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IX. Curious Comparisons.
1. If a pig whose girth is 2 feet weighs 50 lb., what is the weight of a similarly proportioned pig whose girth is 4 feet?
2. If a disk of dough 15 inches in diameter is sufficient for 20 doughnuts, how many such doughnuts can be made from a disk 30 inches in diameter?
3. The bore of a 10-inch gun is how many times as large as the bore of a 2-inch gun?
4. The ball of a 10-inch gun is how many times as large as the ball of a 2-inch gun?
5. A square, a pentagon, a hexagon, an octagon, and a circle have equal perimeters, (a) Which has the greatest area? (b) Which has the least area?
6. The capacity of a cistern 6 feet in diameter and 6 feet deep is about 40 barrels. What is the capacity of a cistern 12 feet in diameter and 12 feet deep?
7. A 2^-inch pipe is how many times as large as a 1-inch pipe?
8. If a man 6 feet tall weighs 190 lb., how much would a similarly proportioned giant 12 feet tall weigh?
9. In a certain orchard the trees are 15 feet apart each way and there are 800 trees. How many trees in an orchard of equal size, the trees being 30 feet apart each way?
10. A ball of yarn 3 inches in diameter is sufficient for one mitten. How many mittens can be made from a ball 6 inches in diameter?
11. A grindstone was originally 30 inches in diameter. It has been worn until it is but 15 inches in diameter. What part of the stone has been worn away?
12. A square and an oblong have equal areas. Which has the greater perimeter?
X. Puzzling Problems.
1. If a person traveling as expeditiously as possible from Boston to San Francisco, should mail a letter to his friend in Boston every day at noon, how often would the letters be received in Boston?
2. If a man and a boy, the boy doing exactly one-half as much work as the man, can hoe one and one-half acres of corn in one and one-half days, how many acres can 6 men hoe in 6 days?
3. John and James sold apples together. The first day they sold 60 apples at the rate of 5 apples for 2 cents, and received 24 cents. The second day they divided the apples. John took 30 of the larger apples and sold them at the rate of 2 for 1 cent. James took the remaining 30 apples and sold them at the rate of 3 for 1 cent. They received 25 cents. Why did they receive one cent more the second day than the first?
4. A pile of four-foot wood stands upon a hill-side. The pile is 8 feet long (measured on the ground), and 4 feet high (measured vertically). Does the pile contain one cord?
5. A man had shingles enough to cover his house if he laid them 4 inches to the weather. He laid them 4|- inches to the weather. What part of the shingles provided remained? Explain.
6. If on a line of railroad connecting Chicago and San Francisco one passenger train leaves Chicago daily at 6 o'clock a. m., and makes the journey to San Francisco in exactly five days, and one train leaves San Francisco daily at 6 o'clock p. m., and makes the journey to Chicago in exactly five days, (a) a person taking the train at Chicago will meet how many passenger trains while going to San Francisco? (b) How many trains of passenger cars required to equip the road?
EXPLANATORY NOTES. Note 1. The forty-five primary facts of addition are as follows:
The nine facts in full-faced type should receive special attention. Pupils seldom fail to memorize the other thirty-six factsNote 2. There are eighty-one primary facts of subtraction; that is, two for every primary fact of addition except the 1st, 3rd, 7th, 13th, 21st, 30th, 37th, 42nd, and 45th. The facts of subtraction should be learned while learning the facts of addition. If a pupil really knows that 8 and 9 equal 17, he knows also that 17 less 8 = 9, and 17 less 9^8.
Note 3. When the sign of multiplication is followed by a fraction, it indicates that a certain part of the number t receding the sign is to be repeated as many times as there are units in the numerator of the fraction following the sign; thus, 12 X 1, means, that 1 fourth of 12 is to be repeated 3 times; 50 X .5, means, that 1 tenth of 50 is to be repeated 5 times.
Note 4. This sign, x, is sometimes so used that it means times. thus, 3 x $6, must be read, three times six dollars. 3x0, may be read, three multiplied by six or three times six. As employed in this book, the sign never means times. Instead of 3 x $6, the author prefers $6 x 3. It is believed that the restriction of this sign to one use and to one meaning, at least in the first years of arithmetical study, will promote clearness of thought and accuracy in expression.
Note 5. Without danger of ambiguity, the sign, x, is sometimes used in this book and elsewhere in place of the word by; thus, 1 pc. of 2 x 4, 12 (to be read, 1 pc. of 2 by 4, 1%) means, a piece of lumber 2 inches thick, 4 inches wide, and 12 feet long.
Note 6. Besides those problems in which either the multiplicand or the multiplier is 1, and which require no effort on the part of the pupil beyond learning to count, there are sixty-four primary facts of multiplication that must be perfectly memorized before the pupil can acquire facility in the process. They are as follows:
Although a knowledge of the "elevens" and "twelves " of the table as it is usually given is convenient and helpful, it will be observed that it is not a necessity in the process of multiplication. The facts given above include all that are essentially fundamental.
Note 7. The sign H-, which is read divided by, has two meanings in concrete problems, which correspond to the two cases in division. In one case it means, find how many times the divisor is contained in the dividend; in the other case it means, find one of a certain number of equal parts hito which the dividend is supposed to be divided. In each case there is division into equal parts. In the first case, the quotient tells the number of parts. In the second case, the quotient tells the size of one part.
818 -=- $2, means, find how many times $2 are contained in $18.
$18 -f- 2, means, find 1 half of $18. (See foot-note, p. 192.)
Note 8. There are, in a sense, 128 primary facts of division,— two for each one of the sixty-four facts of multiplication. These facts are so closely related to the facts of multiplication that they should be learned in connection with the multiplication table. If a child really perceives that five fours ( :: :: :: :: ::) are 20, he will aiso Know that 4 is contained in 20 five times, and that 1 fifth of 20 is 4.
Note 9. If from a square piece of paper, the largest possible circle be cut, a little less than \ of the paper will be cut away. Hence a circle is a little more than J (.78+) of its circumscribed square.
Observe that the diameter of a circle is equal to the side of its circumscribed square.
Note 10. If from a cube of wood, the largest possible sphere be cut, a little less than J of the wood will be cut away. Hence a sphere is a little more than J (.52 +) of its circumscribed cube.
Observe that the diameter of a sphere is equal to the edge of its circumscribed cube.
Note 11. A piece of board 1 inch wide, 1 inch thick, and 12 feet long, is 1 ft. of lumber. Hence the number of feet of lumber in any 12-foot stick, is equal to the number of square inches in its cross-section.
Note 12. If from a square right prism of wood the largest possible cylinder be cut, a little less than J of the wood will be cut away.