122 DIVISION OF DENOMINATE NUMBERS. 8. A printer uses 'one sheet of paper for every 16 pages of an octavo book: how much paper will be necessary to print 500 copies of a book containing 336 pages, allowing 2 quires of waste paper in each ream ? Ans. 24 reams 5 quires 12 sheets. 9. A farmer wishes to divide 108 acres into 8 equal fields : how much will there be in each field? Ans. 13A. 2R. 10. Out of a pipe of wine, a merchant draws 12 bottles, each containing 1 pint 3 gills; he then fills six 5 gallon demijohns; then he draws off 3 dozen bottles, each containing 1 quart 2 gills : how much remained in the cask ? Ans. 82 gal. Ipt. 11. A man lends his neighbour £135 6s 8d and takes in part payment 4 cows at £5 8s a piece, also a horse worth £50 : how much remained due ? Ans. £63 14s Sd. 12. A farmer has 6 T. Scwt. 2qr. 1416. of hay to be removed in 6 equal loads: how much must be carried at each load ? Ans. 1 T. lcwt. Iqr. 2126. 13. A person at his death left landed estate to the amount of £2000, and personal property to the amount of £2803 17s 4d. He directed that his widow should receive one eighth of the whole, and that the residue should be equally divided among his four children. What was the widow's and each child's portion? Widow's portion £600 9s 8d. Each child's portion £1050 16s 11d. Divib Ans. {. RULE OF THREE. $ 104. Ex. 1. If 1 yard of cloth cost $2, how much will 6 yards cost at the same rate ? It is plain that 6 yards of cloth, at the same rate will cost 6 times as much as 1 yard, and therefore, the whole cost is found by multiplying $2 by6giving $12 for the cost. In this example there are four numbers considered, viz. 1 yard of cloth, 6 yards of cloth, $2 and $12: these numbers are called terms. Three of these terms were known or given in the question, and the other was to be found. 1 yard of cloth is the 1st term; 6 yards of cloth is the 2d term ; $2 is the 3d term; and $12 is the 4th term. Now the 2d term 6 contains the first term 1, 6 times, and the 4th term 12 contains the 3d term 2, 6 times that is, the 2d termi is as many times greater than the 1st, as the 4th term is greater than the 3d. Ø 105. This relation between four numbers is called geometrical proportion : 'and generally Pour numbers are in geometrical proportion, when the 2d term is as many times greater or less than the 1st, as the 4th term is greater or less than the 3d. We express that four numbers are in proportion thus : 1: 6 :: 2 : 12 That is, we write the numbers in the same line and place two dots between the 1st and 2d terms, four between the 2d and 3d terms, and two between the 3d and 4th terms. We read the proportion thus, · as 1 is to 6, so is 2 to 12. The numbers 2 :4 :: 5 : 10 are in geo: proportion, since the 2d term is twice the the 4th term twice the 3d. : : answer The 1st and 2d terms of a proportion always express quantities of the same kind, and so likewise do the 3d and 4th terms. As in the first example. yd. yd. 12 Ex. 2. If 416. of tea cost $8, what will. 12b. cost, at the same rate ? 1b. ib. 12 : : 8 12 3 4)96 Ans. $24 $24 the cost of 1216 of tea. It is evident that the 4th term, or cost of 126 of tea, must be as many times greater than $8, the cost of 41b, as 121b. is greater than 41b. But since the quotient of 12 divided by 4 expresses how many times 12 is greater than 4, it follows that the fourth term will be equal to $8 multiplied by this quotient: that sis, equal to $8 multiplied by 3, or equal to 824. But we obtain the same result whether we multiply the 3d term $8 by the quotient 3, or first multiply it by the 2d term and then divide the product by the 1st term; and the same may be shown for every proportion. Hence, we conclude, $106. That the fourth term of every geometrical proportion may be found by multiplying the 2d and 3d terms together, and dividing their product by the first term. § 107. The 1st and 4th terms of a proportion are Ved the two extremes, and the 2d and 3d terms are d the two means. Now, since the 4th term is ined by dividing the product of the 20 and 3d as by the first term, and since the product of the divisor by the quotient is equal to the dividend, it follows, That in any geometrical proportion the product of the two extremes is equal to the product of the two means. Thus in the first example, 1:6 :: 2 : 12 we have, 1x12=6x2=12 and in the second, 4 : 12 ::8: 24 4 x 24=8 x 12=96. § 108. The Rule of Three takes its name from the circumstance that, three numbers are always given to find a fourth, which shall bear the same proportion to one of the given numbers as exists between the other two. GENERAL RULE. $ 109. I. Reduce the two numbers which have different names from the answer sought, to the lowest denomination named in either of them. II. Set the number which is of the same kind with the answer sought in the third place, and then consider from the nature of the question whether the answer will be greater or less than the third term. "When the answer is greater than the third term, write the least of the remaining numbers in the first place, but when it is less, place the greater there. Then multiply the second and third terms together and divide the product by the first term : the quo. tient will be the fourth term or answer sought, and will be of the same denomination as the third term. Et. 3. If 48 yards of cloth cost $67,25, what will 144 yards cost at the same rate ? III. : : yd. yd: 144 6725 96 84 48 360 336 240 240 000 In this example, as the answer is to be dollars, we place the $67,25 in the third term. Then, as 144 yards of cloth will cost more than 48 yards, the fourth term must be greater than the third, and therefore, we write the least of the two remaining numbers in the first place. The product of the 2d and 3d terms is $9684,00: then dividing by the first term we obtain $201,75 for the cost of 144 yards of cloth. Ex. 4. If6 men can dig a certain ditch in 40 days, how many days would 30 men be employed in dig. ging it? men days days 6 Ans. 8 days. e answer must be days, the 40 daysis written rd place. Then as it is evident that 30 men men : answer |