staff, therefore the shadow of the pole must } of the poic. 18 feet is of 27 feet.

e. It would take 2 men 3 times as long would & men.

23. 8 men would do a piece of work 1 half days and it would take 2 men 4 times as lo $ dars

28. He must sell it for 56 dollars in orde dollars. 56 dollars is 7 dollars per barrel.

29. It cost him $5 dollars, and he must sell gain 10 dollars; 45 dollars is 9 dollars a firki: $3. Ans. 56 cents. See Section VI.

$3. If it would last 3 men 10 months, it man 30 months, and 5 men 6 months.

$4. There are 8 times 5 in 40; and since the build as many times 9 as the first does 5, he 8 times 9 or 72 rods.

SECTION VII.

A. 18. of 20 is 4, are 16; 16 being 4, 2 of 14, and 16 is of 14.

16. of 28 are 12; 12 is 2 times 6, and 6 iis of 48), and 48 is 6 times seven and $ of 7

B. 1. of 15 are 12; 12 is 6 times 2; 2 is is of 20); of 21 is 7; 20 is 2 times 7 an 2 of 18 are 24; 24 is § of 27; ↓ of 3., times 5 and 8 of 5.

C. This article contains the multiplicat which the numbers from 10 to 20 are multi first numbers.

SECTION VIII.

A. 1. In one there are 2 halves; in 2

many halves, that is 4 halves,

2. In one there are 2 hals

two halves, that is

3. In one

thirds, that i

of 4 times 23 which is 103;

ollars, of a ton must be of whole would cost 9 times as

of 5 times 78, which is 363.

ich taken from 37 leaves 4.

ave sold it for of what it cost. ollars and 42 cents.

rous Examples.

aff is of the length of the staff; the pole is of the length of the ns. 838 feet.

the cistern in 1 hour. It will be of 60 minutes are 45 minutes Are 40 seconds. Ans. 10 hours, 46

subtract it from 17. Ans. 3.
s 10 yards.

l take as much. Or, 7 yards, 5
1 to 35 yards 1 quarter wide, which
at is 3 quarters wide.

This section are performed in precisely those in the sections to which they realty consists in comprehending, that in figures signify the same thing as words. Make the pupil express them the difficulty will vanish. Let particular d to the explanation of fractions given in

many? expressed in words, is Ans. .

SECTION X.

of 2 is 2 times of 5 is §, or 13.

as much, or
of 6 is §, or 2

A. 21. of 1 is of 4 is, or 13. of 7 is 1, or 2. 27. of 1 is of 2 is . 1 of 3 is g. 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 4 of 1 is, consequently of 38 is 4,

much as of 1, and & is 53. 40. To find

then

74.

of a number, must be found first, and will be 2 times as much.

of 50 is 5, or 55; is 4 times as much; 4 times 5 are 20, 4 times are 20. or 2, which added to 20 make 228.

Note. The manner employed in example 40th is best for small numbers, and that in the 74th for large numbers.

8. of 8 is 12. of 100 is 332.

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 5 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 51 minutes.

SECTION XI.

A. 2. 2 halves of a number make the number, conse quently 1 and 1 half is the half of 2 times 1 and 1 half. which is 3.

15. 4 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

number. of 8 is 23, 23 is of 4 times 23 which is 103; therefore 8 is of 103.

52. If of a ton cost 23 dollars, of a ton must be of 23, that is 4g dollars, and the whole would cost 9 times as much, that is, 41%.

69. of 65 is 7; 73 is of 5 times 73, which is 361. $5 is % of 361.

C. 4. 37 is of 328, which taken from 37 leaves 4. Ans. 4 dollars.

5. 7 feet must be of the whole pole.

6. If he lost, he must have sold it for 3 of what it cost. 47 is of 60%. Ans. 60 dollars and 42 cents.

Miscellaneous Examples.

1. The shadow of the staff is of the length of the staff; therefore the shadow of the pole is of the length of the pole. 67 is of 833. Ans. 834 feet.

2. 9 gallons remain in the cistern in 1 hour. It will be filled in 10 hours and 7; 7 of 60 minutes are 45 minutes and; of 60 seconds are 40 seconds. Ans. 10 hours, 46

minutes, 40 seconds.

10. Find of 33, and subtract it from 17. Ans. 3‡. 11. It will take 3 times 10 yards.

13. 5 is of 3; it will take as much. Or, 7 yards, 5 quarters wide, are equal to 35 yards 1 quarter wide, which is equal to 11 yards that is 3 quarters wide. 15. of 37 dollars.

SECTION XII.

The examples in this section are performed in precisely the same manner as those in the sections to which they refer. All the difficulty consists in comprehending, that fractions expressed in figures signify the same thing as when expressed in words. Make the pupil express them in words, and all the difficulty will vanish. Let particular attention be paid to the explanation of fractions given in the section.

VIII. A. 6. In 7 how many? expressed in words, is n 7 how many sixths? Ans. .

14. Reduce 84% to an improper fraction; that is, in 8 and 3 tenths, how many tenths.~ Ans. §§.

B. 8. 28 are how many times 1? That is, in 23 sev enths how many whole ones? Ans. 34. IX. B. 3. How much is 5 times 64? That is, how much is 5 times 6 and 4 sevenths? Ans. 32.

V. & X. 15. What is of 27?

eighths of 27? Ans. 167.

That is, what is 5

VI. & XI. A. 8. 7 is of what number? That is, 7 and 6 sevenths is 1 eighth of what number? Ans. 624.

SECTION XIII.

THE operations in this section are the reducing of frac tions to a common denominator, and the addition and subtraction of fractions. The examples will generally show what is to be done, and how it is to be done.

4. It will readily be seen that and † are 4.

25. In the fourth square of the second row, it will be seen that 1 half is and in the second square of the fourth row, is, both together make § and make 7.

27. is the same as §.

When these questions are performed in the mind, the pupil will explain them as follows. He will probably do it without assistance. Twenty twentieths make one whole one. of 20 is 5, and of 20 is 8, and of 20 is 2; therefore is, is, and is 2. All the examples should be explained in the same manner 45. One whole one is 8, one eighth of times as much, which is 51. 1 half is, and 61. is, is,

18.

67. is

10

is. is 3

is, which added together make §. is, which added together make

is, which added together make 11; from take, and there remains, or 1.

82. It will be easily perceived that these examples do not differ from those in the first part of the section, except in the language used. They must be reduced to a common denominator, and then they may be added and subtracted as easily as whole numbers. is 18, and f is, and both together make 8 or 1.