in 8 mon.; the remainder in 12 mon.: what the average time of payment? Ans. 9 mon. 6. I buy $200 worth of goods; 1 fifth to be paid now; 2 fifths in 5 mon.; the rest in 10 mon.: what the average time of paying all? Ans. 6 mon. ART. 258. In finding the Average or Mean time for the payment of several sums due at different times, any date may be taken from which to reckon the time. 7. A merchant buys goods as follows, on 60 days credit: May 1st, 1848, $100; June 15th, $200: what the average time of payment? Ans. July 30th. Counting from May 1st, it is 60 days to the first payment, and 105 days to the second. Assuming April 1st as the day from which to count, the period is 120 days, which makes the same day of payment. 8. I bought goods on 90 days credit, as follows: April 2d, 1853, $200; June 1st, $300: what the average time of payment? Ans. Aug. 6th. ART. 259. The preceding rule, generally used, supposes discount and interest paid in advance to be equal; but this (Art. 239, Rem. 2) is not correct. The following, based on true discount (Art. 235), is the TRUE RULE FOR THE EQUATION OF PAYMENTS. Find the present worth of each debt (Art. 237), then find the TIME (Art. 230), at which the sum of the present worths will amount to the sum of the debts: this gives the true equated time. 9. A owes $103 due in 6 mon., and $106 due in 12 mon.: find the true mean time of payment. Ans. 9 mon. REVIEW.-258. To find the mean time, from what date do you reckon? 259. Is the common rule for Equation of Payments strictly accurate? What is the true rule for Equation of Payments? XIX. ALLIGATION MEDIAL. ART. 260. Alligation medial is the method of finding the mean or average price of a mixture, when the ingredients composing it, and their prices, are known. 1. I mix 4 pounds of tea, worth 40 cts. a lb., with 6 lb. worth 50 cts. a lb.: what is 1 lb. of the mixture worth? SOLUTION.-4 lb. at 40 cts. per lb. $1.60, and 6 lb. at 50 cts. $3.00; making the total value of the 10 lb.= $4.60: hence, 1lb. cost of $4.60, or 46 cts. $4.60÷10=46 cts., cost of 1 lb. *2. Mix 6 lb. sugar, at 3 cts. a lb., with 4 lb. at 8 cts. a lb., what will 1 lb. of the mixture be worth? Ans. 5 cts. Rule.-Divide the whole cost by the whole number of ingredients; the quotient will be the average or mean price. NOTE. The principles of this rule may be applied to the solution of many examples not embraced in the definition. 3. Mix 25 lb. sugar at 12 cts. a lb., 25 lb. at 18 cts., and 40 lb. at 25 cts.: what is 1 lb. of the mixture worth? Ans. 19 cts. 4. A mixes 3 gal. water, with 12 gal. wine, at 50 cts. a gal. what is 1 gal. of the mixture worth? Ans. 40 cts. 5. I have 30 sheep: 10 are worth $3 each; the rest, $9 each: find the average value. 12, $4 each; Ans. $5. 6. On a certain day the mercury in the thermometer stood as follows: from 6 till 10 A. M. at 63°; from 10 A. M. till 1 P. M., 70°; from 1 till 3 P. M., 75°; from 3 till 7 P. M., 73°; from 7 P. M. till 6 A. M. of the next day, 55°. What was the mean temperature of the day, from sunrise to sunrise? Ans. 627.° Multiply the number of hours by their mean temperature; divide the sum of the products by 24, the sum of the hours. REVIEW.-260. What is Alligation Medial? What the Rule? 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? ANALYSIS.-1 lemon is of 3 lemons, and will cost as much; but of 12 cents is 4 cents, the cost of 1 lemon. If 4 cents 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? ANALYSIS.—of 28 is 4, and are 3 times 4=12. If 2 cents buy 1 orange, 12 cents will buy as many oranges as 2 cents are contained times in 12 cents, that is 6. Ans. G oranges. of it for cloth at $2 a 9. A man having $40, spent 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? 11. of 45 are how many times 8? 12. 1 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 5×6=30. Hence, 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 of what number? 12 is of what? 15 is of what? of its 15. A farmer sold a wagon for $45, which was cost; he paid for it with sheep at $3 a head: how many sheep were required? 16. 25 is of how many times 4? 17. 21 is of how many times 4? 18. of 40 is of what number? Ans. 21. ANALYSIS.— of 40 is 10, and is 3 times 10=30. If 30 is, of 30=6, is ; and 7, or the required number, áre 7×6=42. 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 yd.? of a bu.? of a yd.? 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. Or thus: Since 1 bl. is of 15 bl., 24 bl. are 24 of 15 bl., and will cost as much. $40×24=$64. Ans. Ans. $35. 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 of 14 yd.? 4. If $8 will purchase 4 yd. of cloth, how many yards will $24 buy? 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. 12 yd. Ans. 267 yd. Ans $3831. 12 days, how SOLUTION. It will require 1 man 5 times as long as 5 men; that is, 5×12 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 ? |