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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

20 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 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) $60$
7.2020 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) $16:3$
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 #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) $600$
10.2018 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.
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 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) $\frac{300}{7}$
15.2014 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) $100\left(\frac{A}{B}\right)$
16.2014 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) $a+7$
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) $14$

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.