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Ratios, Percents & Averages

Ratio, proportion, percent change, means/averages as arithmetic word problems

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Ratios, Percents & AveragesAMC 10 Step by Step

Name: Date: October 28, 2026

all 79 problems, easiest first

What you need to know

A ratio means "the same number of parts." If $a:b:c=p:q:r$ , then $a=pk$ , $b=qk$ , and $c=rk$ for one number $k$ (the size of one part). Work with $3k$ and $5k$ instead of fractions. Example: boys to girls is $3:5$ in a class of $32$ . Then $3k+5k=8k=32$ , so $k=4$ : $12$ boys and $20$ girls.
Percent changes multiply. A $p\%$ increase multiplies by $1+\frac{p}{100}$ , and a $p\%$ decrease multiplies by $1-\frac{p}{100}$ . Several changes in a row multiply together. Example: up $20\%$ and then down $20\%$ is $1.2\times0.8=0.96$ , so you end $4\%$ lower.
A percent is always a percent "of" something, and that something is the base. Example: $50$ is $25\%$ more than $40$ (base $40$ ), but $40$ is $20\%$ less than $50$ (base $50$ ).
Mean $\times$ count $=$ sum. Example: $4$ tests averaging $85$ add up to $340$ .
A combined average is a weighted average. Add the sums and divide by the total count. The result is closer to the mean of the larger group. Example: $10$ students average $80$ and $30$ students average $60$ . All $40$ average $\frac{800+1800}{40}=65$ , which is closer to $60$ .
Chaining ratios. If $a:b=2:3$ and $b:c=4:5$ , scale both so $b$ matches. Multiply the first by $4$ and the second by $3$ : $a:b=8:12$ and $b:c=12:15$ , so $a:b:c=8:12:15$ .

How AMC 10 tests it

Problems 1–6: one percent or ratio computation (a markup followed by a discount, or the ratio of areas after scaling a side).
Problems 4–10: "the average of $n$ numbers is $m$ ; after one number is removed or added, the average becomes ..." Change the averages into sums.
Problems 6–12: chained ratios among three or more groups, leading to a small linear equation.
Problems 8–14: percents where the base is unknown, such as "what percent increase brings the price back to the original?"

Standard approaches

Pick a handy total: $100$ for percents, the LCM of the ratio parts, or parts $pk$ , $qk$ , … with $k$ unknown.
Change every average into a sum right away.
Write each percent change as a multiplier and multiply them. Change back to a percent only at the end.
For "what percent" answers, decide carefully what the base is: the original amount, the new amount, or the other quantity. Example: after a $20\%$ discount, a $25\%$ increase brings the price back, since $0.8\times1.25=1$ .
Check that the answer goes the right way: a discount lowers the price, and a combined average lies between the group averages.

Worked example

In a club the ratio of freshmen to sophomores is $3:5$ . After $12$ freshmen join and $8$ sophomores leave, the ratio becomes $5:6$ . How many members did the club have originally?

(A) $96$ (B) $112$ (C) $128$ (D) $144$ (E) $160$

Say there were $3k$ freshmen and $5k$ sophomores at first. Afterward there are $3k+12$ freshmen and $5k-8$ sophomores, and their ratio is $5:6$ :
$\frac{3k+12}{5k-8}=\frac56.$
Cross-multiply: $6(3k+12)=5(5k-8)$ , so $18k+72=25k-40$ .

Move the $k$ terms to one side: $72+40=25k-18k$ , so $7k=112$ and $k=16$ .

The question asks for the whole club, which is $3k+5k=8k=128$ members.

Check: $48$ freshmen and $80$ sophomores become $60$ and $72$ , and $60:72=5:6$ . The answer is $\boxed{\textbf{(C)}\ 128}$ .

Pitfalls

Taking a percent of the wrong base. A $20\%$ discount and then a $20\%$ increase does not bring you back: $\$100$ becomes $\$80$ and then $\$96$ .
Adding percents that have different bases. Up $10\%$ and then up $10\%$ is $1.1\times1.1=1.21$ , a $21\%$ increase, not $20\%$ .
Averaging group averages without weighting by group size. In the example above, $\frac{80+60}{2}=70$ is wrong; the true average is $65$ .
Giving one part when the whole is asked, or the whole when one part is asked. A ratio of $3:5$ means $8$ parts in all: in the worked example, $48$ is only the freshmen.

