SECTION XIII.

ANALYSIS.

ART. 461. The term Analysis, in physical science, signifies the resolving of a compound body into its elements, or component parts. ANALYSIS, in arithmetic, signifies the resolving of numbers into the factors of which they are composed, and the tracing of the relations which they bear to each other. (Art. 95. Obs. 2.)

OBS. In the preceding sections the student has become acquainted with the method of analyzing particular examples and combinations of numbers, and thence deducing general principles and rules. But analysis may be applied with advantage not only to the development of mathematical truths, but also to the solution of a great variety of problems, both in arithmetic and practical life. Indeed, it is the method by which business men generally solve practical questions. A little practice will give the student great facility in its application.

462. No specific directions can be given for solving examples by analysis. None in fact are requisite. The judgment, from the conditions of the question, will suggest the process. Hence, Analysis may, with propriety, be called the COMMON SENSE RULE. OBS. In solving questions analytically, it may be remarked in general, that we reason from the given number to 1, then from 1 to the number required.

Ex. 1. If 60 yards of cloth cost $240, what will 85 yards cost? Analytic solution.-Since 60 yds. cost $240, 1 yd. will cost of $240; and of $240 is $4. Now if 1 yd. costs $4, 85 yds. will cost 85 times as much; and $4×85=$340. Ans.

Or, we may reason thus: 85 yds. are 5 of 60 yds.; therefore 85 yds. will cost 8 of $240, (the cost of 60 yds.) and 35 of $240 is $240X85 $340, the same as before. (Arts. 210, 212.)

OBS. 1. Other solutions of this example might be given; but our present ob ject is to show how this and similar questions may be solved by analysis. The

QUEST.-461. What is meant by analysis in physical science? What in arithmetic ? To what may analysis be advantageously applied? 462. Can any particular rules be prescribed for solving questions by analysis? How then will you know how to proceed 1 Obs. What is the operation of solving questions by analysis called?

former method is the simplest and most strictly analytic, though not so short as the latter. It contains two steps:

First, we separate the given price of 60 yds. ($240) into 60 equal parts, to find the value of one part, or the cost of 1 yd., which is $4.

Second, we multiply the price of 1 yd. ($4) by 85, the number of yds. whose cost is required, and the product is the answer sought.

2. This and similar questions are usually placed under the rule of Simple Proportion, or the Rule of Three.

3. The operation of solving a question by analysis, is called an analytic solution. In reciting the following examples, each one should be analyzed, and the reason for every step given in full.

2. A man bought a horse, and paid $45 down, which was of the price of it: what did he give for the horse?

Analysis. Since $45 is of the price, the question resolves itself into this: $45 is of what sum?

sum, is of $45; and of $45 is $9.

If $45 is of a certain
Now if $9 is 1 seventh,

7 sevenths are 7 times as much; and $9×7=$63.

Ans. $63.

PROOF. of $63=$9, and 5 sevenths are 5 times as much, which is $45, the sum he paid down for the horse.

Note. In solving examples of this kind, the learner is often perplexed in finding the value of +, &c. This difficulty arises from supposing that if of a certain number is 45, of it must be of 45. This mistake will be easily avoided by substituting in his mind the word parts for the given denominator. Thus, if 5 parts cost $45, 1 part will cost of $45, which is $9. But this part is a seventh. Now if 1 seventh cost $9, then 7 sevenths will cost 7 times as much.

