20 (12 is 6-10ths of 20); and 6-7ths of 7.

1-3d of 21 is 7;

20 is 2 times 7

24 is 8-9ths of 27: 1-7th of 35 is

2. 4-3ds of 18 are 24; 5; 27 is 5 times 5 and 2-5ths 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. 4-7ths, 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.

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

SECTION IX.

A. 2. 1-3d signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, or 2-3ds.

6. 5 times 1-3d is 5-3ds, or 1 and 2-3ds.

10. On the plate in the 3d row, 4 times 2-3ds are 8-3ds, which takes 2 squares and 2 spaces. Ans. 2 and 2-3ds. 24. In the 9th row take 4 spares or 9ths, and repeat them 5 times, which will make 20-9ths, and will take 2 squares and spaces. Ans. 2 and 2-9ths.

B. 4. 3 times 2 are 6, and 3 times 1 half are 3 halves, or 1 and 1 half, which added to 6 make 7 and 1 half.

18. 4 times 3 are 12, and 4 times 3-4ths are 12-4ths, or three whole ones, which added to 12 make 15.

32. 2 times 3 are 6, and 2 times 3-7ths are 6-7ths, which added to 6 make 6 and 6-7ths.

40, 5 times 7 are 35, and 5 times 6-9ths are 30-9ths, or 3 and 3-9ths, which added to 35 make 38 and 3-9ths.

C. 2. 3-4ths to each would be 3 times 3-4ths, or 9-4ths, which are 24 oranges.

3. 10-5ths or 2 bushels.

4. 7 times 3-4ths are 21-4tbs, or 5 gallons and 1-4th. 5. 8 yards and 8-4ths or 2 yards, that is, 10 yards. 6. 4 times 2 are 8, and 4 times 3-5ths are 12-5ths, or 2 and 2-5ths, which added to 8 make 10 bushels and 2-5ths. 12. It would take 1 man 3 times as long as it would 3, men. Ans. 13 days and 4-5ths.

14. 3 men would build 3 times as much as I man; and in 4 days they would build 4 times as much as in 1 day, Ans. 38 rods and 6-9ths.

15. Ans. 12 yards.

SECTION X.

1-3d of 5 is

A. 21. 1 third of 1 is 1-3d. 1-3d of 2 is 2 times as much, or 2-3ds. 1-3d of 4 is 4-3ds, or 1 and 1-3d. 5-3ds, or 1 and 2-3ds. 1-3d of 6 is 6-3ds, or 2. is 7-3ds, or 2 and 1-3d.

1-3d of 7

27. 1-5th of 1 is 1-5th. 1-5th of 2 is 2-5ths. 1-5th of 3 is is 3-5ths. 1-5th of 7 is 7-5ths, or 1 and 2-5ths.

This manner of reasoning may be applied to any number. To find 1-7th of 38: it is 38-7ths, for 1-7th of 38 is 38 times as much as 1-7th of 1, and 1-7th of 1 is 1-7, consequently 1-7th of 38 is 38-7ths, and 38-7ths is 5 and 3-7ths.

40. To find 2-3ds of a number, 1-3d must be found first, and then 2-3ds will be 2 times as much. 1-3d of 7 is 7-3ds, and 2 times 7-3ds are 14-3ds, or 4 and 2-3ds.

74. 1-9th of 50 is 50-9ths, or 5 and 5-9ths; 4-9ths is 4 times as much; 4 times 5 are 20, 4 times 5-9ths are 20-9ths, or 2 and 2-9ths, which added to 20 make 22 and 2-9ths. NOTE. The manner employed in example 40th is best for small numbers, and that in 74th for large numbers.

B. 2. Ans. 1 and 2-3ds apiece.

3. 1-5th of 3 is 3-5ths; 3-5ths of a bushel apiece.

4. 3-5ths of 7 is 4 and 1-5th; he gave away 4 and 1-5th and kept 2 and 4-5ths.

