Printable problems

Sets, Estimation & Miscellaneous

Set problems not primarily counting, estimation, pattern recognition, unit conversion

Print without answers Back to the topic

Sets, Estimation & MiscellaneousAMC 10 Step by Step

Name: Date: October 28, 2026

12 problems, easiest first

What you need to know

Principle of Inclusion-Exclusion (PIE), two sets. (number in $A$ or $B$ ) $=$ (number in $A$ ) $+$ (number in $B$ ) $-$ (number in both). Those in both were counted twice, so subtract them once. Example: of $30$ students, $18$ like math, $15$ like art, and $7$ like both. Then $18+15-7=26$ like at least one, and $4$ like neither.
PIE, three sets. Add the three groups, subtract the three overlaps of two groups, then add back the part in all three. (That middle part was added three times and subtracted three times.) Example: from $1$ to $30$ , the numbers divisible by $2$ , $3$ , or $5$ number $15+10+6-5-3-2+1=22$ .
"Exactly one" and "exactly two" come from the regions of the Venn diagram, not from the PIE total. Example: above, $26$ like at least one subject, but only $(18-7)+(15-7)=19$ like exactly one.
How big can the overlap be? Say there are $N$ things in all. The number in both $A$ and $B$ is at most the smaller of the two group sizes, and at least (size of $A$ ) $+$ (size of $B$ ) $-N$ . If that is negative, the smallest overlap is $0$ . Example: with $30$ students, $18$ in math and $15$ in art, between $18+15-30=3$ and $15$ students are in both.
Packing problems need two things: a lower bound from items that cannot share a container, and an actual packing that matches it. Example: boxes hold $10$ kg, and the items weigh $6,6,6,6,3,3$ kg. The total is $30$ kg, but no two $6$ s fit together, so at least $4$ boxes are needed. And $4$ works: $6+3$ , $6+3$ , $6$ , $6$ .

How AMC 10 tests it

Survey problems (positions 3–12): two or three activities that overlap, plus a group that does none; one region is asked for.
"At least how many do both?" The tool is the bound (size of $A$ ) $+$ (size of $B$ ) $-N$ .
Packing puzzles (positions 8–15): the trap is dividing the total size by the capacity and forgetting that items can't be split.
Estimation problems (positions 1–10): round each number to one or two significant figures.

Standard approaches

Draw the Venn diagram. Call the innermost region (all three) $x$ , and fill in the regions from the inside out.
Use PIE with (number in at least one group) $=N-$ (number in none).
For the largest or smallest possible overlap, decide which bound the question wants.
For packing, list which combinations fit in one container, find the bound, then build a packing that reaches it.

Worked example

In a class of $40$ students, $24$ play soccer, $20$ basketball, and $18$ tennis. Ten play soccer and basketball, $8$ soccer and tennis, $7$ basketball and tennis, and $2$ play none. How many play exactly one sport?

(A) $13$ (B) $14$ (C) $15$ (D) $16$ (E) $17$

Find the middle. Let $x$ students play all three. Since $2$ play none, $40-2=38$ play at least one. By PIE,
$38=24+20+18-10-8-7+x=37+x,$
so $x=1$ .

Find the "exactly two" regions. The $10$ soccer-and-basketball players include the $1$ who plays all three, so $10-1=9$ play only those two. In the same way, $8-1=7$ play only soccer and tennis, and $7-1=6$ play only basketball and tennis.

Subtract. At least two sports: $9+7+6+1=23$ students. So exactly one sport: $38-23=15$ . (Check: the one-sport regions hold $7$ , $4$ , and $4$ .) The answer is $\boxed{\textbf{(C)}\ 15}$ .

Pitfalls

Reading " $10$ play soccer and basketball" as leaving out the ones who also play tennis. Those $10$ include everyone who plays both, tennis or not.
Forgetting the "none" group: in the example, the students in at least one sport number $38$ , not $40$ .
In packing, stopping at the total divided by the capacity, rounded up. Above, $30\div10=3$ , but $4$ boxes are needed.
Rounding every number down in an estimate. Example: $3.9\times4.9\times5.9$ is about $4\times5\times6=120$ (it is $112.749$ ), not $3\times4\times5=60$ .

