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 section. They require nothing but division and multiplication. manner. A. Let the pupil explain each example in the following 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. 14. 1 third of 9 is 3 ; s is 2 times as much as }, therefore ge of 9 is 6. 19. 1 barrel will cost part of 12 dollars; 3 barrels will cost of 12 dollars. 7 barrels will cost 1 of 12 dollars. 37. What is of 327 of 32 is 4, are 5 times 4, or 20. B. 11. of 20 is 4; } are 7 times 4, or 28; and 28 is 4 times 6, and 4 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. 4. 1 yard will cost of what 3 yards cost. 1 of 6 dollars is 2 dollars. 5. 2 yards will cost 1 half of what 4 cost; or 6 dollars. 6. 3 apples will cost of what 9 cost; or 6 cents. 7. 2 is of 3; therefore 2 oranges will cost s of what 3 cost. f of 18 cents are 12 cents. 8. of 25 are 20. The 10 apples cost 20 cents, which was 2 cents apiece. 11. of 42 are 12, and 6 times 12 are 72. 72 dollars. 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. 14. Solved like the 13th. Ans. 15 cents. 15. Since 1 is } of 3, 7 is of 3. 3 of 15 cents are 35 Or, 3 oranges at 15 cents, is 5 cents apiece: 7 times 5 cents are 35 cents. cents. Note. In questions of this kind it is generally the simplest way to find what l article will cost, then it may easily be told how much any number will cost. 19. 4 men would do it in 1 half the time that 2 would do it. Or, you may say, if 2 men would do it in 6 days, 1 man would do it in 12 days, and 4 men in of that time, or 3 days. 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 4 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. 4. 6 is 2 times 3. 16. 2 thirds of any number is twice as much 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. 20. If 6 is f of a number, of 6 or 2 is of the same number; 2 is of 8; therefore 6 is f of 8. 23. It is evident that ț of a pound will cost only fof what will cost. If { cost 6 cents, 4 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 , and then the number is easily found ; 4 being 4,2 is t, and 2 is 4 of 14. 45. 24 being &, of 24 or 3 will be $; 3 is 1 of 27. C. 6. 20 being 4, 5 is 4, and 5 is of 35; and 35 is 5 times 6, and 5 of 6. D. 4. 18 is 3 times 6, and 6 is 1 of 4 times 6, or 24. Ans. 24 dollars. 6. 54 is g of 48; 12 yards at 48 dollars is 4 dollars a yard. He gained 6 dollars. 7. 10 feet is ģ of 15 feet. 8. If are under water, there must be out of the water. 4 is } of 12. 9. If z are under water, there must be fout of the water. 6 is of 10. 10. and & are bear cherries and peaches ; consequently, the 10 which bear plums must be the other : 10 is 4 of 35. 10 bear peaches, and 15 bear cherries. 11. , and , and }, and , are ]; therefore 12 must be the other of the whole. The whole number is 54. 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. f 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. 11. 10 dollars apiece, and 2 dollars a yard. 12. 5 dollars for 1 week, 20 dollars for a month, and 25 dollars for 5 weeks. 14. It would take them 5 times as long to eat 40 bushels, as it would to cat 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 spend 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 s 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 rain 10 dollars ; 45 dollars is 9 dollars a firkin. 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. 31. 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, 4 are 16; 16 being 4, 2 is † ; 2 is + of 14, and 16 is of 14. 16. of 28 are 12; 12 is 2 times 6, and 6 is 3 of 48 (12 is 3 of 48), and 48 is 6 times seven and f of 7. B. 1. $ of 15 are 12; 12 is 6 times 2 : 2 is to of 20 (12 is of 20); } of 21 is 7; 20 is 2 times 7 and of 7. of 18 are 24; 24 is f of 27 ; 4 of 35 is 5; 27 is 5 times 5 and f of 5. C. This article contains the multiplication table, in which the numbers from 10 to 20 are multipled 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 thirds, that is, 6 thirds, or f. 15. Draw two lines on the board thus Each line 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. f 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 23. 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 *, or three whole ones, which added to 12 make 15. 32, 2 times 3 are 6, and 2 times are 4, which added to 6 make 6. 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 f of a dollar. Ans. 382 dollars. C. 2. to each would be 3 times , or 4, which are 27 oranges. 3. 1 or 2 bushels. 4. 7 times are 21, or 54 gallons. 5. 8 yards and or 2 yards, that is 10 yards. 6. 4 times 2 are 8, and 4 times are us, or 2, which added to 8 make 10 bushels. 12. It would take 1 man 3 times as long as it would 3 Ans. 134 days. 14. 3 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. 38% rods. 15. Åns. 12 yards. men. SECTION X. A. 21. $ of 1 is t. of 2 is 2 times as much, or of 4 is , or 13. 1 of 5 is , or 14. 1 of 6 is f, or 2. of 7 is }, or 21 27. 1 of 1 is } } of 2 is . 1 of 3 is t. , of 7 is }, or 13. This manner of reasoning may be applied to any number. To find of 38 : it is 3., for — of 38 is 38 times as inuch as 4 of 1, and of 1 is , consequently of 38 is , and 38 is 54. |