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Page 6

 

Number Magic?
Here's a problem from a Number Magic trick.   One step is missing from this set of directions.  Start with your age as an integer, multiply it by 2, then subtract 3.  Next, multiply that result by 3.   At this point, a step is missing.  Finally, divide by 6, and you obtain your age as your answer.  What is the missing step?   Explain how you figured this out.

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Answer: The missing step is "add 9."  If the final result is your age, then the result just after the missing step must be 6 times your age.   Before the missing step, however, you have doubled your age, subtracted 3, and then multiplied the result by 3.  At that point, your result must be 6 times your age minus 9.  The missing step must then be "add 9."

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Triangles???
What are the possible lengths for the sides of a triangle's if the perimeter is 13 and the lengths of the sides are integers?  Why would segments of lenghts 1, 1, and 11, not form a triangle?

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Answer: There are five possible triagles.   1, 6, 62, 5, 63, 5, 53, 4, 64, 4, 5.   The second question... because any two sides of a triagle must add to be greater than the third.

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The Turtle Race
Mary's three pet turtles are great racers.  In a race, turtle A comes in first, beating turtle B by 12 cm and turtle C by 15 cm.  the race continues for second place, and turtle B beats turtle C by 5 cm.  Assuming that each turtle moves at a constant rate,  how many centimeters long was the race?

 

B

 

A

 

C

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Answer:   The racecourse was 30 cm long.  When A finishes the race, B is 12 cm fro the finish and 3 cm ahead of C.   But B finishes the race 5 cm ahead of C, so B gains an additional 2 cm in the final 12 cm of the race.  In other words,  B gains 1 cm on C for every 6cm of the race.  Because B beats C by 5 cm, the race must have been 5 x 6, or 30 cm long.

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Half a Cookie?

A large cookie is to be divided between two children, Gerald and Johnny.  They are both anxious to have the cutting done fairly so gthat the other doesn't end up with a bigger piece. How can they cut the cookie to be sure that each has recieved a true half of the cookie???

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Answer: This was not actually a math question.  This was a question of logic.  You should decide to have one person break the cookie in half and the other chose which piece to take. 

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What's in the bucket??

Imagine that five buckets of tennis balls contain a total of 100 balls.  How many balls are in each bucket if a) each contains 2 more balls than the previous bucket?  b) each contains 4 more balls than the previous bucket? You must explain your answer.(2pts)

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Answer: a) 16, 18, 20, 22, 24.  b)12, 16, 20, 24, 28.   One approach is to start with 20 balls in each bucket and then redistribute the balls until all conditions are met.

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How much for the Game Ball??

Our middle school needs new game balls for next year.  Footballs cost $80 each, baseballs are $22.50 each, and basketballs are $68 each.  The school needs three times as many baseballs as basketballs, half as many basketballs as footballs. The school needs twelve footballs.  How much will the school spend for new game balls?  Explain your answer.

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Answer: $1773.  Woking backwaerd through the clues, we see that the school needs twelve footballs.  Half as many basketballs would be six basketballs.   Three times as many baseballs as basketballs would be eighteen baseballs.  The school will spend 12 x $80 + 6 x $68 + 18 x $22.50 = $960 + $408 + $405 = $1773

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The Basketball Court Drill

In a practice drill in basketball, players begin at the baseline, run to the closest free-throw line, then back to the baseline, then to half court, back to the baseline, then to the far free-throw line and back to the baseline, and finally, to the opposite baseline and back to the original baseline, this drill is repeated several times during a practice. 

 

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1. The free-throw lines are half the distance to half-court from the baselines closest to them.  Determine the distance, in a basketball-court lengths, that a player runs in one time through the drill?  (1pt)

2.  Basketball courts vary in size.   Beth measured the court at her school and found that the distance from the baseline to the free-throw line is 5.8m.  What distance does a player run in one time through the drill?  (1pt) Write a formula to determine the distance that a player runs in R repetitions of the drill. (2pts)

3.  Beth's coach wants the  team to run at least 800 m during practice using this drill.  How many times must the players repeat the drill? (2pt)

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Answer: 1) 5 court lengths. Running to the free-throw line and back is 2/4, or 1/2, of a court length.  Running to the half-court line and back is 1 court length.  Running to the far free-throw line and back is 3/4 + 3/4, or 1 1/2 court lengths.  Finally, running the length of the court and back is 2 court lengths.  In the drill, a player runs 1/2 + 1 + 1 1/2 + 2 = 5 court lengths.

