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Arithmetic is the science which explains the properties and relations of numbers, and the method of computing by them. No branch of education is of more importance in a great coinmercial nation than arithmetic; without a knowledge of it, the statesman, the merchant, and the mechanic, as well as the philosopher, are left to grope their way in doubt and uncertainty. Too much attention therefore cannot be bestowed upon it in the education of youth, in order to fix upon their minds its principles as a science, and to render them confident, accurate, and expert in their application of it to the various departments of business.
Every teacher ought to fix upon the minds of his pupils its principles, not only as a science, but as a means of developing their intellect, of invigorating their judgment, and of accustoming them from their very entrance on the study of geometry and algebra, to those habits of patient attention and accurate reasoning, on the early acquisition of which their future progress so essentially depends.
It is only of late years that those engaged in the education of youth (with a few exceptions) have given any attention to the exercising of their pupils in mental calculation, or the art of calculating by mind and memory alone, without the aid of slate or paper. It is now deservedly held in high estimation, and taught more or less by almost every teacher of arithmetic. Mental calculation is not only a most useful acquisition of itself, but is of singular utility in assisting pupils to work with rapidity on the slate: it strengthens the memory, improves the powers of reflection, and invigorates the mind.
What is simple proportion? Simple proportion, or the rule of three, is the method of finding a fourth proportional to three given numbers. Why is it called the rule of three ? Because in all the operations which it solves, three terms are given to find a fourth. Why is it called the golden rule? From its excellent and extensive application. What is the ratio of one number to another? The ratio of one number to another is the quotient which arises from dividing the one by the other, and is thus expressed, 8 : 2 or , both expressing the ratio of 8 to 2; but 8 contains 2 four times, therefore 4 is the ratio of 8 to 2, and for the same reason the ratio of 2 : 8 is –, because 2 is the fourth part of 8. What are the terms of a ratio ? The terms of any ratio are the numbers compared, of which the first is called the antecedent, and the second the consequent; thus, in the ratio 8 : 2, the terms are 8 and 2, of which 8 is the antecedent and 2 the consequent. When are two ratios equal ? When their antecedents contain their consequents, or are contained by them, the same number of to the product of the means; thus in the proportion 8: 2 :: 16 : 4, the product of 8 * 4 is equal to the product of 2x16. When the product of any two numbers is equal to the product of any other two numbers, are these four numbers always proportional ? They are proportional in what order soever you take them. Thus, 8X6=2x 24, then 8: 2 :: 24 :6, 6:2 :: 24 :8, and 2:8::6: 24, and 8 : 24: :2:6 and 24 : 6 ::8:2 and 24:8:: 6:2. How do you state the terms in proportion? I write the three given numbers in one line, as follows: I first write that number which is of the same kind with the number required, as the third proportional; I then consider whether the answer must be greater or less than that number; if it must be greater, I write the less of the remaining number as the first proportional, and the other as the second; but if the answer must be less, I write the greater as the first, and the other as the second proportional, How do
thus the ratio of 8 : 2 is equal to the ratio of 24 : 6, because 8 contains 2 the same number of times that 24 contains 6. How is proportion expressed ? By placing a double colon between the ratios. Thus 8 : 2 and 24 : 6 we write 8: 2 :: 24: 6, that is, 8 are to 2 as 24 to 6. Whence are the symbols in proportion derived? They are the abbreviations of the signs of division and equality. Thus, 8: 2 :: 24 : 6, is merely the abbreviation of 8;2=24 :-6. What are the extremes, and what the means? The first and last terms are called the extremes, and the two middle terms the means; thus, 8: 2 :: 24 : 6, the extremes are 8 and 6, and the means and 24. Can you mention any relation which exists between the extremes and means ? Yes. The product of the extremes is always equal
reduce the terins? If the first and second terms consist of different or several denominations, I reduce them to one and the same denomination, and then consider them as abstract numbers. The third term I generally reduce to the lowest name it contains. When the terms are stated and reduced, how do you proceed in order to find a fourth proportional? I mul. tiply the second and third terms together, and divide the product by the first, the quotient is the answer. In what name are the product of the second and third terms, the quotient, and the remainder when there is one? The same name with the third term. Do the quotient and remainder ever require reduction? Yes. What do you mean by a third proportional ? When the second and third terms of an analogy are the same, the fourth term is called the third proportional to the first and second: thus, 8 : 12 :: 12:18; here 18 is the third proportional to 8 and 12. What is 12 here called with respect to 8 and 18? It is said to be a mean proportional between 8 and 18, as it stands in the double relation of consequent and antecedent. When three numbers are proportional, what relation has the first to the third ? The first is always to the third, as the square of the first to the square of the second; thus in the above proportion, where 8 : 12 :: 12 : 18, 8 : 18 :: 64 : 144, where it is evident 64 is the square of 8 and 144 is the square of 12. What is this ratio called? A duplicate ratio; thus, if 4: 8::8:16;
4:16 :: duplicate ratio of 4 : 8,
::42 : 82 ::16 : 64
If 18 yards of broad cloth cost 24 pounds, what will 27 yards cost?
yds. yds. £
£36 Paid for 56 firkins of butter at the rate of 181. 12s. 6 d. for 8 firkins : what did they come to?
fir. fir. £ d
7 1 7
£130 7 94 Bought 36 yards of cloth for 251. 6s. 8d.: how must I sell 9 yards of it to gain at the rate of 41. 108. 4d. upon the whole ?
yds. yds. € s d
Find the value of 80 yards at the rate of 15l. 17s.6d. for 16 yards.
yds. yds. £ d
£ 79 7 6 If a ship's crew have water to serve them 15 days at the rate of 30 gallons per day, what must their daily allowance be that it may serve them 25 days ?
days. days. gall.
25) 450 (18 gal.
200 A piece of calico, 25 inches wide, is valued at 2s. 1d. per yard : what should be the price per yard of another piece of the same quality 30 inches wide ?
inch. inch. 8. d.
2:6 How much money at 31 per cent., will yield as much interest as 4901. at 4 per cent.? £ €
490 2 2