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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

all 105 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 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) $27$
5.2023 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) $9$
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 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) $6 \frac{3}{4}$
8.2021 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) $64$
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 #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) $20$
11.2020 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) $\frac{5}{6}$
12.2020 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) $26$
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 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) $15$
16.2017 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) $\text{8 minutes and 10 seconds}$
17.2017 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) $50$
18.2015 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) $\frac{5}{2}$
19.2015 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) $11$
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) $18$
21.2014 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) $\frac{13}{4}$
22.2014 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) $52$
23.2014 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) $\text{Saturday}$
24.2014 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) $41$
25.2013 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) $11$
26.2013 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) $35$
27.2013 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) $3.75$
28.2012 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) $54$
29.2012 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) $400$
30.2012 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) $80$
31.2012 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) $30$
32.2011 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) $A + B$
33.2011 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
34.2010 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) $55.2$
35.2010 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) $29$
36.2009 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) $5$
37.2009 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) $12$
38.2009 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) $11$
39.2008 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) $700,000$
40.2007 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) $20$
41.2006 AMC 10A #5
Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
42.2006 AMC 10A #1
Sandwiches at Joe's Fast Food cost $\$3$ each and sodas cost $\$2$ each. How many dollars will it cost to purchase $5$ sandwiches and $8$ sodas?
1. A) $31$
2. B) $32$
3. C) $33$
4. D) $34$
5. E) $35$
43.2006 AMC 10B #3
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of $34$ points, and the Cougars won by a margin of $14$ points. How many points did the Panthers score?
1. A) $10$
2. B) $14$
3. C) $17$
4. D) $20$
5. E) $24$
44.2005 AMC 10A #3
The equations $2x + 7 = 3$ and $bx - 10 = -2$ have the same solution $x$ . What is the value of $b$ ?
1. A) $-8$
2. B) $-4$
3. C) $-2$
4. D) $4$
5. E) $8$
45.2005 AMC 10A #5
A store normally sells windows at $\$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
1. A) $100$
2. B) $200$
3. C) $300$
4. D) $400$
5. E) $500$
46.2004 AMC 10A #1
You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?
1. A) $250$
2. B) $300$
3. C) $1500$
4. D) $7500$
5. E) $9000$
47.2003 AMC 10B #2
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $\$1$ more than a pink pill, and Al's pills cost a total of $\$546$ for the two weeks. How much does one green pill cost?
1. A)$7
2. B)$14
3. C)$19
4. D)$20
5. E)$39
48.2003 AMC 10A #2
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?
1. A) $77$
2. B) $91$
3. C) $143$
4. D) $182$
5. E) $286$
49.2003 AMC 10A #4
It takes Anna $30$ minutes to walk uphill $1$ km from her home to school, but it takes her only $10$ minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?
1. A) $3$
2. B) $3.125$
3. C) $3.5$
4. D) $4$
5. E) $4.5$
50.2002 AMC 10A #6
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?
1. A) $15$
2. B) $34$
3. C) $43$
4. D) $51$
5. E) $138$
51.2001 AMC 10 #3
The sum of two numbers is $S$ . Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
1. A) $2S + 3$
2. B) $3S + 2$
3. C) $3S + 6$
4. D) $2S + 6$
5. E) $2S + 12$
52.2001 AMC 10 #2
A number $x$ is $2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie?
1. A) $-4\le x\le -2$
2. B) $-2 < x\le 0$
3. C) $0 < x \le 2$
4. D) $2 < x\le 4$
5. E) $4 < x\le 6$
53.2000 AMC 10 #4
Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $\$12.48$ , but in January her bill was $\$17.54$ because she used twice as much connect time as in December. What is the fixed monthly fee?
1. A)$ 2.53
2. B)$ 5.06
3. C)$ 6.24
4. D)$ 7.42
5. E)$ 8.77
54.2024 AMC 10A #8
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1:00 PM$ and were able to pack $4$ , $3$ , and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group, and Daria was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2:45 PM$ . At what time did Daria join the group?
1. A) $1:25\text{ PM}$
2. B) $1:35\text{ PM}$
3. C) $1:45\text{ PM}$
4. D) $1:55\text{ PM}$
5. E) $2:05\text{ PM}$
55.2023 AMC 10A #8
Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at $110$ degrees Fahrenheit, which is $0$ degrees on the Breadus scale. Bread is baked at $350$ degrees Fahrenheit, which is $100$ degrees on the Breadus scale. Bread is done when its internal temperature is $200$ degrees Fahrenheit. What is this, in degrees, on the Breadus scale?
