Problems:
1
At Patrick's Diner, the appetizer special is a basket of fried mushrooms, onion rings, and
mozzarella sticks. The cook has fried 24 mozzarella sticks, 30 onion rings, and 42
mushrooms to exactly fill several orders. What is the largest number of orders of appetizer
baskets that the cook can prepare? Each basket must contain the same number of each
individual kind of appetizer.
2
While driving the football team to an away game, the careful bus driver who also loved
math noticed that the mileage was 15,951 miles. She noticed that this number was a
palindrome. (It reads the same backward and forward.) She mentioned this fact to an
assistant coach sitting near her, and he said, "I bet it will be a long time before that happens
again!" Two hours later, the driver noticed that the mileage was a palindrome once again.
How fast was the bus traveling during those two hours?
3
How many fractions with a denominator of 23 have values between 0.18 and 0.82?
4
Find the greatest whole number that meets all the following conditions:
a. It is greater than 100.
b. It is less than 200.
c. It is 20 greater when rounded to the nearest 100 than when rounded to the nearest 10.
5
Mrs. Hernandez bought several items, all the same price. The number of items was equal to
the cost of each item in cents. The change that Mrs. Hernandez received from $10 was $1
and 7 coins totaling less than $1. How much did each item cost?
6
The number 1961 has rotational symmetry, because the number looks the same when
rotated 180 degrees. Find the next greatest number that also has this unique feature.
7
Numbers that are palindromes have the same arrangement of digits from left to right as
from right to left, such as 2772 or 27372. Which three-digit and four-digit palindromes
when squared also result in palindromes? Does a method exist to predict which three-digit
or four-digit palindromes will form palindromes when squared?
8
If the mean, median, and mode are all equal for the set {3, 4, 5, 8, x}, what is the value of x?
Describe how you found your answer.
9
The United States acquired Alaska from Russia on October 18, 1867, for about 2¢ an acre.
Currently, Alaska has an area of 570,374 square miles. Use this current area as an estimate
for the 1867 area. About how much did Alaska cost? Rhode Island, the smallest state, is
1045 square miles. At 2¢ an acre, how much would Rhode Island cost? How much would
your state cost? Note: 1 square mile = 640 acres.
10
Adam has won a radio contest that includes a chance to win money. A box is filled with
$100, $50, $20, and $5 bills. Adam will be blindfolded and allowed to draw bills, one at a
time, until he has drawn five bills of the same denomination. What is the largest possible
amount that Adam can win?
11
During a session of Congress, 4000 bills were introduced, and 2200 of them failed to pass
either house. The House approved 1200 bills and the Senate approved 1500; the President
signed 800 bills into law. How many bills were passed by both houses and vetoed by the
President?
12
A typical toy in Santa's Workshop requires 1 hour to assemble, 35 minutes to decorate, and
5 minutes to package. Santa employs 300 elves, and he wants to assign each elf to the
Assembly, Decoration, or Packaging division. How many elves should he assign to each
division so that the work gets done as efficiently as possible?
13
For which integers, n, for 1 ≤n≤ 20, is
n!n+1
an integer?
14
A fictitious news story relates how the Mars probe sent from Earth has disturbed a colony of
Martians. They have come to Earth and have taken over our planet. The supreme ruler of
Mars has decided to change the unit of length used in the United States from 12 inches in a
foot to 10 inches in a foot because the supreme ruler's foot is only 10 inches long. The
Martians are allowing the United States to keep the mile the exact same length so that no
new road signs and mile markers have to be created. Now that the Martians have taken over
our planet, how many feet are in one mile?
15
How many two-digit whole numbers less than 50 are divisible (with no remainder) by the
product of their digits? List each two-digit number that has this trait.
16
A fireman stood on the middle rung of a ladder pouring water on a building. As the smoke
cleared, he stepped up 3 rungs. A sudden flare-up forced him to go down 5 rungs. Later he
climbed 7 rungs to work on the fire. Then he climbed the remaining 6 rungs to the top of the
ladder and stepped off to stand on the roof. How many rungs did the ladder have? Explain
your thinking and your answers.
17
If each letter below stands for a single digit, what are the digits in the division problem?
18
If it is 6:00 A.M. on 13 December 1999, what will be the time and date 2 million seconds
later? Round to the nearest minute.
19
What is the units digit of the least whole number greater than 1000 whose digits are all
different?
20
For how many two-digit numbers is the tens digit larger than the ones digit?
21
Read "Ode to Math," and solve the embedded problem.
A multiple of eleven I be,
Not odd, but even, you see.
My digits a pair,
When multiplied there,
Make a cube and a
Square out of me.