Traps that recur

Approximating each cost against the whole allowance (20% + 5% = 25%) instead of against the stated differences. (2017 AMC 10A #14)
Taking 40% of the 30 swimmers instead of 40% of the 60 soccer players, or dividing the 36 by 100 rather than by the 70 non-swimmers. (2009 AMC 10A #18)
Finding n = 4 coins and 80 cents but then guessing a coin mix (like 25 + 25 + 10 + 10 + 10) without checking it uses exactly four coins. (2004 AMC 10A #14)
Pouring half of the second cup's original 4 ounces (2 oz) back instead of half of its new 6 ounces (3 oz, in a 1:2 coffee-to-cream ratio). (2002 AMC 10A #17)

1.2025 AMC 10B #1
The instructions on a $350$ -gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires $20$ grams of coffee beans. What is the greatest number of properly brewed large mugs of coffee that can be made from the coffee beans in that bag?
1. A) $16$
2. B) $17$
3. C) $18$
4. D) $19$
5. E) $20$
2.2023 AMC 10B #1
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
1. A) $\frac{1}{12}$
2. B) $\frac{1}{4}$
3. C) $\frac{1}{6}$
4. D) $\frac{1}{8}$
5. E) $\frac{2}{9}$
3.2023 AMC 10B #2
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
1. A)$46
2. B)$50
3. C)$48
4. D)$47
5. E)$49
4.2022 AMC 10A #2
Mike cycled $15$ laps in $57$ minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first $27$ minutes?
1. A) $5$
2. B) $7$
3. C) $9$
4. D) $11$
5. E) $13$
5.2021 AMC 10B #3
In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\%$ of the juniors and $10\%$ of the seniors are on the debate team. How many juniors are in the program?
1. A) $5$
2. B) $6$
3. C) $8$
4. D) $11$
5. E) $20$
6.2020 AMC 10B #3
The ratio of $w$ to $x$ is $4:3$ , the ratio of $y$ to $z$ is $3:2$ , and the ratio of $z$ to $x$ is $1:6$ . What is the ratio of $w$ to $y?$
1. A) $4:3$
2. B) $3:2$
3. C) $8:3$
4. D) $4:1$
5. E) $16:3$
7.2020 AMC 10A #2
The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$ . What is the average of $a$ and $b$ ?
1. A) $0$
2. B) $15$
3. C) $30$
4. D) $45$
5. E) $60$
8.2019 AMC 10B #1
Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?
1. A) $\frac{5}{8}$
2. B) $\frac{4}{5}$
3. C) $\frac{7}{8}$
4. D) $\frac{9}{10}$
5. E) $\frac{11}{12}$
9.2018 AMC 10A #2
Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?
1. A)Liliane has $20\%$ more soda than Alice.
2. B)Liliane has $25\%$ more soda than Alice.
3. C)Liliane has $45\%$ more soda than Alice.
4. D)Liliane has $75\%$ more soda than Alice.
5. E)Liliane has $100\%$ more soda than Alice.
10.2018 AMC 10A #6
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$ , and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$ , and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
1. A) $200$
2. B) $300$
3. C) $400$
4. D) $500$
5. E) $600$
11.2016 AMC 10A #3
For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\$12.50$ more than David. How much did they spend in the bagel store together?
1. A)$37.50
2. B)$50.00
3. C)$87.50
4. D)$90.00
5. E)$92.50
12.2015 AMC 10A #4
Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?
1. A) $\frac{1}{12}$
2. B) $\frac{1}{6}$
3. C) $\frac{1}{4}$
4. D) $\frac{1}{3}$
5. E) $\frac{1}{2}$
13.2015 AMC 10A #5
Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$ . After he graded Payton's test, the test average became $81$ . What was Payton's score on the test?
1. A) $81$
2. B) $85$
3. C) $91$
4. D) $94$
5. E) $95$
14.2014 AMC 10A #10
Five positive consecutive integers starting with $a$ have average $b$ . What is the average of $5$ consecutive integers that start with $b$ ?
1. A) $a+3$
2. B) $a+4$
3. C) $a+5$
4. D) $a+6$
5. E) $a+7$
15.2014 AMC 10B #3
Peter drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Peter's trip?
1. A) $30$
2. B) $\frac{400}{11}$
3. C) $\frac{75}{2}$
