rectangles and 2 marks in the fourth. Therefore 11 is 3 times 3, and 2 thirds of another 3. Proceed in the same manner with the other divisions. This being one of the most useful combinations, and one but very little understood by most people, especially when applied to large numbers, the pupil must be made perfectly familiar with it. Ask questions like those in the book for large numbers, and also some like the following: What part of 7 is 18? The answer will be v. C. The first ten figures are here explained. They are used as an abridged method of writing numbers, and not with any reference to their use in calculating. This article is only a continuation of the last. All the numbers from 1 to 100 are introduced into the two articles, and are divided by all the numbers from 1 to 10; except that some of the largest are not divided by some of the smallest. 2. The pupil answers first, how many times 2 is contained in 12, then how many times 3. 45. 63 are how many times 5? In the fifth row count 63. It will take 12 rectangles and 3 marks in the 13th. It will be necessary to count once across the plate, and begin again, and take 2 rectangles and a part of the third. 63 is 12 times 5 and 3 fifths of another 5. D. These examples, which are similar to those in article A of this section, are solved in the same manner. 5. It would take as many hours as 3 miles are contained in 10 miles. 3 hours and f of an hour. 20. They cost as many cents as there are 3 apples in 30 apples; that is, 10 cents. There 21. 12 dollars a month: and 12 dollars a month is 3 dollars a week; that is, 18 shillings a week, which is 3 shillings a day. 26. The whole loss was 35 dollars, which was 7 dollars apiece. SECTION IV. part explair A. This article contains multiplication simply.. It is repeating a number a certain number of times and a part of another time. 14. 6 times 5 are 30, and of 5 are 3, which added to 30 make 33. On the plate in the fifth row, take 6 rectangles and 3 marks in the seventh, and ascertain their sum. of WTI jeir us B. In this article the pupil is taught to change a certain number of twos into threes, threes into fives, &c. This article combines all the preceding oper. ations. 24. 4 cords of wood will cost 28 dollars, and of a cord will cost 2 dollars, which makes 30 dollars. 30 dollars will buy 3 hundred weight of sugar and of another hundred weight. 29. 7 times 8 are 56, and of 8 are 5, which added to 56 make 61; 61 are 6 times 9, and 7 of 9. C. 1. 4 bushels of apples, at 3 shillings a bushel, come to 12 shillings; and 12 shillings are 2 dollars. 2. The 2 lemons come to 8 cents, and 8 cents will buy 4 apples, at 2 cents apiece. This is usually called Barter. The general principle is to find what the article will come to, whose price and quantity are given, and then to find how much of the other article that money will buy. miles 21 7 hour. re 3 8p 6. If 2 apples cost 4 cents, 1 will cost 2 cents, and 4 will cost 8 cents. Or 4 apples will cost 2 times as much as 2 apples. 22. Find how many times 2 pears are contained in 20 pears, which is 10 times. 10 times 3 cents are 30 cents. Or, first find what 20 pears would come to, at 3 cents apiece; and since it is 2 for 3 cents, instead of 1 for 3 cents, the price will be half as much. 23. See how many times you can have 5 cents in 30 cents, and you can buy so many times 3 eggs. 30 is 6 times 5, and 6 times 3 are 18. 18 eggs. 24. 10 dollars a week, and 40 dollars a month. 25. 5 dollars are 30 shillings, which is 10 shillings a day. 26. 5 dollars apiece. 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. 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. To find this answer on the plate ; on the 6th row, the pupil will find 3 times 6 make 18; this will direct him to the third row, where he will find 6 times 3 are 18. Consequently, he will see 18 divided into 6 equal parts. It will be well to let the pupil prove a large number of the examples on the plate. 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; & is 2 times as much as }, therefore of 9 is 6. 19. 1 barrel will cost 4 part of 12 dollars ; 3 barrels will cost of 12 dollars. 7 barrels will cost ? of 12 dollars. 37. What is f cf 32 ? } of 32 is 4, * are 5 times 4, or 20. B. 11. 4 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 10 are 4; 5 and 4 are 9. He gave away nine, and had 1 left. 4. 1 yard will cost of what 3 yards cost. f 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 ef of what 3 cost. of 18 cents are 12 cents. 8. of 25 are 20. The 10 apples cost 20 cents, which was 2 cents apiece. 11. 4 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. şof 15 cents are 35 cents. Or, 3 oranges at 15 cents is 5 cents apiece: 7 times 5 are 35 cents. Note. In questions of this kind, it is generally the simplest way to find what 1 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 three 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 5 of 4 times 2, or 8. Let the pupil answer these questions in the following manner :-4 is 4 of 3 times 4; 3 times 4 are 12. 5 is 7 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 I of the same number. If 4 is of some number, then 1 half of 4, or 2, is of that number : 2 is f of 6; therefore 4 is of 6. 20. If 6 is of a number, 1 of 6, or 2, is of the same number : 2 is 4 of 8; therefore 6 is of 8. 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, o of a number being given, it is necessary to find 7, and then the number is easily found; 4 being 4, 2 is 7, and 2 is 4 of 14. 45. 24 being , 1 of 24 or 3 will be $; 3 is of 27. |