2) 116m.   Problem 1 indicated that the free-throw line is halfway between half-court and the baseline, so the length of the court in Jenny's school is 5.8m x 4 =23.2 m.  One repetition, R = 1, of the drill is five lengths or 23.2 x 5 + 116m.  For R repetitions, the distance formula for Beth's court is d = 23.5 x 5R.

3) Seven repetitions: 7 x 116 = 812 m.

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The Office Building

An office building has 36 stairs between the first and third floors.  If any two floors have the same number of stairs between them, how many stairs must a visitor climb to reach the sixth floor?

 

 

Answer: 90 stairs.  From the first to the third floors are 36 stairs, from the third to the fifth floors are another 36, and from the fifth to the sixth floors are half of 36, or 18.  Adding gives 36 + 36 + 18, or 90.

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What's the Fish Worth?

In the land of Titenium, three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice.  How many bags of rice is one fish worth?

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Answer2 2/3 bags of rice.  One loaf of bread can be traded for four bags of rice, so two loaves can be traded for eight bags of rice. Three fish can be traded for eight bags of rice, so one fish is worth 8 3 bags of rice.

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Girls, Movies, what's it all mean?

1) When 5 new girls joined a class, the percent of girls increased from 40% to 50%.  What is the number of boys in the classs?(2pts)

2) The first row of a movie theater has 11 seats.   Each successive row has one more seat than the previous row.  What is the number of seats if the theater has 30 rows?(1pt)

3)What is the mean of all three-digit numbers that can be created using each of the digitts 1,2, and 3 exactly once? (2pts)

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Answers: 1) 15 boys.  The ratio of girls to boys is 2:3, or 40: 60 reduced.  Adding five girls to the class results in the same number of girls as boys.   We see that 10 : 15 is in that ratio and adding 5 girls to the 10 then we can see how they become even.

2) 765 seats.  Thirty rows of seats uield a total of 11 + 12 + 13 + ...+ 38 + 39 + 40 seats.  A simple way to add these numbers is to add 11 and 40, 12 and 39 and so on.  This process leads to fifteen sums of 51; multiplying gives 51 x 15, or 765.

3) 222.  All possible numbers are 123, 132, 213, 231, 312, and 321.  The average of these numbers is 222.  Note that each digit is used exactly twice in each place value.  The average of the threee digits is 2; therefore, the average of each colum is also 2.

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How many Cookies did They Eat??

Lyndsey, Robert, Yvonne, Eric, and Alicia baked a batch of 36 cookies; 2/3 of these were chocolate chip, and the rest were plain.  They ate some of the cookies.  Only 1 1/2 dozen cookies were left, half of which were plain.  Alicia is allergic to chocolate.  Lyndsey ate twice as many chocolate chip cookies as plain cookies.  Robert and Yvonne each ate as many cookies as Lyndsey and Alicia combined. Robert ate more chocolate chip cookies than Yvonne.  If only the bakers ate the cookies and each of them ate at least one cookie, how many cookies of each kind did each person eat? (3pts)

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Answer:

Name

Cookies Eaten

  Plain Chocolate Chip
Alicia 1 0
Eric 0 6
Lyndsey 1 2
Yvonne 1 3
Robert 0 4

The group made 12 plain cookies and 24 chocolate chip cookies, and they ate 18 cookies.  Since half the remaining 18 cookies were plain, we know that the goup ate 3 plain cookies and 15 chocolate chip cookies.   Alica is allergic to chocolate, so she ate 1, 2, or 3 plain cookies.  Lyndsey ate twice as many chocolate chip as plain cookies, so she ate a least 1 plain cookie.   Robert and Yvonne ate the same number of cookies, but Robert ate more chocolate chip cookies than Yvonne.  So Yvonne must have eaten at least 1 plain cookie.   Now we know that Alicia, Lyndsey, and Yvonne ate 1 plain cookie each.  We know from Lyndsey's information that she ate 2 chocolate chip cookies, for a total of 3 cookies.  Alicaia and Lyndsey together ate 4 cookies.  So Yvonne and Robert each ate 4 cookies, and 1 of Yvonne's was plain.  So Yvonne ate 3 chocolate chip cookies, and Robert ate 4 chocolate chip cookies.  Eric ate the remaining 6 chocolate chip cookies.

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Here's a Fishy Probability...

A fishbowl contains guppies, neon tetras, and gold fish.  When Mr. Girvan reaches in to the bowl, he always touches one of the fish.  If the probability of his touching a guppy is 1/4 and the probability of toching a tetra is 2/5, what is the probability of touching a goldfish?  (2pts)  What percent of the fish are guppies? (1pt)  What percnt of the fish are neon tetras? (1pt) 

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AnswerThe probability of touching a goldfish is 7/20. (1-(1/4 +2/5) = 1 - (113/20) = 7/20.   Percent of guppies is 25%.  The percent of tetras is  40%.

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ICE CREAM...MMMM

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At Mrs. Scott's ice cream shop, five different flavors are offered daily.  Today's flavors include mint chocolate chip, peanutbutter swirl, cookies and creme, coffe chip, and bubblegum.   Homemade waffle, pretzel, and sugar cones are also available.  If Ms. Drozd orders a double-dip cone, from how many different ice-cream and cones combinations can she choose from? (Hint, order of flavors does not matter.)
 

Answer: A total of 45 double-dip cones.  Make an organized list of the fifteen ice-cream combinations (v-v, v-c, v-s, v-cc, v-r, c-c, c-s, c-cc, c-r, s-s, s-cc, s-r, cc-cc, c-cr, r-r).  For each ice-cream combination,   3 different cone options exist,  making a total of 45 double-dip cone options.

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How'd they miss that???

Click To Download

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A single train track runs through a tunnel which goes from east to west.  One afternoon two trains run along the track at the same speed and enter the tunnel, one going east and the other going west.  Niether stops or changes speed, yet they do not crash.  Why not?
Answer: They went through at different times. 

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The varitety pack

#1 Moving Parts??? (2pts)

Two common objects carry out the same function.  One of the objects has many thousands of moving parts and the other has none.  What are the objects? (hint: one is used indoors and the other used outdoors)

#3 Fun Time Questions

(Each Worth 1 pt)

a) Take two apples from 5.  How many do you have?

b)How many animals of each species did Moses take into the ark?

c)What do Kermit the Frog and Attila the Hun have in common?

d)If Mr. Girvan's peacock laid an egg in Mrs. Grillo's yard, who's the rightful owner of the egg?

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#2  Moving Money (2pts)

Ten pennies and ten nickels were arranged alternately in the pattern

PNPNPNPNPNPNPNPNPNPNPN

Find the minimum number of moves needed to move all the pennies to one end and all the nickels to the other end, that is, to change the pattern to

PPPPPPPPPPNNNNNNNNNN

A move consists of exchanging the positions of two adjacent coins.

 

 

Answer

#1 An hour glass and a sun dial. 

#2 45 moves.  Make the problem simpler.   For two pairs of coins, one move is required to change the pattern from PNPN to PPNN.  For three pairs of coins, 1 + 2 = 3 moves are required because the first two moves change PNPNPN to PPPNNN.  One more move gives the desired arrangement.   Four pairs of coins require 1 + 2 + 3 or 6, moves and five pairs require 1 + 2 + 3 + 4, or 10 moves.  Ten pairs of coins, then, require 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 moves. 

#3 a) two ... you take two apples  b) Moses didn't take animals... Noah did.  c)  the middle name  "THE"  d)   Nobody... peacocks don't lay eggs. 

Sorry I had to be Korny this week.

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Page by: Mathman (Bruce A. Titen) 2000

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