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The Garden Grows... 
A
fence encloses a rectangular garden that is 50 feet long and 10 feet wide. To make
the garden larger while using the same fence, its shape is changed to a square. By
how many square feet does this shape enlarge the garden? 


Answer:
400 square feet. The rectangular garden
requires 120 feet of fencing and has an area of 500 square feet. To make the garden
square and still use all the fencing, each side of the square would measure 120 ÷ 4, or
30 feet. The area of the square garden would be 900 square feet, which is 400 square
feet larger than that of the original rectangular design. 
Two
Elevators 
Two
elevators leave the sixth floor at 2:00 PM. The faster elevator takes one minute
between floors and the slower elevator takes two minutes between floors. The first
elevator to reach a floor must stop for three minutes to take on passengers. Which
elevator reaches the lobby on the first floor first? 


Answer: The slower one. 
The Locker Problem
a classic 

When students returned to
school after winter break, they decided to see what would happen if they went into school
one at a time. As each student went into the school they would open or close the
lockers according to a set of directions. The first student 

would open all of the lockers (11000). The second student would
close the even numbered lockers. The third student went to every third locker and
closed the ones that were open and opened the one that were closed. The fourth
student would go to every fourth locker and open the closed ones and close the open
ones. This pattern continued until the 1000th student went through. When this student was done which lockers were still opened???? 
Answer: This is a classic problem and it you do a smaller problem you find
that the lockers that are square numbers are open (ie. 1,4,9,16,25, 36, 49, 64, ...). One
of my students gave a great explanation... here it is:
If a locker is
changed an even amount of times then it will go back to being closed. If it is
changed an odd amount of times, then it will be open. What kind of number has an
even number of factors (how many factors a locker number has determines how many times it
is changed)? Most numbers have an even amout of factors, because for every factor
there is a matching one that when multiplied will equal the locker number. Except
for square numbers because for one of their factor pairs the numbers are the same, and so
those two factors, being exact same, are only counted as one. That would make the
amount of factors for those numbers be odd. 
The
Skateboard Run 

Some of our students are
setting up a skateboarding course for a competition. The course is 80 yards long.
They are using cones to have to skate around as a slalom run. If the cones are
placed 8 yards apart and starting at the start line and the last is placed at the finish
line how many are needed for the entire course? 


Answer: There are 11 cones needed. One every 10 yards and then
one to begin the race with. 
A
Whole Bunch of Problems!!! 

1. For the year 1994, no date
exists for which the product of the month and day equals the number named by the last two
digits of the year. Last year, 33193 met this criterion. During this century
(1900 1999), which year would have the most possibilities?(2pts)
2. While driving the mathematics team to a
contest, the driver noticed that the bus odometer read 21,912 miles. Being a lover
of mathematical curiosities, she noticed that this number was a palindrome, reading the
same backward and forward. The team sponsor said, "I bet it will be a long time
before that happens again!" However, just 105 minutes later, the bus driver
reported that the mileage again was a palindrome! How fast was the bus traveling? (3
pts)
3. A certain kind of folder was priced at
50¢ but found no buyers. The manager of the store decided to reduce the price.
Within a few days, she had sold her entire stock of these folders for a total of
$31.93. What was the reduced price of the folder? (2pts) 






Answers:
1) 1924. It has the
most factors. 2) Approximately 63 mph. the next
palindrome is 22022 or 110 miles. So in 105 minutes or 1.75 hours, they drove 110
miles. Just divide to find the answer. 3) $0.31 or $0.01.
The number must be a factor of 3193. 
2
Flat Tire Problems 
Flat Tire 1:
Mrs. Adams woke up one morning to find that one the wheels
of her car had a completely flat tire. Despite this she set off in her car and drove
100 miles to visit a another school. She then drove 100 miles home. She did
not repair or inflate the flat tire. How did she manage to make the journey? 

Flat Tire 2:
Four high school students arrived late for school, explaining to
Mrs. V, the principal, that their car had suffered a flat tire on the way there. How
did the clever principal immediately discern that the students were not telling the truth?


Answer:
#1 The flat tire was the spare... so
it was not needed to drive at all. I told you to think laterally.
#2 The clever principal took
each student to a different room and asked each one which tire was flat. Of course
each said a different tire and they were caught. 
Plates,
bottles and Glasses 

Here's a classic from the
book Cyclopedia of Puzzles, written by Sam Loyd in 1914.
If a bottle and a glass balance with a pitcher, a bottle balances with a
glass and a plate, and two pitchers balance with three plates, how many glasses will
balance with a bottle? 


Answer: It will take 5
glasses. 
Wendy's Family? 

Wendy's family has fewer than
ten children. She has the same number of sisters as she has brothers, but her
brother Andy hastwice as many sisters as he has brothers. How many boys and how many
girls are in the family 


Answer: There are four girls and three boys.
Because Wendy has the same number of sisters as brothers, we need to test only the
possibilities in which Wendy has the same number of sisters and brothers up to four each.
If Wendy has five sisters and five brothers, then the family has more than ten children.
Once we know Wendy's siblings, we know that Andy has one more sister than Wendy and
one fewer brother. We would then need to check to see whether the number of Andy's
sisters is twice that of brothers. 
Gillian's
Construction Co. 

