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cents are 64 cents. Or, 1 pint will cost 8 cents, and there are 8 pints in a gallon. 8 times 8 are 64.

38. They will be 2 miles apart in 1 hour, 4 miles in 2 hours, &e.

SECTION III.

A. This section contains division.

The pupil will scarcely distinguish it from multiplication. It is not important that he should at first.

Though the pupil will be able to answer these questions by the multiplication table, if he has committed it to memory thoroughly; yet it will be better to use the plate for some time.

9. As many times as 3 dollars are contained in 15 dollars, so many yards of cloth may be bought for 15 dollars. On plate I, in the third row, count fifteen and see how many times 3 it makes. It is performed very nearly like multiplication.

The pupil

B. In this article the pupil obtains the first ideas of fractions, and learns the most important of the terms which are applied to fractions.* has already been accustomed to look upon a collection of units, as forming a number, or as being itself a part of another number. He knows, therefore, that one is a part of every number, and that every number is a part of every number larger than itself. As every number may have a variety of parts, it is necessary to give names to the different parts in order to distinguish them from each other. The parts

* As soon as the terms applied to fractions are fully comprehended, the operations on them are as simple as those on whole nuinbers.

receive their names, according to the number of parts which any number is divided into. If the number is divided into two equal parts, the parts are called halves, if it is divided into three equal parts, they are called thirds, if into four parts, fourths, &c.; and having divided a number into parts, we can take as inany of the parts as we choose. If a number be divided into five equal parts, and three of the parts be taken, the fraction is called three fifths of the number. The name shows at once into how many parts the number is to be divided, and how many parts are taken.

The examples in his book are so arranged that the names will usually show the pupil how the operation is to be performed. In this section, although the pupil is taught to divide numbers into various parts, he is not taught to notice any fractions, except those where the numbers are divided into their simple units, which is the most simple kind.

It will be best to use beans, pebbles, &c. first ; and then plate I.

4. Show the pupil one of the rectangles in the second row, and explain to him that one is 1 half of 2.

7. In the second row count 3 units; it will take all the marks in the first, and I in the second rectangle. Consequently it is 1 time 2, and 1 half of another 2.

15. In the second row count 9. It will take ali the marks in the four first rectangles, and I in the fifth. Therefore 9 is 4 times 2 and one half of another 2.

18. Show the pupil a rectangle in the third row, anu ask him the question, and explain to him that I is 1 third of 3.

20. Since 1 is I third of 3, 2 must be 2 thirds of 3. 34. In the third row count 11. It will take 3

rectangles and 2 marks in the fourth. Therefore 11 is 3 times 3, and 2 thirds of another 3.

Proceed in the same manner with the other dipisicns.

This being one of the most useful combinations, and one but very little understood by most people, especially when applied to large numbers, the pupil must be made perfectly familiar with it. Ask questions like those in the book for large numbers, and also some like the following: What part of 7 is 18? the answer will be 48.

C. The first ten figures re here explained. They are used as an abridged method of writing numbers, and not with any reference to their use in calculating

This article is only a continuation of the last. All the numbers from 1 to 100 are introduced into the two articles, and are divided by all the numbers from 1 to 10 ; except that some of the largest are not divided by some of the smallest.

2. The pupil answers first, how many times 2 is contained in 12, then how many times 3.

45. 63 are how many times 5 ? In the fifth row count 63. It will take 12 rectangles and 3 marks in the 13th. It will be necessary to count once across the plate and begin again, and take 2 rectangles and a part of the third. 63 is 12 times 5 and 3 fifths of another 5.

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 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 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, threcs 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 7 of 9.

C. 1. 4 bushels of apples, at 3 shillings a bushe 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 article that money will buy.

6. If 2 apples cost 4 cents, I 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 30 cents, and you can buy so many times 3 eggs. 30 is 6 times 5, and 6 times 3 are 18.

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.

18 eggs.

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

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