the first term and multiply it by all the other terms as ratios in the order in which they stand, the upper like , member as the numerator and the lower one as denominator. Cor. 2.—When the antecedent of the first equation is the odd term, multiply it by all the other terms inverted as ratios. EXAMPLES. 1. If A can do as much work in 3 days as B in 41 days, B as much in 9 days as C in 12 days, and C as much in 10 days as D in 8 days, how many days work of D's is equal to 5 days of A's. A B 12 D 10 8. Х Х 3 = 8. Ans. 5 days of A. = 8 days of D. 2. If 4 men can do as much work as 5 women, 6 women as much as 9 boys, 15 boys as much as 25 girls, and 27 girls can bind 300 sheaves in an hour, how many sheaves can 18 men bind in the same time? men, 7 sheaves 18 women, 4 5 boys. 6 9 girls. 15 25 sheaves. 27 300 300 X X X X X 18 = = 12 4 = 3. If 1 pound is worth 20 shillings, 1 shilling pence, and 1 penny = 4 farthings, how many farthings are there in 1 pound ? farthings. pence. 1 shilling. 12 1 pound. 20 1 1 ? farthings. 4 x x = 960 farthings = 1 pound. 4 tea. ng sugar. 14 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? Ans. 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? ? soap 27 soap. PROPORTION. REM. 1.-A proportion may be formed of two or more equal ratios ; thus, f = , :: 2 : 3 :: 4:6; that is, as 2 is to 3, so is 4 to 6, and as = 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. to find the denominator of the first Since the first numerator is one-half the second, the first denominator must be one-half the second. 3 6 4 x 3 or 6 : 4:: 3: 2, and = 2, Ans. 2 4 6 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 : ? 2$ 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 ? 28 x 175 3 : 21 :: 75 : ? = 525 miles, Ans. 3 7 %. If 4 gallons of syrup cost $2, what will 64 gallons cost ? 16 64 x 2 4 : 64 :: $2 : $? = $32, Ans. 4 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. 175 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 $10714. 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. |