1. A) $33$
2. B) $34.5$
3. C) $36$
4. D) $37.5$
5. E) $39$
56.2021 AMC Fall 10B #4
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$
1. A) $10$
2. B) $30$
3. C) $60$
4. D) $100$
5. E) $120$
57.2021 AMC Fall 10A #6
Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$ st pole along this road is exactly one mile ( $5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
1. A) $6$
2. B) $8$
3. C) $10$
4. D) $11$
5. E) $15$
58.2021 AMC 10A #6
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?
1. A) $\frac{12}{13}$
2. B) $1$
3. C) $\frac{13}{12}$
4. D) $\frac{24}{13}$
5. E) $2$
59.2018 AMC 10A #8
Joe has a collection of $23$ coins, consisting of $5$ -cent coins, $10$ -cent coins, and $25$ -cent coins. He has $3$ more $10$ -cent coins than $5$ -cent coins, and the total value of his collection is $320$ cents. How many more $25$ -cent coins does Joe have than $5$ -cent coins?
1. A) $0$
2. B) $1$
3. C) $2$
4. D) $3$
5. E) $4$
60.2017 AMC 10A #9
Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$ , a distance of $10$ km all uphill, then from town $B$ to town $C$ , a distance of $15$ km all downhill, and then back to town $A$ , a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$ -km ride than it takes Penny?
1. A) $45$
2. B) $60$
3. C) $65$
4. D) $90$
5. E) $95$
61.2017 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) $4.4$
62.2016 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) $45$
63.2016 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) $160$
64.2016 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) $270$
65.2016 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) $512$
66.2015 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) $8$
67.2014 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) $\frac{10b}{t}$
68.2014 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) $34$
69.2013 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) $55$
70.2013 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) $30$
71.2012 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) $52$
72.2011 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) $66$
73.2011 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) $17$
74.2010 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) $14$
75.2010 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) $80\left(\frac{8}{3} -t\right) + 100t = 250$
76.2010 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) $\dfrac{50}{99}$
77.2010 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) $30$
78.2009 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) $3a - 2b$
79.2008 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) $10$
80.2008 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) $\left(5+7\right)t=1$
81.2007 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) $6$
82.2007 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) $17$
83.2005 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) $8$
84.2004 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) $26$
85.2004 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
86.2003 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) $3$
87.2003 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
88.2002 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) $5$
89.2002 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) $58$
90.2021 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) $126$
91.2014 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) $280$
92.2012 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) $10,000$
93.2011 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) $840$
94.2010 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) $100$
95.2010 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) $33$
96.2009 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) $80$
97.2008 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) $1900$
98.2008 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) $160$
99.2007 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) $\frac 78$
100.2006 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) $50$
101.2004 AMC 10A #17
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
1. A) $250$
2. B) $300$
3. C) $350$
4. D) $400$
5. E) $500$
102.2012 AMC 10A #19
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $8:00$ AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at $4:00$ PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at $2:12$ PM. On Wednesday Paula worked by herself and finished the house by working until $7:12$ P.M. How long, in minutes, was each day's lunch break?
1. A) $30$
2. B) $36$
3. C) $42$
4. D) $48$
5. E) $60$
103.2002 AMC 10B #25
When $15$ is appended to a list of integers, the mean is increased by $2$ . When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$ . How many integers were in the original list?
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $8$
104.2000 AMC 10 #22
One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
1. A) $3$
2. B) $4$
3. C) $5$
4. D) $6$
5. E) $7$
105.2008 AMC 10B #25
Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck travels at $10$ feet per second in the same direction as Michael and stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $8$

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	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.
5	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.
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	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.
8	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.
9	C	Lara's school is one share and Portia's is three shares, so the 2600 students make four equal shares.
10	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.
11	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.
12	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.
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	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.
16	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.
17	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.
18	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.
19	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.
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.
21	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.
22	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.
23	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.
24	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.
25	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.
26	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.
27	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.
28	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.
29	B	Follow the halves one at a time: 50 females, 25 litters, 25 times 4 = 100 kittens, then add the 100 adult cats.
30	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.
31	D	Work out how many cupcakes each person frosts in the 5 minutes on their own, then add the two counts.
32	C	Each person should pay half of the total. LeRoy gives Bernardo the difference between that half and what he already paid.
33	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.
34	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.
35	D	Passing marbles keeps the total at 108. It ends split 2 to 1, so Tyrone keeps 72 and gave away 25.
36	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.
