ART. 261. ANALYSIS is the separation of things into their elements or parts. In Arithmetic, it is the method of solution by reasoning according to the nature of the question, without reference to special rules.

EXAMPLES FOR MENTAL SOLUTION.

1. If 5 oranges cost 15 cts., what cost 4 oranges?

ANALYSIS.-1 orange is of 5 oranges, and will cost as much; of 15 cents is 3 cents, the cost of 1 orange; 4 oranges will cost 4 times as much as 1 orange; 4 times 3 cts. 12 cts. Ans.

=

Here, we first find the price of one, as it is easier to compute from the value of one, than from that of any other number.

2. If 5 sheep cost $20, what will 9 sheep cost? 3. If 5 bl. flour cost $40, what will 3 bl. cost?

4. If 3 lemons cost 12 cts., how many will 28 cts. buy?

as much;

If 4 cents

ANALYSIS.-1 lemon is of 3 lemons, and will cost but of 12 cents is 4 cents, the cost of 1 lemon. buy 1 lemon, 28 cents will buy as many lemons as 4 cents are contained times in 28 cents; that is, 7. Ans. 7 lemons.

5. If 5 barrels of flour cost $15, how many barrels of flour can be purchased for $21?

6. If 6 lb. of sugar cost 30 cts., how many pounds of sugar can be bought for 50 cts.?

7. If 7 yards of cloth cost $28, how many yards of cloth will $40 buy?

8. James had 28 cts., and spent of them for oranges at 2 cts. each: how many oranges did he buy?

12. If 2 cents

ANALYSIS.—of 28 is 4, and 2 are 3 times 4 buy 1 orange, 12 cents will buy as many oranges as 2 cents are contained times in 12 cents, that is 6. Ans. 6 oranges.

9. A man having $40, spent 3 of it for cloth at $2 a

yard: how many yards did he purchase?

10. of 48 are how many times 10?

REVIEW.-261. What is Analysis? What is it in Arithmetic?

261

11. of 45 are how many times 8?

12. I sold a watch for $18, which was of its cost: how much did it cost, and what did I lose?

ANAL. Since $18 is 3 fifths of the cost,

of 18 will be 1 fifth;

and of 18 is 6. If 6 is 1 fifth, 5 fifths will be 5X6=30. Ilence, the cost is $30, and $30-$18-$12, the loss.

Here, we reason from several parts to one part; then from one part to the required number.

REM.-To avoid difficulty in analyzing examples of this kind, observe that the numerator of the fraction shows the number of parts taken. Art. 124. Thus, in the preceding example, 18 is three parts (fifths); and 1 part (fifth) is 1 third of 3 parts (fifths).

13. 8 is of what number?

14. 16 is 4 of what number?

12 is g of what?

15 is of what?

15. A farmer sold a wagon for $45, which was of its cost; he paid for it with sheep at $3 a head: how many

sheep were required?

16. 25 is g of how many times 4?

17. 21 is of how many times 4?

Ans. 21.

ANALYSIS.—1 of 40 is 10, and is 3 times 10-30. If 30 is, of 30=6, is ;

and 7, or the required number, are 7×6=42.

of what number?

of what number?

21. If bushel barley cost 20 cts., what cost bu.? ANALYSIS.-If cost 20 cts., 1 bu. will cost 3 times as much, or 60 cts.; and bu. will cost of this, which is 15 cts. Ans.

22. If yd. muslin cost 24 cts., what cost yd.?

ANALYSIS.— yd. will cost as much as; of 24 cts. is 12 cts., the cost of of a yd.; hence, 1 yd. will cost 3 times 12 cts., or 36 cts. If 1 yd. costs 36 cts., will cost 9 cts., and three times 9 cts., which are 27 cts. Ans.

Here, we reason from several parts to one part; then, from one part to the whole; then, from the whole to a part; lastly, from one part to several parts.

23. If yd. silk cost 20 cts., what cost 24. If bu. wheat cost 14 cts., what cost 25. If yd. linen cost 21 cts, what cost

ART. 262. The solution of a question by analysis, is an analytic solution.. Every operation in analysis depends directly on elementary or self-evident principles.

To solve questions analytically, determine the process to be pursued, by an examination of the conditions of the question, and the relations which the several quantities bear to each other. In general, reason from a given number to unity (1), or one part, and then to the required number.

ART. 263. PROMISCUOUS EXAMPLES.

1. If 15 bl. of flour cost $40, what cost 24 bl.? SOL.-If 15 bl. cost $40, 1 bl. will cost of $40-$23, and 24 bl. will cost 24 times as much as 1 bl. ; 24×$23=$64. Ans.

15

15

Or thus: Since 1 bl. is of 15 bl., 24 bl. are 24 of 15 bl., and will cost 24 as much. $40×24-$64. Ans.

2. If 24 lb. of beef cost $2.16, what will be the cost of 23 lb. ? Ans. $2.07 3. If 13 yd. of cloth cost $32.50, what will be the cost

Ans. $35.

of 14 yd.? 4. If $8 will purchase 4 yd. of cloth, how many yards will $24 buy?