4. D) $40$
5. E) $\frac{300}{7}$
16.2014 AMC 10B #7
Suppose $A>B>0$ and $A$ is $x\%$ greater than $B$ . What is $x$ ?
1. A) $100\left(\frac{A-B}{B}\right)$
2. B) $100\left(\frac{A+B}{B}\right)$
3. C) $100\left(\frac{A+B}{A}\right)$
4. D) $100\left(\frac{A-B}{A}\right)$
5. E) $100\left(\frac{A}{B}\right)$
17.2012 AMC 10A #7
In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
1. A) $\frac{2}{5}$
2. B) $\frac{3}{7}$
3. C) $\frac{4}{7}$
4. D) $\frac{3}{5}$
5. E) $\frac{4}{5}$
18.2012 AMC 10B #5
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?
1. A) $18$
2. B) $20$
3. C) $21$
4. D) $22$
5. E) $24$
19.2011 AMC 10B #2
Josanna's test scores to date are $90, 80, 70, 60,$ and $85$ . Her goal is to raise her test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal?
1. A) $80$
2. B) $82$
3. C) $85$
4. D) $90$
5. E) $95$
20.2011 AMC 10A #5
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of $12$ , $15$ , and $10$ minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
1. A) $12$
2. B) $\frac{37}{3}$
3. C) $\frac{88}{7}$
4. D) $13$
5. E) $14$
21.2011 AMC 10A #8
Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?
1. A) $20$
2. B) $30$
3. C) $40$
4. D) $50$
5. E) $60$
22.2010 AMC 10B #2
Makayla attended two meetings during her $9$ -hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
1. A) $15$
2. B) $20$
3. C) $25$
4. D) $30$
5. E) $35$
23.2009 AMC 10B #3
Paula the painter had just enough paint for $30$ identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for $25$ rooms. How many cans of paint did she use for the $25$ rooms?
1. A) $10$
2. B) $12$
3. C) $15$
4. D) $18$
5. E) $25$
24.2009 AMC 10B #5
Twenty percent less than 60 is one-third more than what number?
1. A) $16$
2. B) $30$
3. C) $32$
4. D) $36$
5. E) $48$
25.2008 AMC 10A #2
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$ . The ratio of the rectangle's length to its width is $2:1$ . What percent of the rectangle's area is inside the square?
1. A) $12.5$
2. B) $25$
3. C) $50$
4. D) $75$
5. E) $87.5$
26.2008 AMC 10A #4
Suppose that $\tfrac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\tfrac{1}{2}$ of $5$ bananas?
1. A) $2$
2. B) $\frac{5}{2}$
3. C) $3$
4. D) $\frac{7}{2}$
5. E) $4$
27.2007 AMC 10A #1
One ticket to a show costs $\$20$ at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?
1. A) $2$
2. B) $5$
3. C) $10$
4. D) $15$
5. E) $20$
28.2006 AMC 10B #9
Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade?
1. A) $129$
2. B) $137$
3. C) $174$
4. D) $233$
5. E) $411$
29.2006 AMC 10A #3
The ratio of Mary's age to Alice's age is $3:5$ . Alice is $30$ years old. How old is Mary?
1. A) $15$
2. B) $18$
3. C) $20$
4. D) $24$
5. E) $50$
30.2005 AMC 10A #1
While eating out, Mike and Joe each tipped their server $\$2$ . Mike tipped $10\%$ of his bill and Joe tipped $20\%$ of his bill. What was the difference, in dollars, between their bills?
1. A) $2$
2. B) $4$
3. C) $5$
4. D) $10$
5. E) $20$
31.2005 AMC 10B #5
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
1. A) $\frac15$
2. B) $\frac13$
3. C) $\frac25$
4. D) $\frac23$
5. E) $\frac45$
32.2005 AMC 10B #2
A positive number $x$ has the property that $x\%$ of $x$ is $4$ . What is $x$ ?
1. A) $2$
2. B) $4$
3. C) $10$
4. D) $20$
5. E) $40$
33.2005 AMC 10B #1
A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?
1. A) $100$
2. B) $200$
3. C) $300$
4. D) $400$
5. E) $500$
34.2005 AMC 10B #3
A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?
1. A) $\frac{1}{10}$
2. B) $\frac{1}{9}$
3. C) $\frac{1}{3}$
4. D) $\frac{4}{9}$
5. E) $\frac{5}{9}$
35.2004 AMC 10A #3
Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
1. A) $0.0029$
2. B) $0.029$
3. C) $0.29$
4. D) $2.9$
5. E) $29$
36.2000 AMC 10 #8
