day for 12 weeks; but had none been lost, they might have had 14 ounces a day for the same time. What was the whole weight, including what was lost, and how much had they left to subsist on ? Ans. The whole weight, 147000lb.; left to subsist on, 126000lb. 6. Bought threescore pieces of Hollands for three times as many dollars, and sold them again for four times as many dollars; but if they had cost me as much as I sold them for, for what should I have sold them to gain at the same rate? Ans. $ 320. 7. Bought 20 pounds of tea at the rate of 13 pounds for a dollar, and 62 pounds more at the rate of 15 pounds for $ 12; and sold the whole at the rate of 3 pounds for $ 4. How much did I gain, or lose? 8. Each of the four sides of a certain field appeared to be 2 furlongs 30 rods and 3 yards in length when measured by a line supposed to be 4 rods long; but the line was found to have been only 64 feet in length. Required the true distance round the field. Ans. 707411 feet. 9. If 5 oxen or 7 cows eat 34 tons of hay in 87 days, in what time will 2 oxen and 3 cows eat the same quantity of hay? Ans. 105 days. 10. If 360 men be placed in a garrison, and have provisions for 6 months, how many men must be sent away at the end of 4 months that the remaining provisions may last the garrison 8 months longer? Ans. 270 men. 11. My tailor informs me it will take 104 square yards of cloth to make me a full suit of clothes. The cloth I am about to purchase is 15 yards wide, and on sponging it will shrink zt in width and length. How many yards of the above cloth must I purchase for my new suit”? Ans. 6163ayd. 12. If the price of a farm of 130A. 2R. 20p. be $6537.50, what will be the price of another, containing 100A. OR. 30p., if 4 acres of the latter be worth 5 of the former? Ans. $ 6267.71+. 13. How much each of copper, tir, zinc, and lead will be required to make a bell weighing 192 tons 17cwt. 161b., the weight of that at Moscow, bell-metal being composed of 800 parts copper, 101 tin, 56 zinc, and 43 lead ? Ans. Copper, 3085724lb.; tin, 389577% lb.; zinc, 21600 lb.; lead, 165853,7lb. 14. The relative heating power per cord of white-oak to white-pine being as 81 to 42, and that of white-pine to pitchpine as 7 to 8; how many cords composed of white-pine and pitch-pine in equal parts will be required to produce as much heat as 20 cords of white-oak? Ans. 36 cords or 18 of each. 15. If a cask of wine, containing 85 gallons, cost $116.95, what would be the value of a hogshead containing 63 gallons, and composed of four parts of the same wine and one part of water? Ans. $ 69.3435 16. If 8 horses in 30 days consume 34 tons of hay, how long will 4,5 tons last 10 horses, 15 cows, and 7 sheep, each cow consuming as much as a horse, and each sheep į as much as a cow ? 17. The relative nutritious matter per pound of potatoes to beets being as 25 to 14, that of beets to carrots as 7 to 5, and that of carrots to turnips as 5 to 2; how many pounds of turnips are equal in nutritious matter to 2000 pounds of potatoes ? and allowing 60 pounds to a bushel, how much cheaper or dearer relatively per bushel are potatoes at 80 cents, and turnips at 20 cents, than either beets at 35 cents or carrots at 25 cents. Ans. 12500 pounds ; potatoes 17 cts. dearer; turnips 10cts. dearer. 18. A and B set out from the same place, and in the same direction. A travels uniformly 18 miles per day, and after 9 days turns and goes back as far as B has travelled during those 9 days; he then turns again, and, pursuing his journey, overtakes B 221 days after the time they first set out. quired to find the rate at which В uniformly travelled. Ans. 10 miles per day. 19. Two men in Boston hire a carriage for $ 25, to go to Concord, N. H., and back, the distance being 72 miles, with the privi. lege of taking in three more persons. Having gone 20 miles, they take in A; at Concord, they take in B; and when within 30 miles of Boston, they take in C. How much shall each man pay? Ans. First man, $ 7.609 183 ; second man, $ 7.609183; A $ 5.873-1903; B, $ 2.86415 ; C, $ 1.041. It is re PERCENTAGE. 342. PERCENTAGE is an allowance for every hundred, consisting of a part of that on which it is reckoned. Per cent., a contraction of per centum, signifies by the hundred. By 1 per cent. is to be understood 1 out of a hundred, or 1 hundredth ; by 2 per cent. is to be understood 2 out of a hundred; or 2 hundredths, &c. 343. The rate per cent. is the number denoting the part allowed for every hundred, as 5 per cent., 6 per cent., &c. 344. The basis of percentage is the number on which percentage is reckoned. 