37	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.
38	E	Ten cans hold only 120 ounces, which is 8 short, so one more can is needed even though it will not be full.
39	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.
40	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.
41	D	Give each slice its own price: every slice costs 1 dollar, and the 2-dollar anchovy charge is shared only by the 4 anchovy slices.
42	A	Find the cost of the sandwiches and the cost of the sodas separately, then add the two amounts.
43	A	Take away the 14-point winning margin from the total 34. What is left, 20, is split equally between the two teams, so the losing team has 10.
44	B	The first equation has no b, so solve it to get x = -2. Put that x into the second equation and solve for b.
45	A	Each group of 5 windows gets one free. Alone, 7 and 8 each earn one free window; together, 15 earns three. Buying together gains one extra free window.
46	A	Count the people first. You plus five friends is six people, so each person raises 1500 divided by 6.
47	D	Divide by the 14 days first: one green plus one pink costs 39 dollars. Two prices that add to 39 and differ by 1 are 20 and 19.
48	B	Work out what one member spends: two pairs of socks and two shirts, which is 26 dollars. Then divide the total bill by 26.
49	A	Average speed is total distance divided by total time: 2 km in 40 minutes, which is 3 km per hour.
50	A	Undo Cindy's wrong steps backwards to find the starting number, 138. Then do the steps the teacher actually asked for.
51	E	Adding 3 to each of two numbers adds 6 to the total, and doubling both numbers doubles the total. So the new total is 2 times (S + 6).
52	C	A number's reciprocal times its opposite is always -1, whatever the number is. So the number equals -1 + 2 = 1.
53	D	January used exactly one extra December's worth of connect time, so the difference between the two bills is the cost of December's time.
54	A	Amy, Bomani and Charlie work the whole 105 minutes and pack 350 packages. Daria packs the other 100, which takes her 80 minutes, so she started 80 minutes before 2:45.
55	D	200 is 90 of the 240 degrees from 110 to 350, which is 3/8 of the way. On the new scale that is 3/8 of 100, or 37.5.
56	C	The gap shrinks by 5 + 3 = 8, so it becomes N - 8. That must be 2 or -2, giving N = 10 or N = 6.
57	B	41 poles make 40 gaps, so each gap is 132 feet. A stride is 132 / 44 = 3 feet and a leap is 132 / 12 = 11 feet.
58	A	Chantal and Jean walk for the same amount of time. Pick an easy halfway distance, such as 12 miles, and add up Chantal's three walking times.
59	C	Call the number of 5-cent coins x. Write the other two counts using x, and the total value becomes one equation in x.
60	C	Time each stretch as distance divided by speed. Penny rides the loop backward, so Minnie's uphill stretch is Penny's downhill one, and the other way round.
61	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.
62	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.
63	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.
64	C	Call the angles 5x and 4x. The smaller angle has the bigger complement, so 90 - 4x = 2(90 - 5x), which gives x = 15.
65	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.
66	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.
67	E	Two conversions are needed: 3 minutes is 180 seconds, and every 3 feet is one yard. Find feet per second first, then scale.
68	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.
69	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.
70	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.
71	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.
72	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.
73	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.
74	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.
75	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.
76	B	There are 100 numbers, not 99. Their sum is 4950 + x, and that sum must equal 100 times 100x, so 9999x = 4950.
77	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.
78	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.
79	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.
80	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.
81	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.
82	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.
83	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.
84	E	The 24 granddaughters come in groups of 6, so only 4 daughters are mothers. Everyone else, 30 - 4 = 26 women, has no daughters.
85	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.
86	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.
87	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.
88	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.
89	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.
90	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.
91	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.
92	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.
93	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.
94	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.
95	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.
96	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.
97	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.
98	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.
99	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.
100	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.
101	C	Together they run half a lap before the first meeting and a full lap between meetings, so Brenda runs 200 more; 200 + 150 is one lap.
102	D	Tuesday's 24% plus Wednesday's 26% equals Monday's 50%. Comparing them gives the helpers' speed compared with Paula's, and then only the lunch break is unknown.
103	A	Call the count n and the mean m, so the total is mn. Each mean change gives one equation: m + 2n = 13 and 2m + n = 14.
104	C	Angela drank one person's share, 1/p of everything, made of a quarter of the milk and a sixth of the coffee. So 1/p is between 1/6 and 1/4.
105	B	Track the truck's lead over Michael. Each 50-second cycle it gains 100 feet while driving and loses 150 while stopped, so the lead shrinks by 50.

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