At Olympic High School, $\frac{2}{5}$ of the freshmen and $\frac{4}{5}$ of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?
1. A)There are five times as many sophomores as freshmen.
2. B)There are twice as many sophomores as freshmen.
3. C)There are as many freshmen as sophomores.
4. D)There are twice as many freshmen as sophomores.
5. E)There are five times as many freshmen as sophomores.
37.2000 AMC 10 #3
Each day, Jenny ate $20\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, $32$ remained. How many jellybeans were in the jar originally?
1. A) $40$
2. B) $50$
3. C) $55$
4. D) $60$
5. E) $75$
38.2025 AMC 10A #4
A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is $15$ . Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from $12$ to $14.$ If Ash plays with the teachers, the average age on that team will decrease from $55$ to $52$ . How old is Ash?
1. A) $28$
2. B) $29$
3. C) $30$
4. D) $32$
5. E) $33$
39.2025 AMC 10A #2
A box contains $10$ pounds of a nut mix that is $50$ percent peanuts, $20$ percent cashews, and $30$ percent almonds. A second nut mix containing $20$ percent peanuts, $40$ percent cashews, and $40$ percent almonds is added to the box resulting in a new nut mix that is $40$ percent peanuts. How many pounds of cashews are now in the box?
1. A) $3.5$
2. B) $4$
3. C) $4.5$
4. D) $5$
5. E) $6$
40.2023 AMC 10A #10
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$ . If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$ . What is the mean of her quiz scores currently?
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $8$
41.2022 AMC 10A #4
In some countries, automobile fuel efficiency is measured in liters per $100$ kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals $m$ miles, and $1$ gallon equals $l$ liters. Which of the following gives the fuel efficiency in liters per $100$ kilometers for a car that gets $x$ miles per gallon?
1. A) $\frac{x}{100lm}$
2. B) $\frac{xlm}{100}$
3. C) $\frac{lm}{100x}$
4. D) $\frac{100}{xlm}$
5. E) $\frac{100lm}{x}$
42.2021 AMC Fall 10B #9
The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?
1. A) $\frac{2}{9}$
2. B) $\frac{3}{13}$
3. C) $\frac{7}{27}$
4. D) $\frac{2}{7}$
5. E) $\frac{1}{3}$
43.2021 AMC 10A #5
The quiz scores of a class with $k > 12$ students have a mean of $8$ . The mean of a collection of $12$ of these quiz scores is $14$ . What is the mean of the remaining quiz scores in terms of $k$ ?
1. A) $\frac{14-8}{k-12}$
2. B) $\frac{8k-168}{k-12}$
3. C) $\frac{14}{12} - \frac{8}{k}$
4. D) $\frac{14(k-12)}{k^2}$
5. E) $\frac{14(k-12)}{8k}$
44.2019 AMC 10B #11
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$ , and the ratio of blue to green marbles in Jar $2$ is $8:1$ . There are $95$ green marbles in all. How many more blue marbles are in Jar $1$ than in Jar $2$ ?
1. A) $5$
2. B) $10$
3. C) $25$
4. D) $45$
5. E) $50$
45.2019 AMC 10B #3
In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
1. A) $66$
2. B) $154$
3. C) $186$
4. D) $220$
5. E) $266$
46.2017 AMC 10B #11
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
1. A) $10\%$
2. B) $12\%$
3. C) $20\%$
4. D) $25\%$
5. E) $33\frac{1}{3}\%$
47.2017 AMC 10B #12
Elmer's new car gives $50\%$ better fuel efficiency. However, the new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline the old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?
1. A) $20\%$
2. B) $26\tfrac23\%$
3. C) $27\tfrac79\%$
4. D) $33\tfrac13\%$
5. E) $66\tfrac23\%$
48.2015 AMC 10A #9
Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?
1. A) $\text{The second height is } 10\% \text{ less than the first.}$
2. B) $\text{The first height is } 10\% \text{ more than the second.}$
3. C) $\text{The second height is } 21\% \text{ less than the first.}$