What number am I?
22
The 107th United States Congress convened in 2001, and the 108th convenes in January
2003. When did the first Congress convene? Which Congress will be in session during
2030?
23
At this time of year, many postal workers have to work overtime, so a supervisor at one
large post office planned a late-night snack for the employees. She ordered 1 extra large
pizza for every two workers, 1 large bag of potato chips for every three workers, and 1 twoliter bottle of cola for every four workers. When the order arrived, 26 items were delivered.
How many employees were working that evening?
24
In 1991, Mike Powell, a long jumper, jumped 8.95 meters. If Powell is 1.75 meters tall, how
many times his body length did he jump? How far would he be able to jump if he could
jump as far in proportion to his height as each creature in the table?
25
What is the tenth number in the following sequence? Give your answer as a mixed number.
15,47,1,1511,11213,…
26
Replace each letter with a different counting number from 1 to 10, inclusive. Each counting
number is to be used only once. Each number must be the difference of the two above it.
For example, E = A – B or B – A, K = H – J or J – H, and so on. Hint: Try K = 3 and F = 9.
27
Mom baked a cake for the entire family. Dad ate 1/6 of the cake. Brother ate 1/5 of what
was left. Sister ate 1/4 of what was left after that. Spot the dog ate 1/3 of what was left after
that. Baby sister ate 1/2 of what was left after that. How much of the original cake was left
for Mom to eat? Who ate the most cake?
28
A high-rise apartment building has four elevators. Each elevator travels upward at an
average rate of 18 feet per second. The elevator travels 12 feet from one floor to the next.
Suppose you get on the elevator on the first floor and travel straight to the fiftieth floor
without stopping. To the nearest second, how long will it take to travel from the first floor
to the fiftieth floor on this elevator?
29
Find the missing value in the fourth grid based on the values in the first three grids.
30
Find the sum of all two-digit multiples of 6 whose digits sum to 6. For example, 24 is a
multiple of 6, and the digits 2 and 4 sum to 6.
31
What is the smallest odd number you can obtain from the product of four different prime
numbers?
Solutions:
1
Six baskets with 4 mozzarella sticks, 5 onion rings, and 7 mushrooms in each basket. To
solve, find the greatest common factor of 24, 30, and 42.
2
55 miles per hour (MPH). The first digit (ten thousands) could not have changed in two
hours, so the first and last digits must each be 1. The 5s must become 6s, giving us 16061,
or 16161, or 16261, and so on. (If the 5s became 7s or higher, the bus would have traveled
more than 1000 miles in two hours.) The bus must have traveled 110; 210; or 310; and so
on miles in the two hours. The possible speeds are 55 MPH (110 miles ÷ 2 hours), 105
MPH (210 miles ÷ 2 hours), and faster. We know that the driver was "careful," so she must
have been traveling at 55 MPH.
3
Fourteen fractions. The fraction 4/23 has a decimal value of approximately 0.174, which is
too low; therefore, 5/23 is the first fraction within the interval. The fraction 19/23 has a
decimal approximation of 0.826, which is too large; therefore, 18/23 is the last fraction in
the interval. Hence, 18 − 5 + 1, or 14, fractions meet the given conditions.
4
184. Since the number rounded to the nearest 100 is greater than the number when rounded
to the nearest 10, the number must be greater than 150. Therefore, the number rounded to
the nearest 100 will be 200. If the number is 20 more when rounded to the nearest 100, then
the number must be 180 when rounded to the nearest 10. Whole numbers greater than 100
that round to 180 when rounded to the nearest ten are 175, 176, 177, 178, 179, 180, 181,
182, 183, 184. Therefore, 184 would be the greatest number.
5
The items cost 29 cents each. From the amount of change received, she must have spent
between $8.00 and $9.00. We are, therefore, looking for a perfect square between 800 and
900. Since 900 = 302, try 29; 29 × 29 = 841 cents, or $8.41. Her change of $1.59 could have
been $1.00, 2 quarters, 1 nickel, and 4 pennies. This solution matches the conditions of the
original problem.
6
6009. This depends on how you write the digits, but we will assume that the digits 0, 1, and
8 have rotational symmetry. Also, 6 and 9 rotate into each other. However, if you consider 2
to have rotational symmetry (as on many calculators), then the answer is 2002.
7
101, 111, 121, 202, 212, 303, 1001, 1111, and 2002. When the numerals making up the
palindrome are sufficiently large to require regrouping in the standard algorithm for
multiplication, the resulting product is not a palindrome.