Traps that recur

Forgetting that 1 through 24 are unconstrained and answering 50 (one from each of fifty imagined pairs). (2005 AMC 10B #25)
Assuming the answer is just the 20% net change, or letting more students switch Yes-to-No than the 30% who answer No at the end. (2010 AMC 10B #12)
Dividing total size 16.8 MB by 1.44 to get 12 disks, ignoring that files cannot be split and that 0.8 and 0.7 cannot share a disk. (2002 AMC 10A #11)
Answering 4 from 52/12 = 4.33 rounded down, or 12 from misreading the question as the number of months. (2011 AMC 10B #11)

1.2023 AMC 10B #4
Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint?
1. A) $162{,}500$
2. B) $162.5$
3. C) $1{,}625$
4. D) $1{,}625{,}000$
5. E) $16{,}250$
2.2019 AMC 10A #4
A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn $?$
1. A) $75$
2. B) $76$
3. C) $79$
4. D) $84$
5. E) $91$
3.2013 AMC 10B #2
Mr. Green measures his rectangular garden by walking two of the sides and finding that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
1. A) $600$
2. B) $800$
3. C) $1000$
4. D) $1200$
5. E) $1400$
4.2011 AMC 10A #6
Set $A$ has 20 elements, and set $B$ has 15 elements. What is the smallest possible number of elements in $A \cup B$ , the union of $A$ and $B$ ?
1. A) $5$
2. B) $15$
3. C) $20$
4. D) $35$
5. E) $300$
5.2010 AMC 10B #3
A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
1. A) $3$
2. B) $4$
3. C) $5$
4. D) $8$
5. E) $9$
6.2008 AMC 10B #2
A $4\times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
$\begin{array}{|c|c|c|c|} \hline 1&2&3&4\\\hline 8&9&10&11\\\hline 15&16&17&18\\\hline 22&23&24&25\\\hline \end{array}$
1. A) $2$
2. B) $4$
3. C) $6$
4. D) $8$
5. E) $10$
7.2011 AMC 10B #11
There are $52$ people in a room. What is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $12$
8.2008 AMC 10B #12
Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
1. A) $2500$
2. B) $3000$
3. C) $3500$
4. D) $4000$
5. E) $4500$
9.2007 AMC 10A #11
The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
1. A) $14$
2. B) $16$
3. C) $18$
4. D) $20$
5. E) $24$
10.2010 AMC 10B #12
At the beginning of the school year, $50\%$ of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$ ?
1. A) $0$
2. B) $20$
3. C) $40$
4. D) $60$
5. E) $80$
11.2002 AMC 10A #11
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
1. A) $12$
2. B) $13$
3. C) $14$
4. D) $15$
5. E) $16$
12.2005 AMC 10B #25
A subset $B$ of the set of integers from $1$ to $100$ , inclusive, has the property that no two elements of $B$ sum to $125$ . What is the maximum possible number of elements in $B$ ?
1. A) $50$
2. B) $51$
3. C) $62$
4. D) $65$
5. E) $68$

Sets, Estimation & Miscellaneous — 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	Change both lengths into centimeters before multiplying: 6.5 mm is 0.65 cm and 25 m is 2500 cm. Then the area comes out directly in square centimeters.
2	B	Build the worst draw with no 15 of a color: 14 of each big color plus every small-color ball, 75 in all. One more forces it.
3	A	Change both sides from steps to feet first, giving 30 feet by 40 feet. Find the area in square feet, then take half of it.
4	C	The union always contains all 20 elements of A. It has exactly 20 when every element of B is already in A.
5	C	Four colors means you can pull four socks that are all different. The fifth sock has to match one of them.
6	B	Write out the new block after reversing the two rows, then add each diagonal. The sums are 50 and 54.
7	D	If every month had at most 4 birthdays, the room would hold at most 48 people. With 52, some month has 5, and 6 can fail.
8	A	Each flip means 100,000 steps. Forty-four flips plus the final 50,000 give 4,450,000 steps, and dividing by 1800 gives about 2470 miles.
9	C	A face and its opposite face use each of the 8 corners once. Their sums add to 36, the total of 1 through 8, so each face sums to 18.
10	D	In a class of 100, Yes grows by 20. At least those 20 switch. At most, all 50 No students switch to Yes and 30 Yes students switch to No.
11	B	Give each big file (0.8 or 0.7) 3 points and each small file 2 points. No disk can hold more than 6 points, but the files total 75 points.
12	C	Numbers 1 to 24 have no partner making 125, so keep them all. Numbers 25 to 100 form 38 pairs; keep one from each: 62.

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.