2. C bought of D as follows: Jan. 1, 1857, 100 yds. cloth, on 2m. Feb. 15, 66 50 shawls, $300. 66 3m. 200. ❝ 60 days, 500. Mar. 1, 66 200bbl. apples, "2m. 400. When shall C pay D the balance of $300? The equated time for the payment of C's bills is found to be April 2, and that for D's, April 27, 25 days later. Since the larger sum is due first, it is evident the balance, $300, must be paid long enough before April 2 to have its interest cancel the interest on the $700 which remain unpaid for 25 days from April 2 to April 27, viz., 58 days (700 × 25 = 17500 and 17500 300 = 581), and 58 days before April 2 will carry us back to Feb. 3, the equated time, Ans. 234. From these illustrations, To equate accounts, RULE.-1. Equate the debit and also the credit side of the account, and find the number of days between the equated times. 2. Multiply the smaller side of the account by this number of days, and divide the product by the difference between the sides; the quotient will be the number of days to be reckoned from the equated time of the larger side—to be reckoned FORWARD if the larger side is due at the later, and BACKWARD if due at the earlier date. 235. PROOF.-The interest on the two sides of the account from the equated time of settlement to the equated time of the sides, severally, will be alike. Proof of Ex. 1: = $16. Interest on $800 from Aug. 20 to Dec. 18, 120 days Proof of Ex. 2 : Interest on $1000 from Feb. 3 to April 2, 58 days = $9.667. = $9.683. The difference between the results in Ex. 2 arises from disregarding the of a day in the equated time. NOTE. When the larger sum is due at the earlier date, the rule may require the settlement to be made before some of the transactions have occurred, as in Ex. 2, a result which is obviously impracticable; but the difficulty will be removed by adding the interest on the balance due from the equated to the actual time of settlement. So also may a balance be paid before it is due, by paying the present worth of it at the time of actual settlement. 3. E bought of F as follows: Jan. 7, 1856, on 1m. credit, a bill of $800 F also bought of E as follows: Jan. 18, 1856, on 2m. credit, a bill of $200 :- 4. G owes H $800, payable June 4, 1857, and H owes G $600, payable Sept. 6, 1857; when is the equated time of settlement, and what should G pay, Sept. 6, 1857? Ans. Aug. 26, 1856; $212.333. § 24. EXAMPLES IN ANALYSIS. 236. We analyze an example when we proceed with it, step by step, according to its own conditions, without the guidance of any particular rule. Ex. 1. If 6 barrels of flour cost $42, what will 11 barrels cost? SOLUTION. If 6bbl. cost $42, then 1bbl. will cost of $42, which is $7; and if 1bbl. cost $7, then 11bbl. will cost 11 times $7, which is $77, the answer. 2. If g of a cask of wine cost $35, what will 7 casks cost? 3. 20 is 4. 51 is 5. 95 is 6. If cost? of what number? of what number? of what number? of a ton of hay cost 95 shillings, what will a ton 7. If 37 of a cask of oil is worth $74, what is the value of 5 casks? 10. A man sold a watch for $63, which was of its cost; what was its cost? 11. A pole is in the mud, & in the water and 6 feet above water; what is the length of the pole? 12. A ship's crew have provisions sufficient to last 12 men 7 months; how long would they last 24 men? 13. A can build 35 rods of wall in 33 days, but B can build 9 rods while A builds 7; how many rods can B build in 44 days? Ans. 60. 16. of 27 is 17. A fox has 39 of how many twelfths of 60? rods the start of a hound, but the hound runs 27 rods while the fox runs 24; how many rods must the hound run to overtake the fox? Ans. 351. 18. A man being asked how many sheep he had, replied, that if he had as many more, as many more and 2 sheep, he should have 100; how many had he? 19. A man being asked how many sheep he had, replied, that if he had twice as many more, as many more and 31 sheep, he should have 70; how many had he? 20. A detachment of 2000 soldiers were supplied with bread sufficient for 12 weeks, allowing each man 14 ounces a day, but finding 105 barrels containing 200lbs. each, wholly spoiled, how many ounces may each man eat daily, that the remainder may last them 12 weeks? 21. A detachment of 2000 soldiers, having of their bread spoiled, were put upon an allowance of 12oz. each per day for 12 weeks; what was the whole weight of their bread, good and bad, and how much was spoiled? 22. A detachment of 2000 soldiers having lost 105 barrels of bread, weighing 200lbs. each, were allowed but 12oz. each per day for 12 weeks; but if none had been lost, they might have had 14oz. daily; what was the weight, including that which was lost, and how much was left to subsist on? 23. A detachment of 2000 soldiers, having lost of their bread, had each 12oz. per day for 12 weeks; what was the weight of their bread, including the part lost, and how much per day might each man have had, had none been lost? 24. A gentleman left his son an estate, of which he spent in 7 months, and of the remainder in 3 months more, when he had only $5000 remaining; what was the value of the estate? 25. The quick-step in marching being 2 paces of 28 inches each per second, what is the rate per hour? and in what time will a detachment of soldiers reach a place 60 miles distant, allowing a halt of 14 hours? 26. Two men and a boy were engaged to reap a field of rye; one of the men could reap it in 10 days, the other in 12, and the boy in 15 days. In how many days can the three together reap it? 27. The commander of a besieged fortress has 2lbs. bread per day for each soldier for 57 days, but, in anticipation of succor, he wishes to prolong the siege to 75 days; in that case, what must be the allowance of bread per day? 28. A merchant bought a number of bales of velvet, each containing 12917yds., at the rate of $7 for 5yds., and sold them out at the rate of $11 for 7yds., and gained $200 by the bargain; how many bales were there? Ans. 9. 29. A merchant bought a number of bales of hops, each bale containing 2461lb., at the rate of $3 for 11lb., and sold them at the rate of $5 for 12lb., and gained $248; how many bales did he buy? Ans. 7. 30. Suppose I pay 33 cents per bushel for carting my wheat to mill, the miller takes for grinding, it takes 4 bushels of wheat to make a barrel of flour, I pay 25 cents each for barrels and $14 per barrel for carrying the flour to market, where my agent sells 60 barrels for $367, out of which he takes 25 cents per barrel for his services; what do I receive per bushel for my wheat? Ans. 873 cents. § 25. RATIO. 237. RATIO is the relation of one quantity to another of the same kind; or, it is the quotient which arises from dividing one quantity by another of the same kind. 238. Ratio is usually indicated by two dots; thus, 8:4 expresses the ratio of 8 to 4. The two quantities compared are the terms of the ratio; the first term being the antecedent, the second the consequent, and the two terms, collectively, a couplet. 239. English mathematicians consider the antecedent a dividend, and the consequent a divisor; thus, but, in the French system, the antecedent is taken as a divisor, and the consequent as a dividend; thus, |