8
5. Since the mode is included in the problem, one of the values 3, 4, 5, 8 must be repeated.
There are five values, and since the mean must be one of the existing numbers (the mean,
median, and mode are equal in this problem), the sum must be divisible by 5. The sum
without x is 20. The only number out of the set that can be added and that will give a new
sum divisible by 5 is 5.
9
About $7,300,000 (the actual price was $7,200,000). Rhode Island would cost about
$13,000. Answers will vary for other states. Multiply the area in square miles times the
number of acres in a square mile times the cost per acre. For Alaska, you get (570,374 mi.2)
× (640 acre/mi.2) × (2¢/acre) = $7,300,787, which should be reported as $7,300,000, with
only two significant digits (at best), because the cost per acre is only given to the nearest
whole cent with one significant digit.
10
$800. The very best that Adam could do is to draw four $5 bills ($20), four $20 bills ($80),
four $50 bills ($200), and five $100 bills ($500) before he has to stop drawing. This would
be a total of $800.
11
100 bills were vetoed.
Using a Venn diagram is one strategy to solve this problem: 800 bills were passed by both
houses and signed into law and V were passed by both houses but vetoed, so 800 + V goes
into the intersection of the two circles.
The House passed 1200 bills, so 400 −V, or 1200 − (800 + V), goes into the "House Passed"
circle.
The Senate passed 1500 bills, so 700 −V, or 1500 − (800 + V), goes into the "Senate Passed"
circle.
Since 1800 bills passed at least one house, the sum of the values in the three areas of the
diagram must be 1800 = (800 + V) + (400 −V) + (700 −V) = 1900 −V; V = 100 bills vetoed.
12
Out of a total of 300 elves, 180 will be in Assembly, 105 in Decoration, and 15 in
Packaging.
Santa needs to assign elves proportionately to the amount of work to be done so that each is
not waiting for one of the preceding steps to be finished.
Completing a toy takes 100 minutes: assembly takes 60/100, or 60 percent, of the time, so
60 percent of 300 elves is 180. Similarly, 35 percent of 300 elves is 105 who must work in
Decoration. Five percent of 300 elves is 15 who will be in Packaging.
This problem can also be set up as a proportion, such as 60/100 = x/300. It can also be
solved by considering that the greatest common factor of 60, 35, and 5 is 5.
Compared with packaging, it takes 12 times as long to assemble a toy and 7 times as long to
decorate it. For each 1 elf who is in packaging, 12 should be assembling and 7 decorating,
which totals a group of 20 elves. Fifteen groups are possible: 180 assembling, 105
decorating, and 15 packaging.
13
5, 7, 8, 9, 11, 13, 14, 15, 17, 19, and 20. The expression is an integer only when n + 1 is a
factor of n!. For example, when n = 8, the numerator is 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
and the denominator is 9. Since 3 and 6 are in the factorial expansion of 8!, we know that 9
is a factor of 8! without calculating its value. Notice that when n + 1 is a prime number, the
expression n!/(n + 1) cannot be an integer. The prime numbers from 2 through 20 are 2, 3,
5, 7, 11, 13, 17, and 19. So values of n that will not yield integers include 1, 2, 4, 6, 10, 12,
and 18. In addition, n = 3 will not work.
14
6,336 feet. Before the ruler was changed, there were 12 inches in 1 foot and 5,280 feet in 1
mile, which meant that there were 63,360 inches in 1 mile. There are still 63,360 inches in 1
mile because the lengths of the mile and the inch were not changed. Since there are now 10
inches in 1 foot and 63,360 inches in 1 mile, divide 63,360 by 10 to get 6,336 feet in 1 mile.
15
Five two-digit numbers. They are 11, 12, 15, 24, 36. For 11, 1 × 1 is 1, which is a factor of
11; for 12, 1 × 2 is 2, a factor of 12; for 15, 1 × 5 is 5, a factor of 15; for 24, 2 × 4 is 8, a
factor of 24; and for 36, 3 × 6 is 18, a factor of 36.
16
23 rungs.
You might think of the ladder as being a number line, with the middle rung signifying zero,
(0). He stepped up 3 rungs (+3). He went down 5 rungs (-2). He climbed 7 rungs (+5). He
climbed 6 rungs to the top of the ladder (+11). Since 0 was at the middle, there are also 11
rungs below the middle. The number of rungs is (11 rungs above 0) + (11 rungs below 0) +
(the 0 rung) = 23 rungs.
17
a = 9, b = 1, and c = 0.
We first know that b = 1, because b in the quotient times divisor a must equal a. (Put 1's in
for b's.)
Then since a subtracted from the two-digit number 1c is 1, we know that c = 0 and a = 9.