345. The amount is the basis of percentage increased by the percentage ; and the remainder is the basis decreased by the percentage. 346. The rates per cent. may be written decimally as in the following TABLE. 1 per cent. .01 .09 2 per cent. .02 .10 3 per cent. 66 .03 .15 4 per cent. .04 .20 5 per cent. .05 .25 6 per cent. .06 .63 7 per cent. .07 1.00 8 per cent. .08 1.16 AND į per cent. may be written .005, or .001, and be read of 1 per cent., or 4 of 1 per cent. 4 per cent. may be written .0025, or .001, and be read 25 of 1 per cent., or of 1 per cent. 5 per cent. may be written .00125, or .00], and be read 1026 of 1 per cent., or ļof 1 per cent. 74 per cent. may be written .075, or .071, and be read 7o per cent., or 74 per cent. NOTE. If a fraction of 1 per cent. cannot be exactly expressed in decimal figures, it may be written as a part of a mixed decimal. Thus, 41 per cent. may be written .04}; 64 per cent. may be written .064; and 11; per cent may be written .11%. 66 66 66 66 EXERCISES. 1. Express decimally 19 per cent. Ans. .19. 2. Express decimally 27 per cent. 3. Express decimally 13 per cent. 4. Express decimally 14 per cent. 5. Express decimally 7 per cent. 6. Express decimally 774 per cent. 7. Express decimally 106 per cent. 8. Express decimally 107 per cent. 9. Express decimally 305 per cent. 10. Express decimally 9991% per cent. 347. To find the percentage any given rate per cent. is of any number or quantity. Ex. 1. Bought į of a ship for $ 15650, and sold the same at a gain of 12 per cent. How much did I make by the transaction ? Ans. $ 1878. Since 12 per cent. equals Basis of percentage, $15 6 5 0 .12 of the original cost, we Rate per cent. .1 2 multiply $ 15650 by the Percentage, $187 8.0 0 Ans. decimal expression 12. RULE. — Multiply the given number by the rate per cent. expressed decimally, and the product will be the percentage. Or, As 100 per cent. is to the given rate per cent., so is the given basis of percentage to the percentage required. OPERATION. EXAMPLES. 2. What is 15 per cent. of 500 bushels ? Ans. 75bu. 3. What is 20 per cent. of 75cwt. ? Ans. 15cwt. 4. What is 30 per cent. of 150 tons ? Ans. 45 tons. 5. What is 75 per cent. of $ 500 ? 6. What is 95 per cent. of 700 chaldrons ? 7. What is 2 per cent. of 40 miles ? Ans. .8 mile. 8. What is 99 per cent. of $ 1000 ? Ans. $ 990. 9. What is 33} per cent. of 144 barrels ? Ans. 48bbl. 10. What is 66s per cent. of 90 hogsheads ? 11. What is 1 per cent. of $ 100 ? Ans. $ 0.25. 12. What is į per cent. of 1728lb. ? Ans. 15.12lb. 13. A certain colonel, whose regiment consisted of 900 men, lost 8 per cent. of them in battle, and 50 per cent of the remainder by sickness. How many had he remaining ? Ans. 414 men. 14. A merchant, having $ 1728 in the Union Bank, wishes to withdraw 15 per cent. ; how much will remain ? 15. A gentleman, who had an estate of $ 25,000, in his will gave to his wife 40 per cent. of his property, and to his son Samuel 30 per cent. of the remainder. The residue he divided equally among his daughters, Marcia, Isabella, and Clara, after having deducted $ 60 as a present to his clergyman. What did each receive ? Ans. Wife, $10,000; son, $ 4,500; daughters, $3,480 each. 348. To find what rate per cent. one given number is of another. Ex. 1. What per cent. of 50 is 12 ? Ans. 24 per cent. Since the percentage equals 295 .24, Ans. the product of the basis of 2 percentage by the number 12 x 1 Ø Ø denoting the rate per cent. 24 per cent. (Art. 343), the quotient 5 Ø arising from dividing the percentage by the number denoting the basis must equal the rate per cent. We therefore divide 12 by the 50, and obtain a 265, which, expressed decimally, equals .24, or 24 per cent. Since the question, evidently, is the same as to find 7 of 100 per cent., we multiply the 12 by 100, or annex two ciphers and divide by 50, and obtain the same result as before. Rule. - Annex two ciphers to the number denoting the percentage, and divide by the number on which the percentage is reckoned ; and the quotient will be the rate per cent. Or, As the given basis of percentage is to the given percentage, so is 100 per cent. to the rate per cent. required. OPERATION. Or, EXAMPLES. 2. What per cent. of 16 is 2 ? Ans. 12 per cent. 3. What per cent. of 110 is 11 ? Ans. 10 per cent. 4. What per cent. of 24 is š? 5. f of 18 per cent. is what per cent. of 24 per cent. ? Ans. 25 per cent. 6. What per cent. of $ 150 is 25 per cent. of $ 36 ? N 22 Ans. 6 per cent. |