Answer: Since the body and tail are 5 times as long as the head, there are really 6 equally long sections of the body. So divide 300 by 6 (which is 50) and one of those 50-decimeter-sections is the head.

D.2. In a 4th grade class 11 children know how to swim, 6 of these children are boys. There are 14 boys all together in the class. 7 girls in the class do not know how to swim. How many students are there in this class?

Answer: If 11 of the kids in the class know how to swim, and 6 are boys, that means that the other 5 are girls. 7 girls in the class do not know how to swim, so, there are 5+7=12 girls in the class. Therefore, the total number of students in the class is 12 girls+14 boys=26.

D.3. If a lion is worth 2 zebras or 3 bears, and a monkey is worth a bear and a zebra, what is the value of each of these animals expressed with the value of each of the other animals?

Answer:

L=2 Z=3 B=6/5 M

Z=1/2 L=3/2 B=3/5 M

B=1/3 L=2/3 Z=2/5 M

M=5/6 L=5/3 Z =5/2 B

D.4. Suppose you stack 45 blocks to make a pyramid with one block on the top, 2 blocks in the next row, 3 in the next row, and so on. How many blocks are in the bottom row? How many rows are there?

Answer: The sum of the whole numbers from 1 to 9 is 45. Therefore, there are 9 rows and there are 9 blocks in the bottom row.

D.5. Mr. Sigman used 36 fence posts to build a fence around his square back yard. He used an equal number of posts on each side of his yard. How many posts were on each side?

Answer: He had to have 1 post at each corner, so, he had 32 more posts to divide evenly between the 4 sides. That is 8, and with the 2 corners on each side, he had 10 posts on each side.

D.6. Arthur had $10 more than Burt, so he gave Burt half of his money. Then Burt had more money than Arthur, so he gave Arthur $4. Then they both had $13. How much money did each person start with?

Answer: Half of Arthur's money plus the $4 (he got from Burt) is $13. So, half of Arthur's money is $9. Therefore, Arthur started with $18, and Burt started with $10 less, which is $8.

E.1. The mass of a jar is 1340 grams when it is full of water, and 720 grams when it is half way. What is the mass of the jar when it is empty?

Answer: The mass of the water filling the jar half way is 1340g-720g=620g. So, the mass of the empty jar is 720g-620g=100g.

E.2. On a column with a rectangular base there are 3 different kinds of sides with the areas of 28, 32 and 56 square centimeters. What are the dimensions of the column?

Answer: 4 cm, 7 cm and 8 cm

E.3. If a cowboy bought 15 more cows he would have twice as much as if he sold 5 of them. How many cows does he have now.

Answer: After buying 15 cows, he would have 10 less than twice the number of cows he has now. So, if the cowboy bought 10 more cows above the 15 he supposedly bought, he would have twice as many cows as he has now. This is 25 more than what he has now, so he must have 25 cows now.

E.4. If 20 people go to a party and they all shake hands with one another, how many handshakes will there be?

Answer: Each person shackes hands with 19 people. That would be 20x19 handshackes, but this way we count every handshacke twice. Therefore, the total number of handshackes is 10x19=190.

E.5. A book has 300 pages numbered 1, 2, 3, and so on. How many times does the number 1 appear?

Answer: For the pages between 1 and 9 one number 1 is needed. For the pages from 10 to 19 eleven numbers 1 is needed. For the pages between 20 and 99 we need another 8 numbers 1's. (21, 31, ..., 91) Therefore for the first 99 pages we use 20 numbers 1's. The same is true for the pages between 200 and 300. For the pages between 100 and 199 we need 20 numbers 1's for the last two digits and another 100 numbers 1's for the first digits. The total number of number 1's needed is 20+20+20+100=160.

E.6. The sum of two numbers is 180, the difference between them is 80. What are the numbers?

Answer: If you add the sum and the difference of two numbers, you get twice the greater nmber. In this case the greater number is (180+80)/2 = 130. The smaller number is 180-130=50. (Check it: 130+50=180 and 130-50=80)

F.1. If it takes an escalator 1 minute to get you down to the platform in a subway/metro station, and it takes you 30 seconds to run down the stairs, how long does it take you to run down to the platform on the escalator that is
a) going down

b) going up?

Answer: a) Running down the stairs you go twice as fast as the escalator. So, if you run down on the escalator you go three times as fast as the escalator. Therefore, it will take you 20 minutes, a third of the time of the escalator.

b) If you are running down twice as fast as the escalator is coming up, you are getting down just as fast as the escalator would take you down you standing on it. So, it will take you 1 minute.

F.2. Imagine that you had a tape measure that you could stretch around the earth at the equator. The tape would fit so tightly that nothing could fit between the tape measure and the surface of the earth, not even a thin blade steak knife.
Now, imagine that you increased the length of the tape measure by only 1 cm, i.e., you increased the circumference by 1 cm. Would that give you enough room so that now you could slide the steak knife between the tape measure and the earth? How much do you have to increase the circumference by in order for a 150 cm tall person to be able to walk under the tape stretched out evenly above the surface of the earth?

Answer: With a 1 cm increase in the circumference of the great circle, the radius increases by 1/(2pi), approximately 0.16 cm. This increase in radius occurs for the entire circle, i.e., the entire earth. Therefore, the stake knife would fit under the tape. You have to increase the circumference by 150x2pi cm, approximately 9.5 meters, in order for a 150 cm tall person to be able to walk under the tape stretched out evenly above the surface of the earth?

F.3. Sara rolled 2 dice and made the following statements about it:
1. Both numbers are greater than 4.
2. One number is even and the other one is odd.
3. Their sum is even.
The second statement is false, but the first one is true. Could the third statement be true? What could Sara roll?

Answer: From the first statement and the denial of the second one you can conclude that on the two dice there could be 5-5 or 6-6. The sum of two even or two odd numbers is always even. Therefore, the third statement is true.