3. A and B hire a pasture for $54: A pastures 23 horses 27 da.; B, 21 horses 39 da.: what will each pay? Ans. A, $23.283; B, $30.711 4. A put in $300 for 5 mon.; B, $400 for 8 mon.; , $500 for 3 mon. they lost $100 find each one's loss. Ans. A's, $24.1911; B's, $51.613; C's, $24.191 : 5. A, B, and C hire a pasture for $18.12: A pastures 6 cows 30 da.; B, 5 cows 40 da.; C, 8 cows 28 da.: what shall each pay? Ans. A, $5.40; B, $6; C, $6.72 6. Three men formed a partnership for 16 mon.: A put in at first $300, and at the end of 8 mon., $100 more; B put in at first $600, but, at the end of 10 mon., drew out $300; C put in at first $500, and, at the end of 12 mon., $400 more; they gained $759: find each man's share. Ans. A's, $184.80; B's, $257.40; C's, $316.80 7. A and B are partners: A put in $800 for 12 mon., and B, $500. What sum must B put in at the end of 7 mon., to entitle him to half the yr.'s profits? Ans. $720. For additional problems, see Ray's Test Examples. XVIII. EQUATION OF PAYMENTS.. ART. 256. Equation or equality of Payments is the method of finding the mean or average time of making two or more payments, due at different times. The rule for finding the mean or equated time, is based on the principle, that The interest of any sum for any given period, is equal to the Int. of half the sum for twice the period; of one-third of the sum for three times the period, and so on. Thus, The Int. of $2 for 1 mon. = Int. of $4 for 5 mon.= Int. of $1 for 2 mon. Int. of $1 for 20 mon. REVIEW.-256. What is Equation of Payments? On what principle is the rule for finding equated time based? Give examples. EXAMPLE.-The Int. of $4 for 5 mon., at 6 per cent., is 10 cents (Art. 224): the Int. of $1 for 20 mon., is also 10 cents (Art. 223). ART. 257. 1. A owes B $2 due in 3 mon., and $4 due in 6 mon. at what period can both sums be paid, neither party being the loser? From Art. 255, it follows, that, Int. of $2 for 3 mon. Int. of $4 for 6 mon. = int. of $1 for 2X3= 6 men. $6 for mon. = int. of $1 for 30 mon. Now find in what time $6 will produce the same Int. as $1 in 30 mon. At $6 is 6 times $1, it will produce the same Int. in of the time (Art. 256); that is, in 30 mon.÷6-5 mon. Ans. PROOF.-Int. of $2 for 3 mon., at 6 %, 2×14= 3 cts. Int. of $4 for 6 mon., Int. of $6 for 5 mon., =4X3 12 cts. =6X215 cts. *2. A owes B $2 due in 4 mon., and $6 due in 8 mon.: find the average time of paying both sums. Ans. 7 mon. COMMON RULE FOR EQUATION OF PAYMENTS. Multiply each payment by the time to elapse till it becomes due; divide the sum of the products by the sum of the payments; the quotient will be the equated time. When one of the payments is due on the day from which the equated time is reckoned, its product is 0; but, in finding the sum of the payments, this must be added with the others. See Ex. 6. 3. A owes B $8, due in 5 mon., and $4 due in 8 mon.: find the mean time of payment. Ans. 6 mon. 4. A buys $1500 worth of goods; $250 are to be paid in 2 mon.; $500 in 5 mon.; $750 in 8 mon.: find the mean time of payment. Ans. 6 mon. 5. A owes B $300; 1 third due in 6 mon.; 1 fourth REVIEW.-257. What is the common rule for Equation of Payments? When one of the sums is to be paid down, how proceed? 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. $100 X 60 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. 9mon. 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? *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 ingre dients; the quotient will be the average or mean price. NOTE. The principles of this rule may be applied to the solu tion of many examples not embraced in the definition. 3. Mix 25 lb. sugar at 12 cts. a lb., 25 lb. at 18cts., 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 four 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. C 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? |