313 a. cost? EXAMPLES IN ANALYSIS. 1. If 6 barrels of flour cost $42, what will 11 barrels 2. If of a cask of wine cost $35, what will 7 casks cost? 3. Twenty is of what number? 5. Ninety-five is 6. If of a ton cost? of what number? of hay cost 95 shillings, what will a ton 7. If 3 of a cask of oil is worth $74, what is the value of 5 casks? 8 Sixty-four is of how many times 12? 9. Seventy-two is of how many times 4? 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? of how many fifths of 55? 14. of 28 is 15. of 44 is 16. of 27 is of how many thirds of 15? of how many twelfths of 60? 17. A fox has 39 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 hare has 32 rods the start of a hound, but the hound runs 12 rods while the hare runs 8; how many rods will the hare run before the hound overtakes him? 19. 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? 20. A detachment of 2000 soldiers was supplied with bread sufficient for 12 weeks, allowing each man 14 ounces a day, but finding 105 barrels, containing 2001b. 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 12 oz. each per day for 12 weeks; what was the whole weight of their bread, good and bad, and how much was spoiled? Fro 336000 22. A detachment of 2000 soldiers having lost 105 barrels of bread, weighing 200 lb. each, were allowed but 12oz. each per day for 12 weeks; but if none had been lost, they might have had 14 oz. daily; what was the weight, including that which was lost, and how much was left to subsist on ?2352000; 20110 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 inénés 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 11⁄2 hours? 26. Two men and a boy engage to reap a field of rye; one of the men can 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. A merchant bought a number of bales of hops, each bale containing 246 lb., at the rate of $3 for 11 lb., and sold them at the rate of $5 for 12lb., and gained $248; how many bales did he buy? 133 Ans. 7. my wheat bushels of 28. Suppose I pay 33 cents per bushel for carting to mill, the miller takes for grinding, it takes 4 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. 87 cents. RATIO. 314. 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. 315. 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. 316. Most mathematicians consider the antecedent a dividend, and the consequent a divisor; but others take the antecedent for the divisor, and the consequent for the dividend; NOTE 1. The first method is often called the English method, and the other the French; but there appears to be no good reason for such a distinction. NOTE 2. The first is a direct ratio; the second is an inverse or reciprocal ratio. The first being considered the more simple and natural, is adopted in this work. 317. The antecedent and consequent being a dividend and divisor, it follows that any change in the ANTECEDENT causes a LIKE change in the value of the ratio, and any change in the CONSEQUENT causes an OPPOSITE change in the value of the ratio (Art. 84, 85, and 131). Hence, 1st. Multiplying the antecedent multiplies the ratio; and dividing the antecedent divides the ratio (Art. 83, a and b). 314. What is Ratio? 315. How indicated? What are the terms? The 1st? The 2d? The two collectively? 316. Which term is divisor? Is the custom uniform? Which method is here taken? Why? What is a direct ratio? An inverse ratio? 317. Explain and illustrate Art. 317 fully. 2d. Multiplying the consequent divides the ratio; and dividing the consequent multiplies the ratio (Art. 83, c and d). 3d. Multiplying both antecedent and consequent by the same number, or dividing both by the same number, does not affect the ratio (Art. 84, a and b). 318. The antecedent, consequent, and ratio are so related to each other, that, if either two of them be given, the other may be found; thus, in 12: 34, we have 319. When there is but one antecedent and one consequent the ratio is said to be simple; thus, 15: 5 = 3, is a simple ratio. 320. When the corresponding terms of two or more simple ratios are multiplied together the resulting ratio is said to be compound; thus, by multiplying together the corresponding terms of the simple ratios, { 6:3 8:2 =4 we have the com 10:2 - 5 pound ratio, 48:4-12 or 480:12 = 40. A compound ratio is always equal to the product of the simple ratios of which it is compounded. NOTE. A Compound ratio is not different in its nature from a simple ratio, but it is called compound merely to denote its origin. Ex. 1. What is the ratio of 20 to 4? 2. What is the ratio of 2 to 9? Ans. 20:45. Ans. 298. 3. What is the inverse ratio of 20 to 4? 4. What is the inverse ratio of 2 to 9? 5. What is the ratio compounded of 8 to 6 and 9 to 2? 197 6. Which is the greater, the ratio of 9 to 7 or of 19 to 14? 7. Which is the greater, the ratio of 5 to 4 or of 15 to 13? 318. What of antecedent, consequent, and ratio? 319. What is simple ratio? 220. Compound ratio? Its value? Its nature? Why called compound? PROPORTION. 321. PROPORTION is an equality of ratios. Two ratios, and .. 4 terms, are required to form a proportion. 322. Proportion is indicated by means of dots; thus, 8:46:3, which is read, 8 is to 4 as 6 is to 3; or, as 8 is to 4 so is 6 to 3; or it may be indicated thus, 8:4 = 6:3, which is read, the ratio of 8 to 4 equals the ratio of 6 to 3. Any 4 num.ers are in proportion, and may be written and read in like manner, if the quotient of the 1st divided by the 2d is equal to the quotient of the 3d divided by the 4th. 323. The 1st and 4th terms are called extremes, and the 2d and 3d, means. The 1st and 3d are the antecedents of the two ratios, and the 2d and 4th are the consequents. The product of the extremes is always equal to the product of the means; thus, in the proportion 8: 4 :: 6 : 3, we have 8 × 3 = 4 × 6. 324. Since the product of the extremes is equal to the product of the means, any one term may be found when the other three are given; for the product of the extremes divided by either mean will give the other mean, and the product of the means divided by either extreme will give the other extreme. Fill the blank in each of the following proportions : 321. What is Proportion? 322. How indicated? Proportion, how read? When are four numbers in proportion? 323. What are the 1st and 4th terms called? 2d and 3d? 1st and 3d? 2d and 4th? The product of the extremes equals what? be found? 324. How many terms must be given? How can the other |