has been gone 3 74. A man travels 7 miles an hour, and after he hours, an express starts in pursuit, miles an hour. In how many hours will the man be overtaken ? and travels 11 75. If I pay one cent for an apple, twice as much for a lemon, and for an orange twice as much as for a lemon, how much do I pay for the three? 76. If a bushel of beans cost $1.25, a bushel of potatoes .6 as much, and a bushel of wheat 2.5 as much as the potatoes, how much do they all cost? 77. If we add a number to itself, we obtain 3 of the number. If 9 is 3 of a number, what is? What is the number? 78. If 95 is 15 of a number, what is the number? 79. If we add 1⁄2 and } of a number to itself, we obtain of the number. 22 is of what number? 80. 41 is of what number? 82. A man being asked how far he had travelled, said that if he had travelled as far again, and as far and as far, he should have gone 119 miles. How far had he travelled? 83. If 3 boys spend 12 cents in 2 weeks, how many boys will spend 30 cents in 3 weeks, at the same rate? How much does 1 boy spend in 2 weeks? In 1 week? In 3 weeks? Then how many boys will spend 30 cents in 3 weeks? 84. If 8 men lay 10 rods of wall in 4 days, how many men will lay 15 rods in 3 days? 85. What number is that, and of which and 1 more, are equal to the number itself? 86. and of a certain number, are 6 less than the number itself. What is the number? CHAPTER X. THE RULE OF THREE,-OR PROPORTION. The RATIO of two numbers, is the quotient of the first by the second. Thus the ratio of 3 to 4, is † ; of 6 to 2, A ratio is usually expressed, by two points written between the numbers; as, 3:4; 6:2, which are read, 3 is to 4; 6 is to 2. = When two ratios are equal to each other, they may be written together, thus: 2:4 3:6, which is read, 2 is to 4, equals 3 is to 6; or thus: 2 : 4 :: 3 : 6, read, 2 is to 4 as 3 is to 6. Four numbers bearing such a relation to each other, are said to be propor tional, and the expression is called a PROPORTION. The first term of every ratio, is called the antecedent, and the second, the consequent. In every proportion, the antecedents and the consequents may exchange places. Thus, 2:6 :: 7 : 21, and 6 : 2 :: 21:7, are each true proportions. The first and fourth terms of a proportion, are called the extremes, the second and third terms, the means. The proportion 3: 12 . If these two fractions were reduced to a common denominator, their numerators would be the same. But the numerators would be found, by multiplying each numerator by the other denominator; therefore, one of the numerators would represent the product of the extremes, the other, the product of the means. Hence in every proportion, the product of the extremes, is equal to the product of the means. Then, when one extreme and the two means are given, to find the other extreme: Divide the product of the means, by the given extreme. = 2: 8 may also be written EXAMPLE FOR THE BOARD. If 7 barrels of flour cost $35, what will 11 barrels cost? It is evident that the ratio of 7 barrels to 11 barrels will be the same, as that of the price of 7 barrels to the price of 11 barrels. We have, then, the three terms of a proportion: Dividing the product of the means, by the given extreme, we find $55 for the price of 11 barrels. Hence we derive 'THE RULE OF THREE. Make that which is of the same kind with the answer, the third term of a proportion. If the answer will be greater, make the greater of the two remaining numbers the second term, and the less the first term. If less, make the less number the second term, and the greater the first term. Divide the product of the second and third terms, by the first term, and you will obtain the fourth term, which will be the Answer. 1. If 11 yards of flannel are worth $2.37, what will be the price of 19 yards? 2. If a cistern discharges 75.09 gallons of water in 1.5 hours, how much will it discharge in 7.93 hours? 3. A man paid $7.96 for the interest of a certain sum of money, for 2.1 years. How much would he pay for the use of the same sum, 5.09 years? 4. How far would a ship sail in 11.64 hours, at the rate of 78.8 miles in 14.2 hours? 5. What must I pay my gardener for 37.25 days' labour, his wages being $10.75 per week? 6. If 12 horses eat 11.375 bushels of oats in a week, how many bushels will 74 horses eat in the same time? 7. How many hours will it take a carrier dove to fly 190.77 miles, at the rate of 46.31 miles in 2.95 hours? 8. How many yards of serge, that is wide, will line a cloak, containing 9.75 yards of broadcloth, that is 1.5 yards wide? 9. The provisions of a garrison are sufficient to supply 736 men 14.5 days. How long will they last 429 men? 10. A family of 7 men use a barrel of flour in 74 days. How many men would use a barrel in 18.5 days? 11. If 49 men can build a wall in 25.8 days, in what time will 65 men build it? 12. How much carpeting, that is yd. wide, will cover a floor 5.25yd. long and 4.75yd. wide? 13. If the interest of $1.50 is $0.0975 for 13mo., what is the interest of $548.63 for the same time? 14. If I can perform a journey in 8.7 days, by travelling 9 hours each day, in what time can I per. form it, by travelling 7.96 hours a day? 15. A bankrupt paid 43 cents for every dollar of his debts. How much would he pay on a debt of $569.31? 16. A grocer bought 11cwt. 3qr. of sugar, for $83.21. For how much must he sell 4cwt. 1qr. 22lb., so as neither to gain nor lose? 17. If the price of 16yd. 3qr. of sheeting is $2.45, what is the price of 4.91yd.? 18. How many men will build a wall in 13.174 days, that 14 men can build in 9.41 days? 19. How many dollars are equivalent to £13 78. 1ld.; $1.00 being equal to 4s. 6d.? 20. How many men can do a piece of work in 28.75 days, that 39 men can do in 57.5 days? 21. The debts of a bankrupt are $19946.25, to meet which, he has property, valued at $7602.375. How much can he pay on a debt of $693.50? 22. A. and B., traded in company. A.'s money, which was $9621.75, remained in the firm 11.8mo. How long was B.'s money, which was $12770.00, in the firm, the gain being divided equally? 23. A. B. & C. invested $16329.00 in merchandise. They sold their goods so as to gain $4101.375. How much of the gain ought B. to receive, he having contributed $7075.15? 24. A tailor made 37 coats, each coat containing 1.87yd. of cloth, 1.5 yards wide. How much silk that is .67 yards wide, will be required to line the whole? 25. When a pole 4ft. high, casts a shadow of 4.25 feet, what is the height of a steeple, that casts a shadow of 197 feet? 26. How much sugar can I buy for $96.25, if I pay $71.50 for 9cwt. 2qr 27lb.? 27. If 2 lb. of indigo cost $4.625, what will be the cost of 9 lb.? EXAMPLE FOR THE BOARD. If 12 men can build a wall 16 feet long, 2 feet thick, and 4 feet high, in a week, how many men can build a wall 24 feet long, 3 feet thick, and 6 feet high, in the same time? This is a question in COMPOUND PROPORTION. The third term is determined in the same way as in Simple Proportion. Of the remaining numbers, we compare any two of a kind, writing one for a first term, and the other for a second, until all are brought down. Then, the product of the first terms, the product of the second terms, the third term, the Answer. 28. If the wages of 24 men for 3.5 days are $96.50, what will be the wages of 19 men for 7.75 days? |