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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)

1.2014 AMC 10B #4
Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?
1. A) $\frac{3}{2}$
2. B) $\frac{5}{3}$
3. C) $\frac{7}{4}$
4. D) $2$
5. E) $\frac{13}{4}$
2.2014 AMC 10A #3
Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $\$2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\$0.75$ for her to make. In dollars, what is her profit for the day?
1. A) $24$
2. B) $36$
3. C) $44$
4. D) $48$
5. E) $52$
3.2014 AMC 10A #2
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
1. A) $\text{Tuesday}$
2. B) $\text{Wednesday}$
3. C) $\text{Thursday}$
4. D) $\text{Friday}$
5. E) $\text{Saturday}$
4.2014 AMC 10B #1
Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies as nickels. In cents, how much are Leah's coins worth?
1. A) $33$
2. B) $35$
3. C) $37$
4. D) $39$
5. E) $41$
5.2013 AMC 10A #1
A taxi ride costs $1.50 plus$ 0.25 per mile traveled. How much does a 5-mile taxi ride cost?
1. A) $2.25$
2. B) $2.50$
3. C) $2.75$
4. D) $3.00$
5. E) $3.75$
6.2013 AMC 10A #5
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$ , Dorothy paid $\$125$ , and Sammy paid $\$175$ . In order to share costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$ ?
1. A) $15$
2. B) $20$
3. C) $25$
4. D) $30$
5. E) $35$
7.2013 AMC 10B #3
On a particular January day, the high temperature in Lincoln, Nebraska, was $16$ degrees higher than the low temperature, and the average of the high and the low temperatures was $3\,^\circ$ . In degrees, what was the low temperature in Lincoln that day?
1. A) $-13$
2. B) $-8$
3. C) $-5$
4. D) $-3$
5. E) $11$
8.2012 AMC 10B #7
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?
1. A) $30$
2. B) $36$
3. C) $42$
4. D) $48$
5. E) $54$
9.2012 AMC 10A #5
Last year $100$ adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was $4$ . What was the total number of cats and kittens received by the shelter last year?
1. A) $150$
2. B) $200$
3. C) $250$
4. D) $300$
5. E) $400$
10.2012 AMC 10B #1
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?
1. A) $48$
2. B) $56$
3. C) $64$
4. D) $72$
5. E) $80$
11.2012 AMC 10A #1
Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes?
1. A) $10$
2. B) $15$
3. C) $20$
4. D) $25$
5. E) $30$
12.2011 AMC 10A #1
A cell phone plan costs $\$20$ each month, plus $5$ ¢ per text message sent, plus $10$ ¢ for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
1. A)$ 24.00
2. B)$ 24.50
3. C)$ 25.50
4. D)$ 28.00
5. E)$ 30.00
13.2011 AMC 10B #4
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip, it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$ . How many dollars must LeRoy give to Bernardo so that they share the costs equally?
1. A) $\frac{A + B}{2}$
2. B) $\dfrac{A - B}{2}$
3. C) $\dfrac{B - A}{2}$
4. D) $B - A$
5. E) $A + B$
14.2010 AMC 10A #3
Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
1. A) $3$
2. B) $13$
3. C) $18$
4. D) $25$
5. E) $29$
15.2010 AMC 10A #4
A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
1. A) $50.2$
2. B) $51.5$
3. C) $52.4$
4. D) $53.8$
5. E) $55.2$
16.2009 AMC 10B #1
Each morning of her five-day workweek, Jane bought either a $50$ -cent muffin or a $75$ -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
17.2009 AMC 10A #1
One can holds $12$ ounces of soda, what is the minimum number of cans needed to provide a gallon ( $128$ ounces) of soda?
1. A) $7$
2. B) $8$
3. C) $9$
4. D) $10$
5. E) $11$
18.2009 AMC 10A #4
Eric plans to compete in a triathlon. He can average $2$ miles per hour in the $\frac{1}{4}$ -mile swim and $6$ miles per hour in the $3$ -mile run. His goal is to finish the triathlon in $2$ hours. To accomplish his goal what must his average speed in miles per hour, be for the $15$ -mile bicycle ride?
1. A) $\frac{120}{11}$
2. B) $11$
3. C) $\frac{56}{5}$
4. D) $\frac{45}{4}$
5. E) $12$
19.2008 AMC 10B #4
A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $\$15,000$ and that the total of all players' salaries for each team cannot exceed $\$700,000.$ What is the maximum possible salary, in dollars, for a single player?
1. A) $270,000$
2. B) $385,000$
3. C) $400,000$
4. D) $430,000$
5. E) $700,000$
20.2007 AMC 10A #4
The larger of two consecutive odd integers is three times the smaller. What is their sum?
1. A) $4$
2. B) $8$
3. C) $12$
4. D) $16$
5. E) $20$

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	B	Calvin's bill is twice Susie's, so 2 muffins and 16 bananas cost the same as 8 muffins and 6 bananas; cancel what they share.
2	E	Follow the loaves in three groups, 24, 16 and 8, to get 88 dollars of sales. Then subtract what all 48 loaves cost to make, 36 dollars.
3	C	One day uses 7/12 of a can. Ten days use 70/12 of the 72/12 in the box, so the last bit is eaten on the 11th morning.
4	C	One more nickel would make the two piles equal, so right now there is one more penny than nickel: 7 pennies and 6 nickels.
5	C	The 1.50 is paid once, but the 0.25 is paid again for every mile, so add 1.50 to 5 times 0.25.
6	B	Everyone should pay the same amount, one third of 405, which is 135. Each person gives Sammy the difference between 135 and what they already paid.
7	C	If two numbers average 3, they add up to 6. The high is 16 more than the low, so the low is 8 below the average.
8	D	Both animals hid the same total. Write the chipmunk's total as 3 times its holes and the squirrel's as 4 times 4 fewer holes, and set them equal.
9	B	Follow the halves one at a time: 50 females, 25 litters, 25 times 4 = 100 kittens, then add the 100 adult cats.
10	C	Each classroom has 16 more students than rabbits. There are 4 classrooms, so there are 4 times 16, which is 64, more in all.
11	D	Work out how many cupcakes each person frosts in the 5 minutes on their own, then add the two counts.
12	D	Split the bill into three parts. Only the extra half hour costs money, and half an hour is 30 minutes at 10 cents each.
13	C	Each person should pay half of the total. LeRoy gives Bernardo the difference between that half and what he already paid.
14	D	Passing marbles keeps the total at 108. It ends split 2 to 1, so Tyrone keeps 72 and gave away 25.
15	B	First decide how many discs are needed: 7 discs hold only 392 minutes, so 8 are needed. Then share 412 minutes equally: 412 / 8 = 51.5.
16	B	Five muffins cost 2 dollars and 50 cents, and each swap to a bagel adds 25 cents, so two swaps reach a whole dollar amount.
17	E	Ten cans hold only 120 ounces, which is 8 short, so one more can is needed even though it will not be full.
18	A	Find how long the swim and the run take, subtract from 2 hours, and use the time that is left for the 15-mile bike ride.
19	C	To make one salary as large as possible, pay everyone else the smallest amount allowed. The star gets whatever is left of the 700,000.
20	A	Two odd numbers in a row are 2 apart. If the bigger one is three times the smaller, the difference of 2 must equal twice the smaller.

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.