Gillian's
Construction Company is building a drainage ditch that is 800 feet long, 6 feet wide, and
5 1/2 feet deep. If a truck can carry 2000 cubic feet of dirt, how many truckloads
will be needed to carry away the dirt? 


Answer:
14 Truck loads. The volume of
dirt taken away is 800 x 6 x 5.5, or 26,400 cubic feet. This amoount divided by 2000
gives 13.2 truckloads of dirt. 
Mr.
Titen and His Long Drive 

Mr. Titen was
traveling with his parents. He noticed a sign that read, "Highland Park 111
miles; Bud Lake 173 miles." He continued his book, figuring that he had
a while before his parents stopped for lunch in Bud Lake. Sometime later, his mother
said, "Look, Bud Lake is exactly twice as far away as Highland Park." How
many miles had they driven while Mr. Titen was reading? 


Answer:
49 Miles. Use a drawing. If Bud Lake is
173 miles away and Highland Park is 111 miles away, then the two towns are 62 miles apart.
To be twice as far from Bud Lake as Highland Park means that my family must be 62
miles away from Highland Park. The distance from the first sign to Highland Park was
111 miles, and 111  62 = 49. 
We
Can Make Any Amount of Change 

Here's
a tough one:
What is the least number of coins that you
need, to pay exactly, for any amount up to but not including $1.00? 


Answer:
Ten coins (not using 50¢
pieces), 4 pennies, two nickels, 1 dime, 3 quarters; nine if
a fiftycent piece is used to replace two of the quarters. 
The
Vacation Package 

#1)
Dharma says that 1634 students attend River View Middle School. Mr. Girvan says that
the school has 72 more girls than boys. How many girls attend River View Middle
School? (1pt) #2) Jona has only 20 minutes before he must
go to bed, but he wants to play a soccer game that he found on the Internet. The
computer downloads at a rate of 58,000 bytes an hour. The game that Max wants to play
consists of 8,000 bytes. How much playing time to the nearest second does Max have
after the game downloads before he has to be in bed?(2pt)
#3) Fluffy was born on 24 June 1983. On her first birthday.
Fluffy's age in cat terms was equivalent to that of a 15yearold human. When
she turned 2, her age in human years was 24. At the end of her third year, her human
age was 28, and when she was 4, her human age was 32. She continued to age 4 years
with each birthday. What year will Fluffy turn 100 in cat years?(2pt)
#4) Because Ms. Drozd wanted to win the Berryville bicycle race,
she trained hard for a week, including the weekend. Each day, she rode 3 miles
farther than she ahd the day before. If she rode a total of 126 miles, how far did
she ride on the last day before the race?(2pt)
#5) Mrs. Scott bought 100 pounds of potatoes that were 99 percent
water. After the potatoes were left outdoors for a day, they were 98 percent water.
What was the weight of the slightly dihydrated potatoes? (3pt) 






Answer:
1) 853. The total number is 1634; let x be the
number of boys, and let x+72 be the number of girls. Then x + x + 72 = 1634;
therefore, x = 781 and 781 + 72 is the number of girls.
2) Jona has 11 minutes and 43 seconds to play.
Dividing 58,000 by 60 minutes gives about 966.7 bytes per minute. Dividing
8,000 by 966.7 gives 8.28 from 20 minutes leaves 11.72 minutes wich is 11 minutes and 43
seconds.
3) On June 24, 2003, Fluffy can blow out 100 candles if she can stay awake long enough.
4) 27 miles on the last day. One strategy to use
is guess and check. An algebraic approach would let x be the number of miles biked
on the first day. Then the equation would be x + (x +3) + (x + 6) + (x+9) + ( x +
12) + (x + 15) + (x + 18) = 126
5) 50 pounds. If the potatoes were originally 99%
water, that means they contained 1% solids, or 1 pound of solids. The amount of
solids stays the same no matter how much water is lost. On the second day, if the
potatoes were 98% water, 2% of their wiehgt is in solids. Because the potatoes
contain one pound of solids, if w = the weight of the potatoes at eh end of the second
day, then 0.02 x w = 1. Solving for w, we get w = 50 
Can
You Crack the Code? 

F 
3R 
2D 
2R 
1L 
2D 
1U 
1L 
2D 
2R 
1L 
3L 
1R 
2U 
1R 
1D 
4L 
1D 
1D 
3R 
1D 
2L 
2U 
5L 
3R 
3U 
3R 
2L 
2U 
1L 

Here is an
unusual safe. Each of the buttons must be pressed only once in the correct order to
open it. (Every button must be pressd.) The last button is marked F.
The number of moves and the direction is marked on each button. Thus 1U would mean
one move up, whilst 1L would mean one move to the left. Which button must be pressed
first? Give its exact location. Hint: it is somewhere in the first row? Give the
order of the buttons. (2pt) 


Answer:
The first Button is 2D in the third column. From there you'll have to figure out the rest... sorry. 
What's
the pattern?? 

1 
2 
3 
4 
5 
4 
5 
1 
2 
3 
2 



1 







? 



Fill up this
square with the numbers 1 to 5 so that no row, column or diagonal line of five squares
uses the same number more than once. What number should replace the question mark? 


Answer:
5 should replace the question mark. 
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Page by: Mathman (Bruce A. Titen) © 2001
