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 1-3d of an hour.

20. They cost as many cents as there are 3 apples in 30 apples; that is, 10 cents.

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.

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 3-5ths 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.

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 2-7ths of a cord will cost 2 dollars, which makes 30 dollars. 30 dollars will buy 3 hundred weight of sugar and 6-8ths of another hundred weight.

29. 7 times 8 are 56, and 5-8ths of 8 are 5, which added to 56 make 61; 61 are 6 times 9, and 7-9ths 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 lemous 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..

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 30cents, 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; 2-3ds is 2 times as much as 1-3d, therefore 2-3ds of 9 is 6.

19. 1 barrel will cost 1-4th part of 12 dollars; 3 barrels

will cost 3-4ths of 12 dollars. 12 dollars.

7 barrels will cost 7-4ths of

37. What is 5-8ths of 32? 1-8th of 32 is 4, 5-8ths are 5 times 4, or 20.

B. 11. 1 5th of 20 is 4; 7-5ths are 7 times 4 or 28; and 28 is 4 times 6, and 4-6ths of 6.

C. 3. 1 half of 10 is 5, 2-5ths of 10 are 4; 5 and 4 are 9. He gave away 9 and had 1 left.

1-3d of 6 5. 2 yards will cost 1 half of what 4 cost; or 6 dollars. 6. 3 apples will cost 1-3d of what 9 cost; or 6 cents. 7. 2 is 2-3ds of 3; therefore 2 oranges will cost 2-3ds of what 3 cost. 2-3ds of 18 cents are 12 cents.

4. 1 yard will cost 1-3d of what 3 yards cost. dollars is 2 dollars.

8. 4-5ths of 25 are 20.

which was 2 cents apiece.

The 10 apples cost 20 cents,

11. 2-7ths of 42 are 12, and 6 times 12 are 72. 72 dollars. 13. 3 is 3-4ths of 4. 3-4ths of 12 dollars are 9 dollars. Or 4 yards at 12 dollars is 3 dollars a yard, and 9 dollars for 3 yards.

Ans. 15 cents.

14. Solved like the 13th. 15. Since 1 is 1-3d of 3, 7 is 7-3ds of 3. 7-3ds of 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 I article will cost, then it may easily be told how much any number will cost.

19. 4 men would do it in 1 balf 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 1-4th 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 1-4th of 4 times 9, or 8.

Let the pupil answer these questions in the following manner: 4 is 1-3d of 3 times 4; 3 times 4 are 12. 5 is 1-7th of 7 times 5; 7 times 5 are 35.

B. 2. 4 is 2 times 2. 6 is 2 times 3.

16. 2 thirds of any number is twice as much as 1-3d of the same number. If 4 is 2-3ds of some number, then 1 half of 4 or 2 is 1-3d of that number; 2 is 1-3d of 6: therefore 4 is 2-3ds of 6.

20. If6 is 3-4ths of a number, 1-3d of 6 or 2 is 1-4th of the same number; 2 is 1-4th of 8; therefore 6 is 3-4ths of 8. 23. It is evident that 1-7th of a pound will cost only 1-3d of what 3-7ths will cost. If 3-7ths cost 6 cents, 1-7th will cost 2 cents, and the whole pound 14 cents.

26. It will probably be perceived by this time, that 2-7ths of a number being given, it is necessary to find 1-7th, and then the number is easily found; 4 being 2-7ths, 2 is 1-7th, and 2 is 1-7th of 14.

45. 24 being 8-9ths, 1-8th of 24 or 3 will be 1-9th; 3 is 1-9th of 27.

C. 6, 20 being 4-7ths, 5 is 1-7th, and 5 is 1-7th of 35; and 35 is 5 times 6, and 5-6ths of 6.

D. 4. 18 is 3 times 6, and 6 is 1-4th of 4 times 6, or 24. Ans. 24 dollars.

6. 54 is 9-8th of 48; 12 yards at 49 dollars is 4 dollars a yard. He gained 6 dollars.

7. 10 feet is 2-3ds of 15 feet.

8. If 2-3ds are under water, there must be 1-3d out of the water. 4 is 1-3d of 12.

9. If 2-5ths are under water there must be 3-5ths out of the water. 6 is 3-5ths of 10.

10. 3-7ths and 2-7ths are 5-7ths. 5-7ths bear cherries and peaches; consequently, the 10 which bear plums must be the other 2-7ths; 10 is 2-7ths of 35. 10 bear peaches and 15 bear cherries.

11. 2-9ths, and 3-9ths, and 1-9th, and 1-9th are 7-9ths; therefore 12 must be the other 2-9ths of the whole. The whole number is 54.

Miscellaneous Examples.

6. The grey-bound gains upon the fox 4 rods in a min It will take him 20 minutes to gain 80 rods.

ute.

8. 3-8ths of 24. Or you may say, I 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 a piece, 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 2 thirds of the length of the staff, therefore the shadow of the pole must be 2-3ds of the length of the pole. 18 feet is 2-3ds 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. 1-5th of 20 is 4, 4-5ths are 16; 16 being 8-7ths, 2 is 1-7th; 2 is 1-7th of 14, and 16 is 8-7ths of 14.

16. 3-7ths of 28 are 12; 12 is 2 times 6, and 6 is 1-8th of 48 (12 is 2-8ths of 48) and 48 is 6 times 7 and 6-7ths of 7.

B. 1. 4-5ths of 15 are 12; 12 is 6 times 2; 2 is 1-10th of