18
9:33 A.M. on 5 January 2000. Compute 2,000,000 seconds ÷ 60 seconds/minute ÷ 60
minutes/hour ÷ 24 hours/day = 23.148 days. The fractional part 148/1000 is equivalent to
3.552 hours, and 552/1000 hour rounded to the nearest minute is equivalent to 33 minutes.
A total of 23 days from 13 December is 5 January, and 3 hours 33 minutes from 6:00 A.M. is
9:33 A.M..
19
3. The number must be greater than 1000. To minimize its value, choose 1 as the first digit.
Hence, 0 can be chosen for the second digit. The third digit, then, cannot be 0 or 1, so use 2.
The fourth digit will then be 3.
20
Forty-five. From 10 to 19, only one number, namely, 10, has a larger first digit. Two such
numbers fall from 20 to 29; and three such numbers, from 30 to 39. This pattern continues
up through nine numbers from 90 to 99. The total is 1 + 2 + 3 + ... + 9 = 9(10)/2, or 45, such
numbers.
21
88.
Follow the clue of each line. The first line indicates that the number is a multiple of 11, or
11, 22, 33, 44, . . . 110, 121, 132, . . . . The second line reveals that the number is even. The
third line indicates that the number contains a pair of digits, which reduces the set to twodigit, even, multiples of 11, or 22, 44, 66, 88. The final task is to multiply the digits and see
which product is both a square and a cube. Try 22; 2 × 2 = 4, which is a perfect square but
not a perfect cube. Try 44; 4 × 4 = 16, which is a perfect square but not a perfect cube. Try
66; 6 × 6 = 36, which is a perfect square but not a perfect cube. Use 88; 8 × 8 = 64, which is
both a perfect square and cube: 82 = 64 and 43 = 64.
22
1789; 121st.
Based on 2-year Congresses, and using the 107th Congress as the last full-term Congress,
2003 − 2(107) = 2003 − 214 = 1789.
The Congress in session in 2030 will convene in 2029: (2029) − 2003)/2 = 13 and 13,
beyond the 108th Congress is the 121st.
23
24 workers.
Suppose there are x workers. The number of pizzas delivered will be (x workers)/( 2
workers/pizza) and similarly for the other two items. The total number of items is then x/2 +
x/3 + x/4 = 26. Solving for x, we find that there must be 24 workers that evening. We could
also reason that the number of workers must be a common multiple of 2, 3, and 4. These
multiples are 12, 24, 36, 48, 60, 72 . . . . Set up a ratio table, giving the number of items for
each possible number of workers, to find that 24 workers result in the given numbers of
items.
24
A little more than 5 times his height; grasshopper, 31.5 m; jerboa, 78.75 m; flea, 385 m.
Multiply Powell's height by each of the ratios obtained by dividing the length of the jump
by the length of the body when both are expressed in the same units. Have students measure
these lengths on the floor in the classroom, hallway, parking lot, or other large open space.
Does it seem possible for a man to jump these lengths? Why or why not?
25
4 8/23. Writing the mixed numbers as improper fractions helps students see the pattern. The
numerators are perfect squares and the denominators are consecutive odd integers starting at
5. In general, the nth term is n²/(2n + 3).
26
Here are two possible answers:
Can you find other solutions?
27
1/6 was left for Mom. Everybody eats the same amount of cake.
(1/6)(1) = 1/6 eaten; 1 – 1/6 = 5/6 of the cake that is left.
(1/5)(5/6) = 5/30 = 1/6; 5/6 – 1/6 = 4/6
(1/4)(4/6) = 4/24 = 1/6; 4/6 – 1/6 = 3/6
(1/3)(3/6) = 3/18 = 1/6; 3/6 – 1/6 = 2/6
(1/2)(2/6) = 2/12 = 1/6; 2/6 – 1/6 = 1/6
28
33 seconds. First, you must determine how many floors you are traveling. Going from the
first to the fiftieth floor is not 50 floors but only 49. (Think of it this way: How many floors
do you go up when going from the first floor to the second floor? Only one floor.) You
travel 12 feet for each floor: 49 × 12 = 588 feet. If the elevator travels an average of 18 feet
per second, then 588/18 = 32.67, or 33 seconds, since the problem says "to the nearest
second."
29
16. The product of the numbers in each grid is 96; 1×2×3× 16 = 96.
30
126. The two-digit multiples of 6 are 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90,
96. The numbers 24, 42, and 60 have digits that sum to 6; 24 + 42 + 60 = 126.
31
1155. Remember that 1 is not a prime number. Two is the first prime number, but if it is one
of the prime numbers used, the product will be even. Therefore, to obtain the smallest value,
you would use 3, 5, 7, and 11, the product of which is 1155.