SECTION V. In this section the principle of fractions is applied to larger numbers, but such as are divisible into the parts proposed to be taken. The pupil, who is familiar with what precedes, will easily understand the examples in this sec tion. They require nothing but division and multiplication. A. Let the pupil explain each example in the following manner. What is 1 sixth of 18? Ans.: 3. Why? Because 6 times 3 are 18; therefore if you divide 18 into 6 equal parts, one of the parts will be 3. The pupil will be very likely to say 3 is the 6th part of 18, because 3 times 6 are 18. Be careful to make him say it the other way, viz., 6 times 3 are 18. B. 11. of 20 is 4; } are 7 times 4, or 28; and 28 is 4 times 6, and of 6. C. 3. 1 half of 10 is 5, of ten are 4; 5 and 4 are 9. He gave away 9, and had 1 left. 5. 2 yards will cost 1 half of what 4 cost; or 6 dollars. 7. 2 is of 8; therefore 2 oranges will cost of what 3 cost. of 18 cents are 12 cents. 8. of 25 are 20. The ten apples cost 20 cents, which was 2 cents apiece. 13. 3 is of 4. of 12 dollars are 9 dollars. Or 4 yards at 12 dollars is 3 dollars a yard, and 9 dollars for 3 yards 15. Since 1 is of 3, 7 is 1 of 3. of 15 cents are 35 cents. Or, 3 oranges at 15 cents, is 5 cents apiece: 7 times 5 cents are 35 cents. Note. In questions of this kind it is generally the simplest way to find wha. 1 article will cost, then it may easily be told how much any number will cost. SECTION VI. A. 4. 2 halves of any number make the whole number Therefore 2 is 1 half of 2 times 2; or 4. It is of 4 times 2, or 8. Let the pupil answer these questions in the following manner: 4 is of 3 times 4; 3 times 4 are 12. 5 is of 7 times 5; 7 times 5 are 35. B. 2. 4 is 2 times 2. 16. 2 thirds of any number is twice as myn as of the same number. If 4 is of some number, then 1 half of 4 or 2 is of that number; 2 is of 6; therefore 4 is of 6. 23. It is evident that of a pound will cost only of what will cost. If cost 6 cents, will cost 2 cents, and the whole pound 14 cents. 26. It will probably be perceived by this time, that of a number being given, it is necessary to find 4, and then the number is easily found; 4 being, 2 is 4, and 2 is of 14. 45. 24 being, of 24 or 3 will be ; 3 is of 27. C. 6. 20 being, 5 is, and 5 is times 6, and of 6. D. 4. 18 is 3 times 6, and 6 is Ans. 24 dollars. of 35; and 35 is 5 of 4 times 6, or 24. 6. 54 is of 48; 12 yards at 48 dollars is 4 dollars a yard. He gained 6 dollars. 7. 10 feet is 4 of 15 feet. 8. If water. are under water, there must be out of the 4 is of 12. 10. and are bear cherries and peaches; consequently, the 10 which bear plums must be the other 4. 10 is of 35. 10 bear peaches, and 15, bear cherries. Miscellaneous Examples. 6. The greyhound gains upon the fox 4 rods in a minute. It will take him 20 minutes to gain 80 rods. 8. of 24. Or you may say, 1 sheep would cost 3 dollars, and 3 sheep 9 dollars. 9. 30 horses will eat 10 times as much as 3 horses. 12. 5 dollars for 1 week, 20 dollars for a month, and 25 dollars for five weeks. 14. It would take them 5 times as long to eat 40 bush. els, as it would to eat 8 bushels. 15. 4 horses would eat 4 bushels in 3 days, and it would take them 9 times as long to eat 36 bushels. Ans. 27 days. 16. If 2 men spend 12 dollars in 1 week, 1 man will pend 6 dollars in 1 week, and 30 dollars in 5 weeks, and 3 men would spend 3 times as much, or 90 dollars. 17. The shadow of the staff is of the length of the staff, therefore the shadow of the pole must be the length, of the pole. 18 feet is of 27 feet. 20. It would take 2 men 3 times as long to do it as it would 6 men. 23. 8 men would do a piece of work 1 half as large in 2 days, and it would take 2 men 4 times as long to do it, or 8 days. 28. He must sell it for 56 dollars in order to gain 16 dollars. 56 dollars is 7 dollars per barrel. 29. It cost him 35 dollars, and he must sell it for 45 to gain 10 dollars; 45 dollars is 9 dollars a firkin. 