1)
In a sequence of numbers, every number (except the first two) is the sum of the two numbers before it.
If the first number is one, and the tenth number is 111, what is the second number?
Be sure to explain and show all your work. If you use variables, be certain to define them. Please include a check to your work. No approximations will be accepted as correct.
Bonus: Form a similarly derived sequence of ten numbers, except choose the first two numbers to be any two integers. Find the total sum of the ten numbers. Then take the seventh number in your sequence and multiply it by eleven. What was the result? Prove algebraically that what you discovered will always work.
2)
Shanika and Monica were in their favorite store, checking out the current sales. One item they liked was listed at $24.99 with a 5% discount.
Monica commented that she'd still need $24.99 to buy it since tax in their city was also 5%.
Shanika disagreed. She claimed the item would be less than $24.99.
Do you agree with Shanika or Monica? Why? Be sure to show your calculations for the final price in your Explanation.
3)
Sasha wants to take the train to visit her friends over the holidays. Amtrak's Train 29, the Capitol Limited, leaves Union Station in Washington, D.C., at 4:05 p.m. Eastern Time.
It arrives at Chicago's Union Station at 8:45 a.m. Central Time the next morning.
How long will her trip take? Explain how you found the answer.
Extra 1: Use the schedule to find out which station is closest to half of the distance from Washington to Chicago.
Extra 2: Where will the train be when half of the time has passed?
Be sure to explain how you know.
Times in the Central Time Zone (CT) are one hour earlier than in the Eastern Time Zone (ET). When it is 2 p.m. in Washington, it is 1 p.m. in Chicago. Visit http://www.timeanddate.com/worldclock/ to see current times in cities around the world.
4)
This one is for all you pet lovers out there!
A hog, a dog, and a frog all want to cross a bog.
The hog slogs across at a speed of five feet per second.
The dog logs a time of 20 minutes for its crossing.
The frog jogs across twice as fast as the hog but half as fast as the dog.
What is the distance in feet across the bog?
Extra: Why do these funny fellows want to cross this ol' bog in the first place?
5)
Valentine's Day Chocolates
When Tristan visits Isolde on Valentine's Day, he's planning to bring her a one pound box of chocolates. He knows that when they share a one pound box of chocolates, it usually lasts for two weeks. When he eats a one pound box of chocolates on his own it lasts for six weeks.
How long should a one pound box of chocolates last if Isolde eats it alone?
Remember to explain how you solved the problem and show how you know you are correct.
Note: Assume they don't eat faster or slower when they are alone.
6)
Ann and Sue bought identical boxes of stationery. Ann used hers to write 1-sheet letters, and Sue used hers to write 3-sheet letters.
Ann used all the envelopes and had 50 sheets of paper left, while Sue used all of the sheets of paper and had 50 envelopes left.
How many sheets of paper and how many envelopes were in the box to begin with?
7)
I'm thinking of a particular integer. It is a multiple of three, five, and seven. No digit occurs more than once. Can you find my number?
With just that information there are lots of possibilities, so let me narrow the problem down with a few more clues.
The digit in the tens place is a square number.
The digit in the hundreds place is a cube.
The digit in the hundred thousands place is both a square and a cube.
Only the digit in the hundreds place is larger than the digit in the ones place.
What is my integer?
Bonus: List the prime factors of my integer.
8)
There was a surge in math club membership after the fun Halloween party. When the principal asked Mrs. Germain how many students were currently in the club, she replied, "Three times our number plus a third of our number plus a fourth of our number plus you and me is two hundred sixty."
The principal smiled and nodded, then added, "If you can pick up another eight students before your Christmas party you will have exactly 10% of the student body involved in the club."
How many students are enrolled in the math club? How many students are in the school?
9)
When planting a garden of flowers and vegetables in central New Jersey, you must surround it with a fence to protect it from the deer. Last summer I planted a long, skinny garden all the way along a 30 foot wooden fence, so that I would only need to put a netting fence up along three sides of it. Because my garden was so long, I only made the sides of the garden 1 foot wide.
This year I decided to move my garden to another spot in the yard where there isn't a permanent fence and I changed the shape. I made my garden 6 feet by 8 feet.
Question 1: How much larger is the area of my garden this year?
My garden is larger, so I was thinking I had to go to the store to buy more netting. I did some calculations to figure out what I would need to buy.
Question 2: How does the amount of netting I'll need this year compare to what I needed last year?
Extra: So that my snow peas can grow in rich, fertile soil, I like to add a layer of mushroom soil that is 6 inches deep before I plant my seeds. How many cubic feet of soil will I use this year?
10)
I reached into my pocket and found that I have only dimes and quarters.
I have $2.00.
I have 11 coins.
How many dimes do I have?
Remember to explain how you found the answer and how you can be sure it is correct.
Extra: From my coin jar I exchanged half my quarters for the same number of dimes. How much money do I have in my pocket now?