4. D) $\text{The first height is } 21\% \text{ more than the second.}$
5. E) $\text{The second height is } 80\% \text{ of the first.}$
49.2014 AMC 10B #11
For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:
(1) two successive $15\%$ discounts
(2) three successive $10\%$ discounts
(3) a $25\%$ discount followed by a $5\%$ discount
What is the smallest possible positive integer value of $n$ ?
1. A) $27$
2. B) $28$
3. C) $29$
4. D) $31$
5. E) $33$
50.2014 AMC 10A #11
A customer who intends to purchase an appliance has three coupons, only one of which may be used:
Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$
Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$
Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$
For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$ ?
1. A)$179.95
2. B)$199.95
3. C)$219.95
4. D)$239.95
5. E)$259.95
51.2014 AMC 10B #6
Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?
1. A) $33$
2. B) $34$
3. C) $36$
4. D) $38$
5. E) $39$
52.2014 AMC 10A #6
Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?
1. A) $\frac{bde}{ac}$
2. B) $\frac{ac}{bde}$
3. C) $\frac{abde}{c}$
4. D) $\frac{bcde}{a}$
5. E) $\frac{abc}{de}$
53.2013 AMC 10A #10
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
1. A) $15$
2. B) $30$
3. C) $40$
4. D) $60$
5. E) $70$
54.2013 AMC 10A #9
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\%$ of her three-point shots and $30\%$ of her two-point shots. Shenille attempted $30$ shots. How many points did she score?
1. A) $12$
2. B) $18$
3. C) $24$
4. D) $30$
5. E) $36$
55.2013 AMC 10B #8
Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?
1. A) $10$
2. B) $16$
3. C) $25$
4. D) $30$
5. E) $40$
56.2009 AMC 10A #7
A carton contains milk that is $2$ % fat, an amount that is $40$ % less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
1. A) $\frac{12}{5}$
2. B) $3$
3. C) $\frac{10}{3}$
4. D) $38$
5. E) $42$
57.2009 AMC 10B #8
In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$ ?
1. A) $12$
2. B) $17$
3. C) $20$
4. D) $25$
5. E) $35$
58.2009 AMC 10A #8
Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$ % discount as children. The two members of the oldest generation receive a $25\%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $\$6.00$ , is paying for everyone. How many dollars must he pay?
1. A) $34$
2. B) $36$
3. C) $42$
4. D) $46$
5. E) $48$
59.2008 AMC 10A #12
In a collection of red, blue, and green marbles, there are $25\%$ more red marbles than blue marbles, and there are $60\%$ more green marbles than red marbles. Suppose that there are $r$ red marbles. What is the total number of marbles in the collection?
1. A) $2.85r$
2. B) $3r$
3. C) $3.4r$
4. D) $3.85r$
5. E) $4.25r$
60.2008 AMC 10B #6
Points $B$ and $C$ lie on $AD$ . The length of $AB$ is $4$ times the length of $BD$ , and the length of $AC$ is $9$ times the length of $CD$ . The length of $BC$ is what fraction of the length of $AD$ ?
1. A) $1/36$
2. B) $1/13$
3. C) $1/10$
4. D) $5/36$
5. E) $1/5$
61.2008 AMC 10A #8
Heather compares the price of a new computer at two different stores. Store $A$ offers $15\%$ off the sticker price followed by a $\$90$ rebate, and store $B$ offers $25\%$ off the same sticker price with no rebate. Heather saves $\$15$ by buying the computer at store $A$ instead of store $B$ . What is the sticker price of the computer, in dollars?
1. A) $750$
2. B) $900$
3. C) $1000$
4. D) $1050$
5. E) $1500$
62.2008 AMC 10A #6
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
1. A) $3$
2. B) $4$
3. C) $5$
4. D) $6$
5. E) $7$
63.2007 AMC 10A #10
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$ , the father is $48$ years old, and the average age of the mother and children is $16$ . How many children are in the family?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
