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the solution of a great variety of examples, both in arithmetic and practical life.
In the preceding sections the pupil has become acquainted with the method of analyzing particular principles, and thence deducing general truths or rules.
We will now illustrate the application of analysis to the solution of examples or problems.
Obs. Business men have a method of solving practical questions, which is frequently shorter and more expeditious than that of arithmeticians fresh from the schools. If asked, by what rule they perform them, their reply is, “ they do them in their head,” or by the no rule method.”
Their method consists in analysis. It might with propriety be called the common sense rule. A little practice will give the learner great facility as well as satisfaction in its application.
Ex. 1. If 8 barrels of flour cost 40, how much will 5 barrels cost?
Analysis.--1 is 1 eighth of 8: therefore 1 barrel will cost 1 eighth as much as 8 barrels ; and 1 eighth of $40 is $5. Now it is obvious that 5 barrels will cost 5 times as much as 1 barrel; and 5 times $5 are $25, the answer required.
Or, we may reason thus : 5 barrels are of 8 barrels ; 5 barrels will therefore cost as much as 8 barrels. Now 1 eighth of $40 is $5, and 5 eighths is 5 times $5, which is $25.
2. If 7 lbs. of tea cost 42 shillings, what will 10 lbs. cost ?
3. If 9 sheep are worth $27, how much are 15 sheep worth?
4. If 10 barrels of flour cost $60, what will 12 barrels cost ?
5. Suppose 30 gallons of molasses cost $15, how many dollars will 7 gallons cost ?
QUEST.-Obs. How do business men solve practical questions? What do they call it? In what does it consist? What might it with propriety be called ?
6. If a man earns 54 shillings in 6 days, how much can he earn in 15 days ?
7. If 12 men can build 48 rods of wall in a day, how many rods can 20 men build in the same time?
8. A gentleman divided 90 shillings equally among 15 beggars : how many shillings did 7 of them receive?
9. Suppose 75 pounds of butter lasts a family of boarders 25 days, how many pounds will supply them for 12 days?
10. If 7 yards of cloth cost $30, how much will 9 yards cost ?
11. If 10 barrels of beef cost $72, how much will 8 barrels cost ?
12. If 7 acres of land cost $50, what will 12 acres cost ?
13. A farmer buying an ox cart, paid $15 down, which was ic of the price of it: what was the price of the cart; and how much does he owe for it?
Analysis.-The question to be solved is simply this: 15 is of what number? If 15 is so, tb is $ of 15, which is 5. Now if 5 is 1 tenth, 10 tenths is 10 times 5, which is 50.
S $50 is the price of the cart, and
$50—$15=35, the sum unpaid. Note. In solving examples of this kind, the learner is often perplexed in finding the value of to, &c. This difficulty arises from supposing that if To of a certain number is 15, bo of it must be to of 15. This mistake will be easily avoided by substituting in his mind the word parts for the given denominator.
Thus, if 3 parts cost $15, 1 part will cost $ of $15, which is $5. But this part is a tenth. Now if 1 tenth cost $5, then 10 tenths will cost 10 times as much.
14. A man bought a yoke of oxen, and paid $56 cash down, which was f of the price of them: what did they cost?
15. A merchant bought a quantity of wood and paid $45 in goods, which was ģ of the whole cost : how much did he pay for the wood ?
16, A whale ship having been out 24 months, the Cap
tain found that his crew had consumed of his provisions : how many month's provision had he when he embarked, and how much longer would his provisions last ?
17. How many times 7 in of 35 ?
Analysis of 35 is 7, and 4 is 4 times 7, which is 28. Now 28 is 4 times 7. Ans.
18. How many times 6 in Ğ of 45 ?
times 8? 28. j of 180 are how many times 10? 29. ja of 84 are how many times 9 ?
30. How many yards of cloth, at $7 per yard, can be bought for 7 of $54?
31. How many barrels of flour, at $5 per barrel, can be bought for of $60 ?
32. A man had $64 in his pocket, and paid of it for 10 barrels of flour: how much was that
barrel ? 33. 40 is of how many times 6 ?