Ans. 12 yd.

5. If 159 yd. of muslin cost $20.67, how many yards can be bought for $34.71?

6. If 34 yd. cost $147, what cost 9 yd.?

7. If 5 men perform a piece of work in

many days will it take 3 men?

Ans. 267 yd.

Ans $3831:

12 days, how

SOLUTION. It will require 1 man 5 times as long as 5 men; that is, 5X12 days: 60 da. Again, it will require 3 men as long as 1 man; of 60 days 20 days. Ans.

=

REVIEW.-261. What is Analysis? 262. On what does every operation in it directly depend? How ascertain the process to be pursued?

8. If 17 men can do a job of work in 25 days, in what time will 10 men do it? Ans. 424 da. 9. If 6 men can build a wall in 10 days, how many men can build it in 15 days? Ans. 4 men.

10. If 5 men consume a barrel of flour in 12 days, how long would it last 4 men?

Ans. 15 da.

11. A man performs a piece of work in 7 days, working 10 hours a day if he labors 12 hr. a days will be required?

12. If 18 men can reap 72 acres of how many days will 8 men require? 13. If 5 hogs are worth 9 sheep, how pay for 54 sheep?

day, how many Ans. 55 da. wheat in 7 days, Ans. 15 da. many hogs will Ans. 30.

14. If 3 gal. wine are worth 7 gal. cider, how many gal. of cider are worth 42 gal. wine? Ans. 98 gal.

15. If a 3 cent loaf weigh 8 oz. when flour is $4 a bl., what should it weigh when flour is $5 a bl.? Ans. 63 oz. 16. If 3 stacks of hay will feed 12 horses 5 mon., how long will they feed 20 horses? Ans. 3 mon.

17. If

cost of

of a yard of gingham cost 40 cents, find the of a yard. Ans. 60 cts. 18. If of a yard of cloth cost $2, what will be the

cost of

19. If tun cost?

3

of a yard?

Ans. $4.50

17

of a tun of hay cost $4.25, what will ¦¦ of a Ans. $3.85

20. If C. of wood cost $2.52, what will 3 of a C. cost?

21. If 3 lb. of tea cost $1, what cost & lb.?

Ans. $2.86

Ans. $14. Examples in Art. 263 are often placed under Sim. Proportion. Solve, now, by Analysis, all the questions in Art. 203.

ART. 264. 22. If 7 men eat 56 lb. of bread in 16 days, how many lb. will 21 men eat in 6 days?

SOLUTION. If 7 men eat 56 lb., 1 man will eat as much, which is 8lb.; then, if 1 man in 16 days eat 81b., in 1 day he will eat of 8 lb. 8-1 lb. And, 21 men will eat 21=101⁄2 lb. in 1 day, and in 10×6=63 lb. Ans.

23. If 2 men earn $16 in 4 days, how much will 7 men earn in 3 days?

Ans. $42. 24. If 2 men build 12 rods of a wall in 9 days, how many rods can 7 men build in 6 days?

25. If 15 oxen plow 11 A. in 5 da., will plow 33 A. in 9 da.?

Ans. 28 rd. how many oxen Ans. 25 oxen.

Examples in Art. 264 are usually placed under Comp. Prop. Those in Art. 205 should now be solved analytically.

ART. 265. 26. A cistern, of 250 gal., has two pipes; the first fills 41 gal. an hour, and the second empties 6 gal. an hour: if both pipes are open, how long will the cistern be in filling? Ans. 7 hr. 8 min. 34 sec.

27. A cistern of 600 gal. can be filled by pipe A, in 8 hours; and by pipe B, in 12 hours: in what time can it be filled by both pipes? Ans. 4hr.

28. A cistern, of 900 gal., can be filled by pipe A, in 10 hr., and emptied by pipe B, in 12hr.: in what time will it be filled, if both pipes are open?

Ans. 60 hr.

29. A can do a job of work in 2 days, and B in 3 days: in what time can both do it, working together?

SOLUTION.-If A does it in 2 days, he does of it in 1 day: if B does it in 3 days, he does of it in 1 day; therefore, both do 1+1 in one day, and the whole in 1÷8=1 da. Ans.

Or, Suppose the work to consist of any number of equal parts, say 6; then, if A does it in 2 days, he will do 3 parts in 1 da.; and if B does it in 3 da., he will do 2 parts in 1 da.; hence, both do 3+2=5 parts in 1 da., and the whole in 6÷÷5=11da.

30. A can perform a piece of work in 20 da., B, in 15 da., and C, in 12 da.: in what time can the three do it, working together? Ans. 5 da. alone can do it in 24 da.: how Ans. 4 da. 4 B in what Ans. 4,4, da. and A alone in

31. I hired 4 men to build a wall: A in 12 da.; B, in 15 da.; ́C, in 18 da.; D, long will it take to do it, all working?

32. A mows of a field in a day, and time can both do it?

33. A and B can mow a field in 12 da., 20 da.: in what time can В mow it?

Ans. 30 da