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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)

1.2022 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) $\frac{100lm}{x}$
2.2021 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) $\frac{1}{3}$
3.2021 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) $\frac{14(k-12)}{8k}$
4.2019 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) $50$
5.2019 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) $266$
6.2017 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) $33\frac{1}{3}\%$
7.2017 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) $66\tfrac23\%$
8.2015 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) $\text{The second height is } 80\% \text{ of the first.}$
9.2014 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) $33$
10.2014 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
11.2014 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) $39$
12.2014 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) $\frac{abc}{de}$
13.2013 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) $70$
14.2013 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) $36$
15.2013 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) $40$
16.2009 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) $42$
17.2009 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) $35$
18.2009 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) $48$
19.2008 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) $4.25r$
20.2008 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) $1/5$

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.