64.2007 AMC 10B #3
A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
1. A) $22$
2. B) $24$
3. C) $25$
4. D) $26$
5. E) $28$
65.2007 AMC 10B #14
Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\%$ of the group are girls. How many girls were initially in the group?
1. A) $4$
2. B) $6$
3. C) $8$
4. D) $10$
5. E) $12$
66.2007 AMC 10A #6
At Euclid High School, the number of students taking the AMC 10 was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ and 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase?
1. A) $2002 \text{and} 2003$
2. B) $2003 \text{and} 2004$
3. C) $2004 \text{and} 2005$
4. D) $2005 \text{and} 2006$
5. E) $2006 \text{and} 2007$
67.2007 AMC 10B #16
A teacher gave a test to a class in which $10\%$ of the students are juniors and $90\%$ are seniors. The average score on the test was $84.$ The juniors all received the same score, and the average score of the seniors was $83.$ What score did each of the juniors receive on the test?
1. A) $85$
2. B) $88$
3. C) $93$
4. D) $94$
5. E) $98$
68.2007 AMC 10A #7
Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\$10500$ for both taxes. How many dollars was his inheritance?
1. A) $30000$
2. B) $32500$
3. C) $35000$
4. D) $37500$
5. E) $40000$
69.2006 AMC 10B #13
Joe and JoAnn each bought $12$ ounces of coffee in a $16$ ounce cup. Joe drank $2$ ounces of his coffee and then added $2$ ounces of cream. JoAnn added $2$ ounces of cream, stirred the coffee well, and then drank $2$ ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
1. A) $\frac{6}{7}$
2. B) $\frac{13}{14}$
3. C) $1$
4. D) $\frac{14}{13}$
5. E) $\frac{7}{6}$
70.2005 AMC 10B #6
At the beginning of the school year, Lisa's goal was to earn an $A$ on at least $80\%$ of her $50$ quizzes for the year. She earned an $A$ on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an $A$ ?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
71.2004 AMC 10B #14
A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only $\frac{1}{3}$ of the marbles in the bag are blue. Then yellow marbles are added to the bag until only $\frac{1}{5}$ of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?
1. A) $\frac{1}{5}$
2. B) $\frac{1}{4}$
3. C) $\frac{1}{3}$
4. D) $\frac{2}{5}$
5. E) $\frac{1}{2}$
72.2004 AMC 10B #7
On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$ ?
1. A) $5$
2. B) $6$
3. C) $7$
4. D) $8$
5. E) $9$
73.2004 AMC 10A #11
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased?
1. A) $10$
2. B) $25$
3. C) $36$
4. D) $50$
5. E) $60$
74.2001 AMC 10 #9
The state income tax where Kristin lives is charged at the rate of $p\%$ of the first $\$28000$ of annual income plus $(p + 2)\%$ of any amount above $\$28000$ . Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income?
1. A)$28000
2. B)$32000
3. C)$35000
4. D)$42000
5. E)$56000
75.2017 AMC 10A #14
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?
1. A) $9\%$
2. B) $19\%$
3. C) $22\%$
4. D) $23\%$
5. E) $25\%$
76.2009 AMC 10A #18
At Jefferson Summer Camp, $60\%$ of the children play soccer, $30\%$ of the children swim, and $40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
1. A) $30\%$
2. B) $40\%$
3. C) $49\%$
4. D) $51\%$
5. E) $70\%$
77.2004 AMC 10A #14
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse?
1. A) $0$
2. B) $1$
3. C) $2$
4. D) $3$
5. E) $4$
78.2002 AMC 10A #17
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
1. A) $\frac{1}{4}$
2. B) $\frac13$
3. C) $\frac38$
4. D) $\frac25$
5. E) $\frac12$
79.2002 AMC 10B #21
Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first?
1. A) $\text{Andy}$
2. B) $\text{Beth}$
3. C) $\text{Carlos}$
4. D) $\text{Andy and Carlos tie for first.}$
5. E) $\text{All three tie.}$

