1)
My name is Shane. I am a surfer, and I have this really, really gnarly surf shack. It's a bummer, though - my floor needs re-tiling because one day this crazy west swell came out of nowhere and dropped a 15- footer on a couple of trees. My bad luck, the trees went through my roof and massacred my floor.
Anyhow, I have an area like this that I have to fill:
Here is the tile pattern that I want to use.
Using this pattern, which fits the floor plan perfectly, how many of each tile will I need?
The Surf's Up Tile Shop told me that all of the tiles would cost $2019.60 and that the small tiles cost $1.49 each, but I've forgotten the cost of the 8-inch tiles. Can you help me determine the cost of one of them?
2)
Accounting for Age
Step 1. Pick a number.
Step 2. Double it.
Step 3. Add 5.
Step 4. Multiply by 50.
Step 5. If you've had your birthday this year, add 1770, or, if you've not yet had your birthday this year, add 1769
Step 6. Subtract the year of your birth.
You should end up with a number that is a combination of the number you started with followed by your age. For example, if you are 12 and you start with the number 27, the result you should end up with is 2712.
Your tasks:
Let n represent the number in Step 1 and show what happens through Step 6.
Explain why the result is always the original number followed by your age.
3)
I am thinking of making a quilt from an illustration that appeared on the cover of an old Scientific American magazine. Here's a sketch of what it looks like:
I know that:
all of the pieces are squares
the area of C is 64 square inches
the area of D is 81 square inches
Question: Will the finished shape be a square?
Extra: What's the area of the quadrilateral?
4)
Two people are walking through a garden. One of them points to a rectangular garden and says, "If I had made that bed 2 feet broader and 3 feet longer it would have been 64 square feet larger; but if it had been 3 feet broader and 2 feet longer it would then have been 68 square feet larger. What are its length and breadth?"
Make sure you carefully explain how you got your answer!
5)
Last year right when summer was winding down, Mr. and Mrs. Mallery decided to take their family to the beach one last time. They loaded their kids, Olivia and Liam, into the car and headed off early in the morning.
At 7:05 Olivia asked, "Are we there yet?"
"We're one-third of the way there," replied Mr. Mallery.
At 7:25 Liam asked, "Are we there yet?"
"We're 75% of the way there," said Mrs. Mallery. "Now, can you two figure out what time we should get to the beach?"
Assuming the Mallerys maintain a constant speed during their trip, can you help Olivia and Liam determine what time they will arrive?
Extra: If the distance from their house to the beach was 32 miles, what was their average speed in miles per hour during the car trip?
6)
Lily and her little brother Mikey eat different kinds of cereal, but their cereals come in the same size box. Every morning at breakfast Lily eats four times as much cereal as Mikey does.
One week an interesting thing happened. On Monday Lily started with a new full box, while Mikey still had half a box to finish. Later in the week they each had the same amount left in their cereal boxes.
What fraction of a box did each child have left?
7)
This week we bring you another "proof" problem. As we had in a problem last semester, the square numbers figure prominently in the result. Using your experience on that other problem, this one should not be so difficult.
It is a curious, but not-so-well-known, fact that:
The product of any four consecutive integers, increased by one, is always a square number.
Give at least three instances of that statement.
Prove that this will always occur by finding an algebraic expression for that "square number."
Remember: citing many examples of the truth of the above statement does NOT constitute an algebraic proof. It merely convinces you of the probable truth - but it undoubtedly would guide you as to how to proceed with your proof.
8)
The ratio of girls to boys in Mrs. Franks’ 7th grade class is ¾ (excluding Mrs. Franks). On Monday, 4 girls and 6 boys got to go on a special field trip for honor roll students. Since no other students were absent on Monday, the ratio of girls remaining in the class to boys remaining in the class was 4/5. How many students are in Mrs. Franks’ class when everyone is present?
On the field trip, Abbey, Bethany, Christine and Darcy bought a total of 4 hot dogs and 3 sodas, which cost the girls $17.00 before tax. Edward (one of the boys on the field trip) bought 2 hot dogs and 1 soda from the same lunch stand and it cost him $7.80 before tax. How much would 1 hot dog and 1 soda cost at the lunch stand before tax?
On the same day, 32 students from the 6 th grade got to go on a field trip to the zoo. When they returned to the school after the field trip the students were asked to answer a couple of survey questions. One of the questions asked, “Which animal did you like best, the elephants or the lions?”. When all of the surveys had been turned in, 20 students had selected elephants and 22 had selected lions. If all of the students selected at least one of the two choices, how many students selected both lions and elephants?
9)
Nancy and Tim have an apple cider stand at their school every morning during the fall to raise money for their club. Since September 23 is the Autumnal Equinox, the first day of fall and their first day of business will be September 23.
Several weeks ago, in preparation for their 2019 opening, Nancy and Tim went to pick their apples from a local orchard (all of their cider is homemade). Nancy picked apples at a rate of 30 apples per hour and Tim picked apples at a rate of 25 apples per hour. Tim and Nancy picked the same number of apples. Nancy picked apples for 5 hours, so how many hours must Tim have spent picking apples?
Each gallon of cider produced requires one bushel of apples (40 apples), which cost Nancy and Tim $30 at the orchard. Anxious to calculate how much they will make, Nancy decides to calculate their profits, based on serving 6 oz cups of cider (in cups that were donated by a friend) and charging $2.00 per cup. Assuming they sell all of the cider they can make, how much profit will they make from the apples they picked? Note: There are 128 oz in 1 gallon.
Hoping to make more than what the calculations show, Nancy ponders what will happen if they add some water to the cider. She decides to calculate how much they would profit if they added 1.5 gallons of water to the 7.5 gallons of cider. If they sell the diluted cider for the same price as they had planned to sell for ($2.00 per 6-ounce cup), how much additional profit will be made?
10)
Lately Sam has had trouble with math, and has been having strange, math-related dreams. "I hope I don't have any more of those bizarre math dreams," he mumbles to himself as he starts to doze off, but he wakes the next morning and can remember the following dream vividly.
"My name is Fred. Welcome to the Land of Make Believe," a voice says. Sam turns around and sees a boy who looks about seven years old. "Let me give you a tour," Fred says.
First they visit Fred's home, and Sam notices that all the clocks in the house are missing the digits 8 and 9, and that the phone is also missing the digits 8 and 9.
"Why don't your clocks have the numbers 8 or 9 on them?" Sam asks.
"What is an 8 or 9?" Fred responds. "The only digits we use are 0, 1, 2, 3, 4, 5, 6 and 7."
"What comes after 7?" Sam asks.
"That's easy," Fred replies. "10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30..."
Sam scratches his head in bewilderment. "If you don't have the digits 8 or 9 in your number system, how do you know if you are adding right?" he wonders.
"Well, in school we were taught that you can check your addition by doing the following. First, add up the digits in the answer. Then subtract that sum from the answer. If the result is divisible by 7, then you did it right." (Fred is obviously a good math student.)
"But why does that work - and can I do the same thing in my world where we have eights and nines?" Sam queries.
Right then, before Fred can answer his questions, Sam wakes up. Maybe you can help Sam.
1. First, come up with an addition example from the Land of Make Believe (remember that they only use the digits 0-7) and show that the addition checking rule works.
2. Second, for any base b, show that if you subtract the sum of its digits from a number, the result is divisible by b - 1.
