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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

19 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)

1.2008 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) $1500$
2.2008 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) $7$
3.2007 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) $6$
4.2007 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) $28$
5.2007 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) $12$
6.2007 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) $2006 \text{and} 2007$
7.2007 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) $98$
8.2007 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) $40000$
9.2006 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) $\frac{7}{6}$
10.2005 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) $5$
11.2004 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) $\frac{1}{2}$
12.2004 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) $9$
13.2004 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) $60$
14.2001 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
15.2017 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) $25\%$
16.2009 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) $70\%$
17.2004 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) $4$
18.2002 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) $\frac12$
19.2002 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) $\text{All three tie.}$

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.