Ratios, Percents & Averages — answersAMC 10 Step by Step

Name: Date: October 28, 2026

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Answers

#	Answer	The idea it turns on
1	B	Each mug needs a full 20 grams. Seventeen mugs use 340 grams, and the 10 grams left over cannot make another proper mug.
2	C	Find the level every glass must end at first: the total juice split four ways. Then each full glass gives up the difference between full and that level.
3	B	A 20% discount means times 0.8 and a 7.5% tax means times 1.075. Together Carlos pays 0.86 of the original price, so divide 43 by 0.86.
4	B	At a steady speed, laps go up in step with time. 27 minutes is 9/19 of the ride, so he did 9/19 of the 15 laps, a little over 7.
5	C	A quarter of the juniors equals a tenth of the seniors, so for every 2 juniors there are 5 seniors. Split 28 into 2 and 5 parts: 8 juniors.
6	E	Give x a handy value that works with every ratio (x = 6 makes z = 1), work out w and y, then divide.
7	C	Five numbers averaging 15 add up to 75. Take away 3 + 5 + 7 = 15 to get a + b = 60, then halve it.
8	D	The same water fills 5/6 of the first container and 3/4 of the second. Pick an amount of water that makes both containers whole numbers.
9	A	Pick an easy amount for Jacqueline, like 100. Then Liliane has 150 and Alice has 125, and you just compare 150 with 125.
10	B	Out of every 100 votes, 65 are likes and 35 are dislikes, so each 100 votes adds 30 to the score. A score of 90 needs 300 votes.
11	C	David spends 75 cents for each dollar Ben spends, so the 12.50 dollar gap is one quarter of Ben's total. Ben spent 50 dollars and David 37.50.
12	B	Only the ratio matters, so give Mia 1 egg, Sofia 2 and Pablo 6. Everyone should end with 3, so Sofia gets 1 of Pablo's 6 eggs.
13	E	Turn each average into a total: 14 tests averaging 80 add to 1120, and 15 tests averaging 81 add to 1215. The difference, 95, is Payton's score.
14	B	The average of five consecutive whole numbers is the middle one. So b is a + 2, and the second list averages b + 2, which is a + 4.
15	E	Gravel and dirt together are 1/3 + 1/5 = 8/15 of the trip, so the 20 paved miles are the remaining 7/15 of it.
16	A	Percent greater is measured against the number you start from, B: find how much bigger A is (A - B), divide by B, and multiply by 100.
17	C	Pretend the bag holds 5 marbles: 3 blue and 2 red. Doubling the red gives 4 red, and the bag now holds 7 marbles.
18	D	The tax and the tip are both taken from the meal price, so every dollar of meal costs 1.25 dollars in total. Divide 27.50 by 1.25.
19	E	Her average now is 77, so she wants 80. Six tests averaging 80 must add up to 480, and she already has 385.
20	C	The grade sizes are in the ratio 4 : 2 : 1, so pretend there are 4, 2 and 1 students. Add up all their minutes and divide by 7.
21	C	Pretend there are 100 birds. Leaving out the 25 swans leaves 75 birds, and 30 of them are geese, so the answer is 30 out of 75, which is 40%.
22	C	Put everything in minutes. The meetings take 45 + 90 = 135 minutes, and the work day is 9 times 60 = 540 minutes, exactly four times as long.
23	C	The three lost cans were exactly the paint for the 5 rooms she can no longer do, so 3 cans paint 5 rooms, and 25 rooms need 15 cans.
24	D	Twenty percent less than 60 is 48. A number plus a third of itself is 4 thirds of it, so 4 thirds make 48.
25	A	Only ratios are given, so give the square a side of 1. Then the rectangle is 2 wide and 4 long, and the square fills 1 out of 8.
26	C	Two thirds of 10 bananas is 20/3 bananas, and that equals 8 oranges. So one banana is worth 6/5 of an orange, and 5/2 bananas are worth 3 oranges.
27	C	Turn each discount into a price per ticket first: 25% off 20 dollars is 15 dollars, and 30% off is 14 dollars. Then multiply by the number of tickets.
28	B	The batch weighs 600 grams and has 411 calories. A 200-gram glass is one third of the batch, so it has 411 divided by 3 = 137.
29	B	Think of the ages as 3 equal parts and 5 equal parts. Alice's 5 parts are 30 years, so one part is 6 years.
30	D	The same 2 dollar tip is a smaller percent of a bigger bill. Work backward from each tip to each whole bill, then subtract.
31	C	If one third of the CDs costs one fifth of her money, then all the CDs cost three times as much: three fifths. Two fifths is left.
32	D	The phrase 'x percent of x' means x divided by 100, then times x. Setting that equal to 4 gives x times x = 400, so x = 20.
33	A	Find the price of one bar when buying (40 cents) and when selling (50 cents). Each bar earns 10 cents, and there are 1000 bars.
34	D	Using one third of the paint leaves two thirds. This happens twice, so what remains is two thirds of two thirds, which is 4/9.
35	E	The answer must be in cents, so change 20 dollars into 2000 cents first. Then 1.45 percent of 2000 is 29.
36	D	Two fifths of the freshmen equals four fifths of the sophomores. The group with the smaller fraction taking part must be the bigger group, twice as big.
37	B	Eating 20% leaves 80%, which is four fifths. Two days multiply the start by four fifths twice, so 32 is 16/25 of the start.
38	A	Turn each average into a total. Then Ash's age is 14 plus 2 per student, and also 52 minus 3 per teacher. Both agree at 7 students.
39	B	Follow only the peanuts. The peanut pounds must be 40 percent of the new total weight; that tells how much mix was added, and then count the cashews.
40	D	Turn each mean into a total: n quizzes with mean m add up to n times m. Each story then gives a simple equation.