6. 1 half dollar a yard, or 50 cents.

7. 1-5th of 7 is 7-5ths, or 1 and 2-5ths; 2-5ths of a dollar is 2-5ths of 100 cents, which is 40 cents.

lar and 40 cents a bushel.

Ans. 1 dol

2-6ths of 100 is 33 and

8. 1-6th of 8 is 1 and 2-6ths. and 2-6ths. Ans. 1 dollar and 33 cents, and 2-6ths of a cent, or it is 1 dollar and 2 shillings.

9. If 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars and 2-3ds, and 2 bushels will cost 5 dollars and 1-3d. Ans. 5 dollars and 2 shillings, or 33 cents and 2-6ths of a

cent.

13. If 7 pounds cost 40 cents, 1 will cost 5 cents and 5-7ths; 10 pounds will cost 57 cents and 1-7th.

16. 1 cock would empty it in 6 hours, and 7 cocks would empty it in 1-7th of 6 hours, or 6-7ths of 1 hour, which is 6-7ths of 60 minutes; 6-7ths of 60 minutes is 51 minutes and 3-7ths.

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. 3 and 2-7ths is 1-5th of 5 times 3 and 2-7ths, which is 16 and 3-7ths.

17. 4 and 3-7ths is 1-9th of 9 times 4 and 3-7ths, which is 39 and 6-7ths.

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B. 4. 4 is 3 times 1-3d of 4, which is 4-3ds, or 1 and 1-3d.

30. If 5 is 3-4ths of some number, 1-3d of 5 is 1-4th of the same number. 1-3 of 5 is 1 and 2-3ds, 1 and 2 3ds is 1-4th of 4 times 1 and 2-3ds, which is 6 and 2-3ds; therefore 5 is 3-4ths of 6 and 2-3ds.

40. If 5 is 6-7ths, 1-6th of 5 is 1-7th; 1-6th of 5 is 5-6ths 5-6ths is 1-7th of 35-6ths, or 5 and 5-6ths; therefore 5 is 6-7ths of 5 and 5-6ths.

52. If 5-9ths of a ton cost 23 dollars, 1-9th of a ton, must be 1-5th 23, that is 4 and 3-5th dollars, and the whole would cost 9 times as much, that is, 41 and 2-5ths.

69. 1-11th of 65 is 5 and 10-11ths; 5 and 10-11ths is 1-5th of 5 times 5 and 10-11ths, which is 29 and 6-11ths. 65 is 115ths of 29 and 6-11ths.

C. 4. 37 is 9-8ths of 32 and 8-9ths, which taken from 37 leaves 4 and 1-9th. Ans. 4 dollars and 1-9th.

5. 7 feet must be 2-5ths of the whole pole.

6. If he lost 2-9ths, he must have sold it for 7-9ths of what it cost. 47 is 7-9ths of 60 and 3-7ths. Ans. 60 dollars and 42 cents, and 6-7ths of a cent.

Miscellaneous Examples.

1. The shadow of the staff is 4-5ths of the length of the staff; therefore the shadow of the pole is 4-5ths of the length of the pole. 67 is 4-5ths of 832. Ans. 834 feet.

2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and 7-9ths; 7-9ths of 60 minutes are 46 minutes and 6-9ths; 6-9ths of 60 seconds are 40 seconds. Ans. 10 hours, 46 minutes, 40 seconds,

10. Find 2-5ths of 33, and subtract it srom 17. Ans. 3 & 4-5ths.

11. It will take 3 times 10 yards.

Or 7

13. 5 is 5-3ds of 3; it will take 5-3ds as much. yards, 5 quarters wide, are equal to 35 yards 1 quarter wide, which is equal to 11 and 2-3ds yards, that is 3 quarters wide.

15. 3-5ths of 37 dollars.

16. 5-4ths as much.

SECTION XII.

The examples in this section are performed in precisely the same manner, as those in the sections to which they re