SECTION IV.

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 g of 5 are 3, which added to 80 make 33.

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 operations.

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 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 two 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 ar ticle that money will buy.

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 pairs 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 6 shillings is equal to one dollar in 14 of the 29 states of the Union.

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 multipli

cation.

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.

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

14. 1 third of 9 is 3;

fore of 9 is 6.

are 18.

is 2 times as much as }, there

19. 1 barrel will cost part of 12 dollars; 3 barrels will cost of 12 dollars. 7 barrels will cost of 12 dollars. 37. What is § of 32 ? † of 32 is 4, § are 5 times 4, or 20. B. 11. of 20 is 4; 7 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 9, and had 1 left.

4. 1 yard will cost of what 3 yards cost. 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 of what 8 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. of 15 cents are 85 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 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.

of the

16. 2 thirds of any number is twice as much as 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 of a number, of 6 or 2 is of the same number; 2 is 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 of a number being given, it is necessary to find, and then the number is easily found; 4 being 4, 2 is 4, and 2 is of 14. 45. 24 being, of 24 or 3 will be ; 8 is of 27.

C. 6. 20 being 4, 5 is 4, and 5 is 4 of 35; and 35 is 5 times 6, and 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 of 48; 12 yards at 48 dollars is 4 dollars a yard. He gained 6 dollars.

7 10 feet is of 15 feet.

8. If are under water, there must be ter. 4 is of 12.

ter.

9. If

10.

are under water, there must be 6 is of 10.

out of the wa

out of the wa

and are bear cherries and peaches; consequently, the 10 which bear plums must be the other 4; 10 is 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 minate. 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 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 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.

83. 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, are 16; 16 being, 2 is; 2 is of 14, and 16 is & of 14.

16. of 28 are 12; 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 is of 20); of 21 is 7; 20 is 2 times 7 and 2. of 18 are 24; 24 is of 27; 4 of 35 is times 5 and of 5.

of 20 (12 of 7.

5; 27 is 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.

A. 1. IN one there are 2 halves; in 2 there are twice as many_halves, that is four halves or 4.

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 .

Each line

15. Draw two lines on the board thus 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 ilustrations for himself at his seat.