Analysis.--Since 40 is , is of 40, or 8; and is 9 times 8, or 72. Now 6 is contained in 72, 12 times.
Ans. 12 times. 34. 56 is , of how many times 7 ? 35. 81 is so of how many times 30 ? 36 72 is ii of how many times 9 ? 37. 96 is, of how many times 12 ? 38. 64 is of how many times 20 ? 39. 54 is g of how many times 24 ? 40. 108 is ii of how many times 12?
41. Frank sold 10 peaches, which was of all he had ; he then divided the remainder equally among 5 companions : how many did they receive a piece ?
42. Lincoln spent 60 cents for a book, which was of his money ; the remainder he laid out for oranges, at 4 cents apiece: how many oranges did he buy? 43. A man paid away $35, which was of all he
he then laid out the rest in cloth at $2 per yard : how many yards did he obtain ?
44. A farmer bought a quantity of goods, and paid $20 down, which was of the bill ; how many cords of wood, at $3 per cord, will it take to
the balance ? 45. A man bought a horse and paid $60 in cash, which was of the price : how many barrels of flour, at $6 per barrel, will it take to pay the balance ?
46. of 27 is $ of what number?
Analysis.- of 27 is 9. And if 9 is of a certain number, 4 of that number is 3; and 1 is 4 times 3, which is 12, the number required.
of 30 is of what number? 48. f of 40 is of what number? 49. of 35 is so of what number? 50. ğ of 54 is 15 of what number?
EXERCISES FOR THE SLATE.
296. It will be seen from the preceding examples that no particular rules can be prescribed for solving questions by analysis. None in fact are requisite. The process will be easily suggested by the judgment and the conditions of the question.
Obs. The operation of solving a question by analysis, is called an analytic solution. In reciting the following examples, the pupil
QUEST.-296. Can any particular rules be given for solving questions by analysis ? Is it necessary to give them? How then will you know how to proceed? Obs. What is the operation of solving questions by analysis, called ?
should be required to analyze each question, and give the reason for each step, as in the preceding mental exercises.
Ex. 1. If 40 barrels of beef cost $320, how much will 52 barrels cost ?
Analytic Solution.—Since 40 lbs. cost $320, 1 bl. will cost of $320. And do of $320 is $320-40=$8. Now if i bl. costs $8, 52 bls. will cost 52 times as much; and $8 X 52=416, which is the answer required.
Or thus : 52 bls. are 3 of 40 bls. ; therefore 52 bls. will cost of $320 ; (the cost of 40 bls.;) and as of $320 is $320 x =$416, the same result as before. (Arts. 132, 133.)
Note.-1. Other solutions might be given ; but our present object is to show how this and similar examples may be solved by analysis. The former method is the simplest and the most strictly analytic, though not so short as the latter. It contains two steps:
First, we separate the given price of 40 lbs., ($320,) into 40 equal parts, to find the value of one part, or the cost of 1 bl., which is $8.
Second, we multiply the price of 1 bl. ($8,) by 52, the number of barrels, whose cost is required, and the product is the answer sought.
2. In solving questions analytically, it may be remarked in general, that we reason from the given number to 1, then from 1 to the required number.
3. This and similar questions are usually placed under Simple Proportion, or the “Rule of Three ;" but business men almost invariably solve them by analysis.
2. If 30 cows cost. $360.90, how much will 47 cows cost, at the same rate ?
3. If 25 barrels of apples cost $15, how much will 37 barrels cost?
4. If 15 hogsheads of molasses cost $450, how much will 21 hogsheads cost ?
5. If 31 yards of cloth cost $127, how much will 89 yards cost?
6. If 55 tons of hay cost $660, what will 17 tons come to?
7. An agent paid $159 for 530 pounds of wool : how much was that per 100 ?
8. A man bought 30 cords of wood for $76.80 ; how much must he pay for 65 cords?