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together are worth 16 dollars, but the money is worth 7 times as much as the purse; how much money was there in the purse ? and what is the value of the purse ?
168. A man, being asked the price of his horse, answered, that his horse and saddle together were worth 100 dollars, but the horse was worth 9 times as much as the saddle. What was each worth ?
169. A man, having a horse, a cow, and a sheep, was asked what was the value of each. He answered, that the cow was worth twice as much as the sheep, and the horse 3 times as much as the sheep, and that all together were worth 60 dollars. What was the value of each ?
170. A man bought an apple, an orange, and a melon, for 21 cents; for the orange
he as much as for the apple, and for the melon he gave twice as much as for the orange. How much did he give for each?
171. If 80 dollars' worth of provision will serve 20 men 24 days, how many days will 100 dollars' worth of provision serve 30 men?
172. There is a pole 4 and 1 under water, and 10 feet out; how long is the pole?
173. In an orchard of fruit trees, f of them bear apples, 1 of them bear plums, šof them pears, 7 of them peaches, and 3 of them cherries, how many trees are there in the whole, and how many of each sort?
174. A farmer, being asked how many sheep he had, answered, that he had them in 4 pastures; io the first he had of his flock; in the second t; i.. the third }; and in the fourth 15; how many sheep had he?
175. A man, driving his geese to market, was met by another, who said, Good morrow, master, with your hundred geese ; pays he, I have nos
hundred; but if I had half as many more as I now have, and two geese and a half, I should have a hundred; how
had he? 176. What number is that, to which if its half be added, the sum will be 60 ?
177. What number is that, to which if its third be added, the sum will be 48 ?
178. What number is that, to which if its fifth be added, the sum will be 54 ?
179: What number is that, to which if its half and its third be added, the sum will be 55 ?
180. A man, being asked his age, answered, that, if its half and its third were added to it, the sumi would be 77; what was
his 181. What number is thạt, which being increased by its half, its fourth, and eighteen more, will be doubled ?
182. A boy, being asked his age, answered, that, if and of his age and 20 more, were added to his age, the sum would be 3 times his age. What was his age?
18. A man, being asked how many sheep he had, answered, that, if he had as many more, as many more, and 27 sheep, he should have 100. How
many had he
The Key contains an explanation of the plates, and the manner of using them. The manner of solving the examples in each section is particularly explained. All the most difficult of the practical examples are solved in such a manner, as to show the principles by which they are performed. Care has been taken to select examples for solution, that will explain those which are not solved. Many remarks, with regard to the manner of illustrating the principles to the pupils, are inserted in their proper places.
Instructers, who may never have attended to fractions, need not be afraid to undertake to teach this book. The author flatters himself that the principles are so illustrated, and the processes are made so simple, that any one, who shall undertake to teach it, will find himself familiar with fractions before he is aware of it, although he knew nothing of them before; and that every one will-acquire a fasility in solving questions which he never before possessed.
The reasoning used in performing these small examples is precisely the same as that used upon large ones. And when any one finds a difficulty in solving a questinn, he will remove it back sooner, and much more effectually, by taking a very small example of the same kind, and observing how he does it, than by recurring to a rule.
The practical examples at the commencement of each section and article are generally such as to show the pupil what the combination is, and how he is to perform it. This will learn the pupil gradually to reason upon abstract numbers. In each combination, there are a few abstract examples without practical ones. to exercise the learner in the combinations, after he knows what these combinations are. It would be an excellent exercise for the pupil to put these into a practical form when he is reciting. For instance, when the question is, How many are 5 and 3? let him make a question in this way: If an orange cost 5 cents, and an apple 3 cents, what would they both come to? This may be done in all cases.
The examples are often so arranged, that several depend on each other, so that the preceding explains the following one. Sometimes, also, in the same example, there are several questions asked, so as to lead the pupil gradually from the simple to the more difficult. It would be well for the pupil to acquire the habit of doing this for himself, when difficult questions occur.
The plates should be used for young pupils, but they are not necessary for the older ones, The plates for fractions, however, will frequently be useful to these. The first plate need not be used much, after the pupil is familiar with the multiplication table.
The book may be used in classes, where it is convenient. The pupil may answer the questions with the book before him or not, as the instructer thinks proper. A very useful mode of recitation is, for the instructer to read the example to the whole class, and then, allowing sufficient time for them to perform the question, call upon some one to answer it. In this manner every pupil will be obliged to perform the example, because they do not know who is to answer it. In this way it will be best for them to answer without the book.
It will often be well to let the elder pupils hea. the younger. This will be a useful exercise for them, and an assistance to the instructer.
Explanation of Plate I. This plate, viewed horizontally, presents ten rows of rectangles, and in each row ten rectangles.
In the first row, each rectangle contains one mark, each mark representing unity or one.
In the seco ond row, each rectangle contains two marks; in the third, three marks, &c.
The purpose of this plate is, first, to represent unity either as a unit, or as making a part of a sum of units : secondly, to represent a collection of units, either as forming a unit itself, or as making a part of another collection of units; and thus to compare unity and each collection of units with another collection, in order to ascertain their ratios.
All the examples as far as the eighth section can be solved by this plate. The manner of using it is explained in the Key for each section in its proper place.
The pupil, if very young, should first be taught to count the units, and to name the different assemblages of units in the following manner:
The instructer, showing him the first row, which contains ten urits insulated, requests the pupil to put his finger on the first, and say, one; then on the second, and sav, and one are two; and on the third, and say, and one are three ; and so on to ten : then, com