his three children together lose 3N. Writing 'Tom then = twice the children then' gives T = 5N.
2	D	The three blank problems give 4.5 points for free, so the right answers must bring at least 95.5 more. Fifteen right is 90, too few; sixteen right is 96, enough.
3	B	Josh rides twice as long at four fifths the speed, so he covers 8 fifths of Mike's distance. Mike's 5 parts plus Josh's 8 parts make 13 miles.
4	E	The 24 granddaughters come in groups of 6, so only 4 daughters are mothers. Everyone else, 30 - 4 = 26 women, has no daughters.
5	A	Each nickel swapped gains 5 cents and each dime swapped loses 5. A 70-cent gain means 14 more nickels than dimes: 17 and 3.
6	C	Because of the 4-inch overlap, each pass mows only 24 inches = 2 feet of new grass. Walking 5000 feet an hour then clears 10000 square feet an hour.
7	C	Treat Betty and Clare as one team whose money doubles. If Al starts with a, the team has 1000 - a, and the total 1500 gives a = 400.
8	E	Cross-multiply. The x squared terms cancel, leaving (k-5)x = 2k-6. At k = 5 the left side is always 0 but the right side is 4.
9	B	The two trips differ by 6 minutes. That tells you the distance, 12 miles; then the on-time trip takes 15 minutes, so the speed is 48.
10	A	Walking with the ship, Emily gains one ship length at the difference of the speeds; walking against it, they close one ship length at the sum.
11	C	Look only at the distance left after the first hour. Driving it at 35 instead of 50 miles per hour takes exactly one and a half hours longer.
12	C	Two runners are together whenever the faster has gained a whole lap, 500 m. Find when each pair meets, then the first time on both lists.
13	C	Miles per gallon for the whole trip is total miles divided by total gallons. Only the miles after the first 40 use any gas.
14	A	For price p, Coupon A saves 0.15p, B saves 30, C saves 0.25(p - 100). A ties B at p = 200 and ties C at p = 250.
15	C	With c correct answers, Jesse needs 4c - 99 wrong ones to get 99 points. Correct plus wrong is at most 50, so c is at most 29.
16	D	The gap closes 1 km per minute, split 3 to 1, so Lauren rides 1/4 km per minute. After 5 minutes, she alone rides the last 15 km.
17	B	Measure speed in bricks per hour. Brenda lays one ninth of the chimney each hour and Brandon one tenth; together, minus 10 bricks, they must lay one fifth.
18	D	Expanding Han's distance shows Ian's speed plus 5 times Ian's time is 65. Jan's extra distance is twice that amount plus 20, which is 150.
19	B	Set the walking speed to 1 and biking to 7. Walking straight takes as long as walking home plus biking the whole way from home to the stadium.
20	D	Both runners take the same time for one lap, 2π/5 minutes. Running toward each other, they meet twice per lap time, once every π/5 minutes.

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.