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them fill in 1 hour? How much would both together? How long would it take them both to fill it ?
41. A cistern has 2 pipes. The first can fill it in 3 hours, the second in 4 hours. How much of it can both fill in an hour? In what time can both fill it?
42. A cistern has 3 pipes. The first can fill it in 4 hours, the second in 6, and the third in 8. In what time would all three fill it?
43. A cistern has 3 pipes ;, 2 to fill, and one to empty it. The first can fill it in 3 hours, the second in 4, the third can empty it in 6 hours ? In what time would it be filled if all three were set running ?
44. A and B can do a piece of work in 10 days, and with the assistance of C they can do it in 8 days. How long would it take C alone to do it ? 18 5. Ans. 40 days. Why?
45. A man and his wife consume together a bushel of meal in 2 weeks; but when the wife was alone, it would last her 5 weeks. How much of it did the man consume in 1 week ?
46. If a man could build a piece of wall in 5 days, and another in 6 days, how long would it take both to build it ?
47. If a staff 4 feet long cast a shadow 6 feet long, how long a pole could cast a shadow 12 feet long at the same time? How long a pole could cast a shadow 15 feet long ?
48. A man bought a horse and wagon for 160 dollars. The horse cost 1} times the price of the wagon. What was the price of the wagon ? 3e Why?
49. A boy being asked how many cents he had, replied, if you multiply the number by 5, take 8 from the product, divide the remainder by 9, and add 2 to the quotient, the amount will be 10. How many cents had he ?
50. A mason, 12 journeymen, and 4 laborers, receive together $72 wages for a certain time. The mason receives a dollar daily, each journeyman į dollar, and each laborer & dollar. How many days must they have worked for the
? 51. In an orchard of fruit trees, 1 bore apples, į plums, pears, 7 peaches, and 3 cherries. How many trees are there in the whole, and how many of each sort?
52. Sold 16 pounds of butter, at 25 cents a pound, and received in pay sugar, at 8 cents a pound. How many pounds of sugar
would I receive ?
53. Sold 24 chickens at 50 cents a pair, for 20 yards of delaine. What was the price per yard ?
54. If 5 men in 40 days can make 300 pairs of boots, how many can 1 man make in a day? How many could 15 men make in 4 days ?
55. A, B, and C, divide $100 among them, so that B should have $8 more than A, and C $9 more than B. How much does each receive ?
[In reviewing, omit the leading questions in the three following examples.]
56. If 49 pounds of bread be sufficient for 14 men for 7 days, how much will suffice for 1 man 1 day? How much for 20 men 4 days ?
57. If 4 men can reap 48 acres in 12 days, bow many acres can 1 man reap in 1 day? How many can 8 reap in 16 days?
58. If $75 be the wages of 5 men for 3 weeks, what will be the wages of 1 man for 1 week? What will be the wages of 8 men for 4 weeks?
59. How many yards of baize, 4 quarters wide, will line 8 yards of camblet, 3 quarters wide ?
60. Four mechanics completed a piece of work, for which they received as wages $36, and it was agreed to divide this in proportion to the time they had been respectively employed. Now, the first had worked 3 days, the second 4, the third 5, and the fourth 6 days. What were their respective shares of the wages ?
61. Two men purchased some goods, and sold them again at a profit of $24. The first advanced $100, and the second $200. What share of profits should each receive ?
62. Two merchants traded together for a year. The first advanced a capital of $3000 for 12 months, and the other a capital of $6000 for 4 months. The profit was $3000. How should it be divided ?
63. How many pounds of coffee, at 9 cents a pound, with twice the weight of sugar, at 6 cents a pound, may be
purchased for 4 dollars and 20 cents ?
64. If 30 men can do a piece of work in 8 days, how many men could do the same piece of work in 12 days?
65. When 20 men can do a piece of work in 12 days, in what time may 30 men do the same work ?
All the above questions should be proved by the class.
ADDRESSED TO TEACHERS.