3. If 40 cords of wood cost $120, how much will 100 cords cost?

4. Bought 35 tons of hay for $700: how much will 16 tons cost?

5. What cost 37 gallons of molasses, at $21 a hogshead?

6. What cost 1500 pounds of hay, at $14 per ton?

7. What cost 18 quarts of chestnuts, at $3 a bushel?

8. If 55 tons of hemp cost $660, what will 220 tons cost at the same rate?

9. If 165 bushels of apples cost $132, how much will 31 bushels cost?

10. If 72 bushels of peanuts cost $253.44, what will a pint cost at the same rate?

11. If 150 acres of land cost $7000, what will a square rod cost?

12. If 2 pipes of wine cost $315, what is that per gill?

13. A farmer bought a yoke of oxen, and paid $40 in work, which was of the cost: what did they cost?

14. Bought a house, and paid $630 in goods, which was of the price of it: what was the cost of the house?

15. A young man lost $256 by gambling, which was of all he was worth: how much was he worth?

16. A man having $1500, paid 2 of it for 1124 acres of land: how much did his land cost per acre?

17. If a stack of hay will keep 350 sheep 90 days, how long will it keep 525 sheep?

18. If 440 bbls. of flour will last 15 men 55 months, how long will the same quantity last 28 men?

19. If 136 men can build a block of stores in 120 days, how long will it take 15 men to build it?

20. If of a pound of tea cost 40 cents, what will of a pound cost?

21. If of a yard of broadcloth cost $2.50, how much will of a yard cost?

22. Bought of a ton of hay for $3.42: how much will of a ton cost?

23. Bought of a hogshead of molasses for $38.19: how of a hogshead cost?

much will

24. If of an acre of land cost $108, how much will of an acre cost?

25. If of a barrel of beef cost $6.48, how much will of a barrel cost?

26. Paid $4200 for of a sloop: how much can I afford to sell of the sloop for?

27. Sold 18 baskets of peaches for $34: how much would 651 baskets come to?

28. If I pay $60.50 for building 20 rods of wall, how much must I pay for 215 rods?

29. A man can hoe a field of corn in 6 days, and a boy can hoe it in 9 days: how long will it take them both together to hoe it?

Analysis. Since the man can hoe the field in 6 days, in 1 day he can hoe of it; and since the boy can hoe it in 9 days, in 1 day he can hoe of it; consequently in 1 day they can both hoe + of the field. (Art. 202.) Now if of the field requires them both 1 day, 1 of it will require them of a day, and 18 will require them 18 times as long, or 18 of a day, which is equal to 3 days. Ans.

30. If A can chop a cord of wood in 4 hours, and B in 6 hours, how long will it take them both to chop a cord?

31. A can dig a cellar in 6 days, B in 9 days, and C in 12 days: how long will it take all of them together to dig it?

32. A man bought 25 pounds of tea at 6s. a pound, and paid for it in corn at 4s. a bushel: how many bushels did it take?

Analysis.If 1 lb. of tea costs 6s., 25 lbs. will cost 25 times as much, which is 150s. Again, if 4s. will buy 1 bushel of corn, 150s. will buy as many bushels as 4s. is contained times in 150s.; and 150s.÷4=37. Ans. 37 bushels.

463. The last and similar examples are frequently arranged under the rule of Barter.

Barter signifies an exchange of articles of commerce at prices agreed upon by the parties.

OBS. Such examples are so easily solved by Analysis that a specific rule for them is unnecessary.

33. A farmer bought 110 lbs. of sugar at 18 cents a pound, and paid for it in lard at 5 cents a pound: how much lard did it take?

34. How much butter, at 12 cents a pound, must be given for 250 lbs. of tea, at 75 cents a pound?

35. How many cords of wood, at $2 per cord, must be given for 56 yds. of cloth, at $44 per yard ?

36. How many pair of boots, at $4.50 a pair, must be given for 50 tons of coal at $9 per ton?

37. A, B, and C, united in business; A put in $250; B, $270; and C, $340; they gained $258: what was each man's share of the gain?

Analysis.-The whole sum invested is $250+$270+$340= $860. Now since $860 gain $258, it is plain $1 will gain of $258, which is 30 cents. And

If $1 gains 30 cts. $250 will gain $250×.30=$75, A's share,

Or, we may reason thus: Since the sum invested is $860,

A's part of the investment is equal to 358, or 35;

S6

A must receive of the whole gain $258=$75;

464. When two or more individuals associate themselves together for the purpose of carrying on a joint business, the union is called a partnership or copartnership.

OBS. The process by which examples like the last one are solved, is often called Fellowship.

38. A and B join in a speculation; A advances $1500 and B $2500; they gain $1200: what was each one's share of the gain?

39. A, B, and C, entered into partnership; A furnished $3000, B $4000, and C $5000; they lost $1800: what was each one's share of the loss?

40. A's stock is $4200; B's $3600; and C's $5400; the whole gain is $2400: what is the gain of each?

41. A's stock is $7560; B's $8240; C's $9300; and D's $6200; the whole gain is $625: what is the share of each ?

42. A bankrupt owes one of his creditors $400; another $500; and a third $600; his property amounts to $1000: how much can he pay on a dollar; and how much will each of his creditors receive?

OBS. The solution of this example is the same in principle as that of Ex. 37,