30. Ans. 56 cents. See Section VI. 33. If it would last 3 men 10 months, it would last 1 man 30 months, and 5 men 6 months. 34. There are 8 times 5 in 40; and since the other would build as many times 9 as the first does 5, he would build 8 times 9 or 72 rods. SECTION VII. A. 13. of 20 is 4, are 16; 16 being, 2 is; 2 is of 14, and 16 is of 14. 16. of 28 are 12; 12 is 2 times 6, and 6 is is of 48), and 48 is 6 times seven and of 7. is B. 1. of 15 are 12; 12 is 6 times 2; 2 is of 48 (12 of 20 (12 20 is 2 times 7 and of 7. of 20); of 21 is 7; 2. of 18 are 24; 24 is of 27; times 5 and of 5. C. This article contains the multiplication table, in which the numbers from 10 to 20 are multiplied by the ten first numbers. SECTION VIII. A. 1. In one there are 2 halves; in 2 there are twice as many halves, that is 4 halves, or . 2. In one there are 2 halves, in 3 there are three times two halves, that is, 6 halves, or §. 3. In one there are 3 thirds, in 2 there are twice 3 'hirds, that is, 6 thirds, or . Each line 15. Draw two lines on the board thus call one; divide the upper line into two equal parts, each part is one half of one, or one half; divide the lower line in the same way, and then call for the answer. The form of the question may be varied by asking how many half apples there are in two apples. 37. In 5 and 2 thirds, how many thirds? Draw five parallel equal lines, divide them into thirds, and draw another parallel with the others and two thirds as long, let the scholar count the thirds. In all these cases of illustration on the board, it is presumed that the scholar will soon take the hint, and devise illustrations for himself at his seat. SECTION IX. A. 2. signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, or . 6. 7 times is, or 21. B. 4. 4 times 2 are 8, and 4 times 1 half are 4 halves, or 2, which added to 8 make 10. 18. 4 times 3 are 12, and 4 times are 42, or three whole ones, which added to 12 make 15. 40. 10 barrels at 3 dollars and a barrel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times is 50, or 8 and 3 of a dollar. Ans. 382 dollars. C. 2. to each would be 3 times, or, which are 21 oranges. 3. 10 or 2 bushels. 4. 7 times are 4, or 5 gallons. 5. 8 yards and or two yards, that is 10 yards. 6. 4 times 2 are 8, and 4 times added to 8 make 10 bushels. are, or 2, which 12. It would take 1 man 3 times as long as it would 3 Ans. 13 days. men. 14. men would build 3 times as much as 1 man; and in 4 days they would build 4 times as much as in 1 day. Ans. 388 rods. A. 21. of 1 is of 4 is or of 7 is, or 2}. 27. of 1 is of 2 is. of 3 is . of 7 is 7, or 13. This manner of reasoning may be applied to any number. To find of 38: it is, for of 38 is 38 times as and of 1 is, consequently of 38 is, of 1, much as then 74. of a number, must be found first, and of 50 is 5, or 5; is 4 times as much; 4 times 5 are 20, 4 times are 30. or 24, which added to 20 make 228. Note. 20 The manner employed in example 40th is best for small numbers, and that in the 74th for large numbers. B. 2. Ans. 14 apiece. 3. of 3 is ; of a bushel apiece. 4. of 7 is 4; he gave away 4, and kept 24. 7. of 7 is for which is 40 cents. 1; of a dollar is of 100 cents, Ans. I dollar and 40 cents a bushel. 8. of 8 is 12. of 100 is 338. 9. If 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars and, and 2 bushels will cost 5 dollars. 13. If 7 pounds cost 40 cents, 1 will cost 54 cents; 10 pounds will cost 574 cents. 16. 1 pipe would empty it in 6 hours, and 7 pipes would empty it in of 6 hours, or of 1 hour, which is of 60 minutes; of 60 minutes is 514 minutes. SECTION XI. A. 2. 2 halves of a number make the number, consequently 1 and 1 half is the half of 2 times 1 and 1 half, which is 3. 15. 44 is of 5 times 4 and 4, which is 224. B. 4. 5 is 3 times of 5, which is §, or 13. 30. If 8 is of some number, of 8 is of the same |