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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)

1.2011 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) $60$
2.2010 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) $35$
3.2009 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) $25$
4.2009 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) $48$
5.2008 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) $87.5$
6.2008 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) $4$
7.2007 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) $20$
8.2006 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) $411$
9.2006 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) $50$
10.2005 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) $20$
11.2005 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) $\frac45$
12.2005 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) $40$
13.2005 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) $500$
14.2005 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) $\frac{5}{9}$
15.2004 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) $29$
16.2000 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.
17.2000 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) $75$
18.2025 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) $33$
19.2025 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) $6$
20.2023 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) $8$

Ratios, Percents & Averages — 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	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%.
2	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.
3	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.
4	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.
5	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.
6	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.
7	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.
8	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.
9	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.
10	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.
11	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.
12	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.
13	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.
14	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.
15	E	The answer must be in cents, so change 20 dollars into 2000 cents first. Then 1.45 percent of 2000 is 29.
16	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.
17	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.
18	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.
19	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.
20	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.

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.