Note. In questions of this kind it is generally the simplest way to find what I 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 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.

then 1 half of 4 or 2 is 6: therefore 4 is 2 of 6.

If 4 is of some number, of that number; 2 is of

20. If 6 is of a number, of 6 or 2 is of the same number; 2 is of 8; therefore 6 is 3 of 8. 23. It is evident that of a pound will cost only of what will cost. If cost 6 cents, 2 cents, and the whole pound 14 cents.

will cost

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, 2 is 4, and 2 is 4 of 14.

45. 24 being 3, of 24 or 3 will be ; 3 is } 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 of 4 times 6, or 24. Ans. 24 dollars.

6. 54 is of 48; 12 yards at 48 dollars is 4 dollars a yard. He gained 6 dollars.

7. 10 feet is of 15 feet.

3. If are under water, there must be out of the water. 4 is of 12.

9. If are under water there must be out 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 of 35. 10 bear peaches and 15 bear cherries.

11. 2, and, and †, and †, are ; therefore 12 must be the other of the whole.

number is 54.

Miscellaneous Examples.

The whole

6. The grey-hound 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.

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 eat S 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 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,

2 is of 14, and 16 is 16. of 28 are 12;

are 16; 16 being †, 2 is 4 ; of 14.

12 is 2 times 6, and 6 is † of 48, (12 is of 48) and 48 is 6 times 7 and of 7.

B. 1. of 15 are 12; 12 is 6 times 2; 2 is 'T of 20 (12 is of 20); ¦ of 21 is 7; 20 is 2 times of 7.

7 and

2.

of 18 are 24; 24 is of 27; of 35 is 5; 27 is 5 times 5 and 2 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.

Explanation of Plate II.

PLATE I, which has been used in the preceding sections, presents each unit as a simple object and undivided. Plate II, presents the units as divisible objects, the different fractions of which form parts, and sums of parts of unity.

This plate is divided into ten rows of equal squares, and each row into ten squares.

The first row is composed of ten empty squares, which are to be represented to the pupil as entire units. The second row presents ten squares, each divided into two equal parts by a vertical line, each of these parts of course represents one half. In the third row, each square is divided into three equal parts, by two vertical lines, each part representing one third, &c. to the tenth row, which is divided into ten equal parts, each part representing one tenth of unity.

N. B. In plates II and III, the spaces and not the marks are to be counted.

Be careful to make the pupil understand, 1st, that each square on the plate is to be considered as

an entire unit, or whole one. 2d, explain the divisions into two, three, four, &c. parts. 3d, teach him to name the different parts. Make him observe that the name shows into how many parts one is divided, and how many parts are taken, in the same manner as it does when applied to larger numbers. 4 for example, shows that one thing is to be divided into 7 equal parts, and 4 of those parts are to be taken. 4th, make the pupil compare the different parts together, and observe which is the largest. Ask him such questions as the following: Which are the smallest halves or thirds? Ans. Thirds. Why? Because, the more parts a thing is divided into, the smaller the parts must be.

A. 15. On plate II., count two squares in the second row, and then ascertain the number of spaces or halves in them. There are 4 halves.

21. In the 2d row take 3 squares and 1 space in the 4th square; then count the spaces. Ans. 7 halves.

37. In the 3d row take 5 squares, and 2 spaces in the 6th; then count the spaces or thirds. Ans. 17 thirds.

54. In the 5th row take 6 squares, and 4 spaces in the 7th square; then count the spaces or fifths. Ans. 34 fifths.

B. 2. This operation is the reverse of the last. In the 2d row count 4 spaces or halves, and see how many squares or whole ones it takes. It will take 2.

It

38. In the 9th row count 48 spaces or 9ths, and see how many squares or whole ones it takes. will take 5 squares and 3 spaces in the 6th. Ans. 5 whole ones and §.