= ? soap 14 3. If 1 pound is worth 20 shillings, 1 shilling = 12 pence, and 1 penny = 4 farthings, how many farthings are there in 1 pound? farthings. pence. 4 = 1 shilling. 12 1 pound. 20 1 1 ? farthings. 4 x 8 x 9 = = 960 farthings = 1 pound. 4. If 3 lbs. tea are worth y lbs. coffee, 14 lbs. coffee worth 48 lbs. sugar, and 18 lbs. sugar worth 27 lbs. soap, how many lbs. soap are 6 lbs. tea worth? 7 sugar. 48 Ans. 72 lbs. soap. 18 5. If 1 French crown = 80 pence Holland, 83 pence Holland = 48 pence English, 4 pence English = 70 pence Hamburg, 64 pence Hamburg = 1 florin Frankfort, how many florins Frankfort = 166 French crowns? Anus. 2100. 6. If 30 acres of land in Frederick are worth 40 acres in Washington, 60 acres in Washington are worth 90 acres in Allegheny, 100 acres in Allegheny worth 40 acres in Carroll, 50 acres in Carroll worth 75 in Montgomery, how many acres in Frederick are worth 450 in Montgomery? acres. acres. 30 F. = 40 W. acres. 60 W. = 90 A. 100 A. acres. 75 M. Ans. 375 acres. 450 M. =? F. 27 soap acres. = 40 C. 50 C. PROPORTION. REM. 1.-A proportion may be formed of two or more equal ratios ; thus, j = 1, :: 2 :3 :: 4 :6; that is, as 2 is to 3, so is 4 to 6, and as a = 1, by multiplying each numerator by the other denominator, the equation becomes 12 = 12 ; that is, the product of the first and last, called the extremes, is equal to the product of the second and third, called the means ; hence, any three of these terms being given, the fourth is readily found. REM. 2.-The left-hand member of the equation is the product of the means, and the right member of the extremes. 3 1. Given ? fraction. PROBLEMS. 6 to find the denominator of the first 4 Since the first numerator is one-half the second, the first denominator must be one-half the second. 3 6 4 X 3 6 6 hence 3:2::6: ? ? 2 x 6 = 4, Ans. 3 6 hence 4:6:: 2:? 4' 6 x 2 = 3, Ans. 4 COR.-Arrange the proportion so that the unknown is to be the fourth term; and as the product of the two means is equal to the product of the two extremes, therefore the unknown is equal to the quotient obtained by dividing the product of the two means by the given extreme. 4. If 5 lb. of sugar cost 50 cents, what will 25 lb. cost ? 5 As 5 : 25 :: 50 : ? 25 x 50 = 250 cts., Ans. REM.-It is evident that the ratio of the money must be the same as that of the sugar. 5. If 4 hats cost $12, what will 15 hats cost? 3 As 4 : 15 :: 12 : ? 15 x 12 = $45, Ans. A 6. If a man walk 75 miles in 3 days, how many miles will he walk in 21 days ? 21 x 75 3 : 21 :: 75 : ? = 525 miles, Ans. 7. If 4 gallons of syrup cost $2, what will 64 gallons cost? 4: 64 :: $2 : $? 64 x 2 = $32, Ans. A REM. 1.-In a proportion each ratio is composed of two like quantities termed a couplet; the couplets, however, are generally unlike quantities. If the like quantities are of different denominations, they must be reduced to the same denomination. REM. 2.-In any number of proportions the odd terms are called .antecedents and the even terms consequents. REM. 3.—The proportion is correct, although the ratios are inverted by alternating the antecedents and consequents. EXAMPLES. 1. If a railroad car runs 21 miles in 50 minutes, how far will it run in 5 hours and 50 minutes ? As 50 m. : 5 h. 50 m. :: 21 miles : ? miles. 350 x 21 50 : 350 :: 21 : ? = 147 miles, Ans. 50 2. If 45 acres of land can be purchased for $900, what is the cost of 175 acres at the same rate ? Ans. $3500. 3. If a man can do a piece of work in 24 days, working 10 hours a day, how long will it take him to do the same, if he works 12 hours a day? Ans. 20 days. 4. If the wages of 5 men for 20 days is $125, what would be the wages of 8 men for 24 days? Ans. $240. REM.—5 men for 20 days is the same as 100 men for one day, and 8 men for 24 days is the same as 192 men for one day. 5. If a man travel 117 miles in 15 days of 9 hours each, how far will he travel in 20 days of 12 hours each? Ans. 208 miles. 6. If 25 lb. butter purchase 40 lb. cheese, how many pounds butter will purchase 120 lb. cheese ? Ans. 75 lb. butter. %. Two men engage in business; A puts in $7500 and B $3000; the profits are in proportion to the stock except that B is to receive $500 for his special attention to the business—the profits are $2500; what does each receive ? Ans. A receives $14284 and B $1071%. 8. If $200 gain $12 in one year, what will $600 gain in 9 months ? Ans. $27. REM.—Examples like this should be solved by proportion, by ratio, and by analysis. PERCENTAGE. Per Cent, means per hundred, and is generally expressed fractionally; thus, 5 per cent., 6 per cent., marked 5% and 6%, is expressed T60, T6, etc., or .05, 06; thus, Too of 100 = 1ØØ x Tha = 5, and 18 of 100 is 6. 188 = 100%, 1 = 50%, f = 331%, t = 25%, t = 20%, *= 164%, o = 2%. EXAMPLES. 2 1. What is 5% of 200 ? 200 x mia = 10, Ans. What is 5% of 300 ? Ans. 15. What is 5% of 400 ? Ans. 20. 2. What is 5% of 245 ? 2.45 X = 12.25, Ans. The 100 is canceled in the 245 by pointing off two places of decimals. 3. What is 6% of 300 ? 800 x = 18, Ans. What is 6% of 400 ? Ans. 24. What is 6% of 500 ? Ans. 30. 4. What is 6% of 368? 3.68 = 22.08. 3 COMMISSION, OR BROKERAGE. The business of a commission merchant or broker is to make purchases and sales, on which he receives a percentage. PROBLEM I. A purchase of $100 worth of goods, at 1% commission, will cost $101; that is, 180 of the amount of the purchase. |