Mrs. Hernandez bought several items that each cost the same number of cents as the number of items purchased. She received $1 and 7 coins in change from $10, so each item must have cost 3 cents. Alice has a 20 gallon tub of water. She will pour out half the water, then half of what remains, and half of the remainder, repeating until the tub is empty. This will take her 4 pours to empty the tub. A 10% raise followed by a 10% cut results in the original salary, while a 10% cut followed by a 10% raise results in a 1% higher salary. Therefore, it is better to take the raise first, then the cut.

1. October Problems
MP2 Reason abstractly and quantitatively

2. Mrs. Hernandez’ Change
10/1
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?

3. Watching Water Evaporate
10/2
Alice has just finished washing clothes in a 20-gallon
tub and must now throw out the wash water. She
pours half the water on the ground to evaporate.
After it evaporates, she will again pour out half the
water that is left in the tub. How many times will she
pour out the water before the tub is empty?

4. Raise-Cut ~ or ~ Cut-Raise?
10/3
Suppose your salary could be raised 10% and then a
month later reduced by 10%. Or suppose that you may
choose to have the cut first, followed by the raise one
month later. Which option is better? Why?

5. Will Wants His Watch Back
10/4
Two boys are discussing money.
Will:“How about lending me $10?”
Tyler: “I can’t; I spent some of it.”
Will:“How much did you spend?”
Tyler:“Exactly 1/4 of what I have left.”
Will: “Good. That leaves you with just what I need
to get my watch back from the watchmaker.”
How much money did Tyler have left?

6. What Are The Rules, Anyway?
10/5

7. Reel It In!
10/9
Jake caught a fish. To win a contest, his fish had to
weigh more than the biggest one, which weighed 4
pounds. Jake’s fish was hard to weigh. See if you can
figure out its total weight. The tail weighed 9 ounces,
the head weighed as much as the tail and half the
body, and the body weighed as much as the head and
tail together. What was the weight of the fish?

8. 2 Keep'n It Real! +
4 5 -
10/10
3
Using only the digits 2, 3, 4, and 5 and the two
mathematical symbols “+” and “–” each once and
only once, create a true mathematical sentence.
For example, 2 + 3 = 45 uses all the digits and
symbols once and only once, but it is not a true
mathematical sentence. You may not use any other
mathematical symbols or digits.

9. Minimize it!
10/11
Let A, B, and C represent different digits greater than 0.
Determine the minimum value of the expression below.
(Note: for ABC, A simply signifies a number that is the hundreds digit,
B is the tens digit, and C is the ones digit—they are not multiplied.)

10. Pocket Change
10/12
I have 6 coins in my pocket
totaling $1.15, but I cannot make
change for a dollar, half dollar,
quarter, dime, or nickel. What
coins do I have in my pocket?

11. Counting Rectangles
10/15
The figure below is composed of congruent squares.
How many rectangles are in the figure?

12. Circumference River and
10/16 Square Root Bridge
The width of the Circumference River is 3100 meters.
The Square Root Bridge spans the Circumference
River. If 1/8 of the bridge stands on land on one side of
the river, and 1/10 of the bridge stands on land on the
other side, how long is the Square Root Bridge?

13. A Paint Predicament
10/17
A box (shown here as a rectangular
prism) is 3 units by 4 units by 5 units.
If the box is composed of unit cubes
and completely dipped in paint, how
many unit cubes will have no paint on
any of their faces?

14. What’s the Relation?
10/18
If the radius of a circle is
doubled, what happens to
the area? What happens
to its circumference?

15. Sports Confusion
10/19
There are 40 kids in gym.
• 10 play football, soccer and basketball
• 15 play football and soccer
• 24 play football only
• 22 play soccer only
• 14 play football and basketball
How many kids are only on the basketball team?

16. The Prime Difference
10/22
Which of the following numbers:
1, 2, 7, 8, or 10 cannot be the
difference of two prime numbers?
Explain your reasoning, and
provide a counterexample for
each number that can be the
difference between two primes.

17. Tennis, Anyone?
10/23
Think about a typical can containing three
tennis balls. Which is greater, the height of
the can or the circumference of the base of
the can? (Ignore the thickness of the plastic.)
Make an estimated guess first. Then use
mathematics formulas and/or actual
measurement to verify your guess.

18. Prime Number Constraints
10/24
Find the sum of the least and
greatest two-digit prime numbers
whose digits are also prime.
Hint: 0 and 1 are not prime numbers.

19. Extend the Sequence
10/25
Find the next three numbers in the special
sequence of numbers.
3, 1, 4, 1, 5, 9, 2, 6, 5, ___, ___, ___

20. Decoding the Riddle
10/26
Remove “twelve letters” to
reveal two hidden numbers.
What are the two hidden
numbers? You may have to
reorder the letters.
(Hint: In this puzzle, “twelve letters” ≠ 12)

21. These Shoes Were Made
for Walking
10/29
You begin walking on a road. You travel
78 feet during the first minute, 85 feet the
second minute, 92 feet the third minute,
increasing by 7 feet each minute. If the
total time you traveled is 8 minutes, how
far did you walk?

22. Find the Missing Number
10/30
Based on the numbers in the first three 2 ×2 grids,
determine the missing number in the fourth number grid.

23. Fraction Frustration
10/31
Find the value of the expression
above. Your answer must be in
simplified fractional form.