WRITTEN Arithmetic is the art of calculating by means of written characters. As already shown, when treating of Oral Arithmetic, there are only two operations that can be performed with numbers, namely, Increase and Decrease. To perform these with skill and rapidity, comprises the whole of Arithmetic. Both may be accomplished by numeration. But such a method is entirely too slow for practice in the extensive operations of civilized society. Shortened processes have therefore been invented, chiefly by omitting superfluous steps, which have effected wonderful savings both of time and labor. Indeed, so numerous and important are the abbreviations in constant use, that Arithmetic may not inaptly be defined the art of increasing and decreasing numbers by SHORTENED pro
In the following exercises it is of vast importance that pupils should avoid the use of all unnecessary words, either in speech or thought. For instance, when 4 and 2 are to be added, there is no occasion to think of, far less to naine, those numbers. In reading, we never think of nor use the name nor sound of the letters which compose the syllables or words. The sound of the syllable or word itself occurs at the first glance. It will be the same with the numeral characters, if proper care be taken from the outset ; for it is much more difficult to reform a bad habit than to form a good one. The thought of 6, and the
writing of it, will be almost, if not quite, simultaneous with the sight of the figures 4 and 2; and, after a little practice, the mind will readily grasp, in the same way, three, four, five, and more figures, at a time.
The same remark applies to all the other processes. There is no occasion, for instance, to say, or even to think of, four times five make twenty. When properly trained, the pupil will think of twenty the instant he sees the 4 and the 5. The tedious talk about carrying, also, should be altogether dispensed with. Thus, in the following example,
420 in place of saying, or thinking, five times four are twenty, nothing and carry two ; five times eight are forty, and two are forty-two ; the words twenty, forty-two (two words out of eighteen), are all that are necessary. The rest are mere hindrances. And not only does such verbiage lengthen the process, but experience has shown that it actually increases the chance of error, by allowing time for the mind to wander from the subject. By strict attention to this matter, multiplication and subtraction will be performed as fast as the figures can be written, division will be much shortened, and addition need not occupy more than a fourth of the time usually required.
The mode of performing the elementary processes now universally used in our schools, is the same in principle with the obsolete plan of teaching children to read by spelling every word. It differs, however, in one important respect. In reading, the spelling process generally lasts but a few months. In arithmetic, unfortunately, it clings to most persons through life. Let it be wholly abjured from the first, then, and the profit and ease both to teacher and pupil will be found to be immense. The spelling process merely « darkeneth by words.”' Nor is this all. The advantage will be reflected on all other studies. For, by the improved method, it is obvious the mind is kept continually on the alert, and the slovenly, dreamy, mental habits, which are now the bane of our schools, are entirely avoided.
The whole subject of Written Arithmetic may be suitably arranged in five chapters.
I. A DESCRIPTION of the CHARACTERS, or Notation and Numeration.*
II. The shortened processes of Increase and Decrease, applied to Integers and Decimal Fractions. Under the head of Increase come Addition and Multiplication (or addition of equal numbers), including Involution (or multiplication of equal numbers). Under the head of Decrease come Subtraction and Division (subtraction of equal numbers), including Evolution (division of equal numbers).
III. The same processes applied to Common (or Vulgar) and Denominate Fractions.
IV. Practical application of the methods of increase and decrease, promiscuously arranged.
V. The Comparison of Numbers, or Ratio and Proportion.
To these are added a SUPPLEMENT, containing a few subjects more properly belonging to Algebra, though commonly inserted in books of arithmetic, such as Progression by Differences and Progression by Ratios, usually, though improperly, called Arithmetical and Geometrical Progression, Compound Interest, Permutation, &c.
The rapid method of performing the elementary operations necessarily requires considerable practice in Oral Arithmetic. Such pupils, therefore, as have not already studied Part I. of this treatise, should now recite from it once or twice a day, simultaneously with the study of this part of the work. If this is faithfully attended to, adding in such numbers as 25, 27, 28, &c., 35, 36, 37, &c., will soon be found as easy as the addition of single digits.
* The art of writing the characters is Notation ; that of reading them Numeration.