41	E	Follow one trip of 100 kilometers: change it to miles, then to gallons of fuel used, then to liters.
42	C	Pick 42 knights: 12 red, 30 blue, 7 magical. If a fraction x of blue knights is magical, then 12 times 2x plus 30 times x equals 7.
43	B	Turn every mean into a total. The class total is 8k, the twelve scores total 168, and the other students share what is left.
44	A	Both jars hold the same number of marbles. Green is 1/10 of Jar 1 and 1/9 of Jar 2, so a jar holding 90 marbles is the natural unit.
45	B	Two of the facts are about students who do not play. Count the non-players among seniors and non-seniors, and set the total equal to 46.8% of 500.
46	D	Picture 100 students. 12 like dancing but say they dislike it, and 36 dislike it and say so. Of the 48 who say dislike, 12 actually like it.
47	A	Try real numbers: a trip needing 3 liters in the old car needs 2 in the new one. At 1.20 dollars instead of 1, cost drops from 3.00 to 2.40.
48	D	Volume uses the radius squared. A radius 1.1 times as big gives 1.21 times the base area, so the first cylinder must be 1.21 times as tall.
49	C	Discounts one after another multiply what is left of the price. The three deals take off 27.75%, 27.1% and 28.75%, so n must be more than 28.75.
50	C	Write each coupon's savings for a price P. Beating the 20-dollar coupon needs P above 200; beating the 18 percent coupon needs P below 225. Only 219.95 fits.
51	C	Measure money in regular balloon prices. On sale, a pair costs 1 + 2/3 = 5/3 prices, so 30 prices buy 18 pairs, which is 36 balloons.
52	A	First find how much one cow gives in one day, b/(ac) gallons. Then multiply by the number of cows d and the number of days e.
53	E	Pretend there are 100 flowers: 60 pink (40 of them carnations) and 40 red (30 of them carnations). That is 70 carnations.
54	B	A three-point try averages 0.2 times 3 = 0.6 points and a two-point try averages 0.3 times 2 = 0.6, so 30 tries give 18.
55	B	Pick an easy distance, like 40 miles each. Ray uses 1 gallon and Tom uses 4, so 80 miles on 5 gallons is 16 miles per gallon.
56	C	Taking 40% away from the whole-milk amount leaves 60% of it, so 60% of the whole-milk amount is 2, and the whole-milk amount is 2 divided by 0.6.
57	B	Follow a price of 100 dollars month by month. It reaches 120 after March, and dropping back to 100 is 20 out of 120, about 17 percent.
58	B	A senior ticket is 75% of the full price, so the full price is 6 divided by 0.75, which is 8 dollars; a child ticket is half of that.
59	C	25% more red than blue means red is 1.25 times blue, so blue is red divided by 1.25, which is 0.8r. Green is 1.6r, and the total is 3.4r.
60	C	Measure both points from D. BD is one fifth of AD and CD is one tenth of AD, so BC is the difference, one tenth of AD.
61	A	Store B takes off 10% more than store A, but A gives 90 dollars back. The 90 beats that 10% by 15, so 10% of the price is 75.
62	D	Average speed is total distance over total time. With 60 km legs, the times are 20, 3 and 6 hours, so the average is 180 over 29, about 6.
63	E	Turn averages into totals. The family totals 20 times its size, the mother and children 16 times theirs; the gap is the father's 48.
64	B	Average mileage is all the miles divided by all the gallons: 240 miles on 4 + 6 = 10 gallons.
65	C	The group size never changes, so the drop from 40 percent to 30 percent girls is the 2 girls who left: 10 percent of the group is 2.
66	A	A percent increase compares the gain to the starting number. From 60 to 66 is exactly 10%, and every other year's gain is less than 10% of its start.
67	C	Pretend the class has 10 students: 1 junior and 9 seniors. The class total is 840 and the seniors earned 747, so the junior scored the other 93.
68	D	The state tax is 10% of what is left, not of the whole. So the taxes are 20% plus 8%, or 28% of the inheritance.
69	E	Joe adds cream after drinking, so he keeps all 2 ounces. JoAnn drinks 2 of her 14 stirred ounces, so she loses one seventh of her cream.
70	B	Her goal is 40 A's out of all 50 quizzes. She has 22, so she needs 18 of the last 20 quizzes to be A's. Only 2 can be lower.
71	C	Only the last step matters. One fifth blue means 1 blue per 4 others; doubling blue gives 2 per 4 others, which is one third.
72	A	Every 7 U.S. dollars becomes 10 Canadian, a gain of 3. She spent exactly her gain, 60, so she changed 20 groups of 7 dollars: d = 140.
73	C	Making the jar 5/4 as wide makes the base 25/16 as big, so the height must shrink to 16/25 of what it was. That is a 36 percent drop.
74	B	See the tax as p% on everything plus an extra 2% above $28000. The p% parts cancel, so that extra 2% equals 0.25% of her income.
75	D	Write both sentences as equations: ticket is a fifth of (A minus soda), soda is a twentieth of (A minus ticket). With A = 99, costs are 19 and 4.
76	D	Pretend there are 100 children: 60 play soccer, 24 of those swim, so 36 soccer players do not swim, out of 70 children who do not swim.
77	A	Adding one 25-cent coin raises the average by exactly 1 cent only if there were 4 coins, worth 80 cents in all. That forces three quarters and a nickel.
78	D	Keep track of coffee and cream separately. Stirred liquid keeps the same mix, so the 3 ounces poured back are 1 ounce of coffee and 2 of cream.
79	B	Only comparisons are given, so pick easy numbers: lawns 6, 3, 2 and mowing speeds 3, 2, 1. Then times are 2, 1.5 and 2, so Beth finishes first.

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.