3 -- 8. If 4 11 96 lb. of indigo cost L. what quantity will 21. 3s. 4d. buy? Ans. 14lb. 2oz. 14dr.fr. 9. Paid 21. 2s. 8d. for the carriage of 1cwt. 2qr. 14lb.; what sum will pay for 214 cwt. carried the same distance? Ans. 31. 16s. 6d.3. 6 of a calf be worth of a pound, what is the whole Ans. 21. 1s. 1d.ş. 7 11. If 7 men can mow a hay-field in 4 days, how long would 4 men require to mow the same? Ans. 817 days. 3 12. Paid 11 shillings for of a gallon of brandy; what must 8 be given for 4 gallons at the same rate? Ans. 71. 2s. 5d.ž. of a block of mahogany cost 2§L. what sum will 13. If 2 11 purchase the remaining 9 11 ܕ܂ Ans. 11l. 16s. 3d. 14. Suppose 15 feet of plank 9 inches wide sufficient to make a side-board, how many feet of plank 14 foot wide will be required to make another of equal dimensions? Ans. 7 feet 6 inches. 15. If 9 English ells of linen cost 11. 17s. 6d. what cost 10 Flemish ells? Ans. 1l. 5s. 16. What is the value of 5 fother of lead, at 16s. 4d. per cwt.? Ans. 41. 11s. 6d.. 17. A corn-chandler charges 91. 5s. 6d. for a load of barley; what must be given for 34 bushels at that price? Ans. 15s. 2d.ff. 18. Suppose 120 men can build a ship in 153 weeks, how long will it be building if 87 men only are employed? Ans.21w. 5d.. 19. If 5l. 6s. 8d. be paid for the use of 1731. 2s. 6d. what sum must be paid for the use of 310L. for the same time? Ans. 81. 19s. 6d. 27. 2 20. After using of a Cheshire cheese, 2 of the remainder 3 sold for 13s. 5d.+; what was the whole cheese worth? Ans. 11. 2s. 5d.. 21. If 3 yards of cloth 1 yard wide will make a suit of clothes, what quantity of cloth to make a similar suit? Ans. 10yd. 1qr. 0‡n. how many tickets can be had for a thousand pounds? Ans. 51 tickets. 2251. 10s.; what is the privateer worth? 24. How many yards of carpet 14 yard wide will be required to cover a floor 87 yards long, and 65 yards wide? 211. COMPOUND PROPORTION IN VULGAR FRACTIONS". EXAMPLES. 1 10 sum will pay the interest of 10L. for 1. If L. will pay the interest of 2: L. for of a year, what 12 First, 24L. = L. 5 This rule depends on the same reasons as Compound Proportion in whole numbers, and its operations are performed in the same manner, as far as relates to the general principle, differing only as a fractional differs from an integral process. • It will be found more convenient to place the fractions all in one line, with the divisors inverted; thus, 10 5 5 12 X X X X (which by cancelling, Art, 203, becomes) = 1 10 Explanation. Having stated the question, and marked the divisors, &c. the complex fraction which arises from the operation is reduced to a simple one, by multiplying 5 by 60, and 14 by 7; the result is next reduced to its lowest terms, and then to its proper quantity, which is the answer. 2. If 12 men earn 24L. in 4 days, what sum will 20 men earn in 124 days? 4. If 9 students spend 103L. in 18 days, how much will 20 students spend in 30 days? Ans. 391. 18s. 4d.. 5. If 100L. gain 44 L. interest in a year, what sum will gain 5 6 5 L. interest in of a year? Ans. 138t. 18s. 11d.. 6. A trench is required to be dug 100 feet in length, and 10 men have been employed 63 days in digging 49 feet of it; how long will 12 men require to finish the remainder? Ans. 5d. 8h. at 12 hours to the day. 7. If 13 feet of deal 9 inches wide cost 3s. 4d. what sum will pay for the floor of a room 15 feet long and 12 wide? Ans. 31. 38. 8d.. DECIMAL FRACTIONS. 212. If unity be supposed to be divided into 10, 100, 1000, &c. equal parts indefinitely, that is, into ten parts, or any multiple of ten, that which expresses any number of such parts less than that the unit is divided into, is called a Decimal Fraction. NOTATION OF DECIMALS. 213. It is well known, that the value of any figure is increased tenfold by removing it one place to the left hand of the place it occupies; thus 1 removed one place to the left (by placing a cipher to the right of it) becomes 10, or ten times 1; remove it one place farther, and it becomes 100, or ten times 10; remove it one place farther, and it becomes 1000, or ten times 100; and so on without end. Hence likewise it follows, that if any figure be removed in a contrary direction from the left hand towards the right, its value will be decreased tenfold at every step of such removal; thus, by removing 1 one place to the right, 1000 becomes 100, or one tenth of 1000; 100 becomes 10, or one tenth of 100; 10 becomes 1, or one tenth of 10; and, since I is the least whole number, this is as far as we can go in whole numbers. 214. But it is plain, that 1 may be removed step by step indefinitely towards the right; if therefore we put a point to the right hand of the place of units to distinguish it, and then place the 1 to the right hand of the point, thus .1, it is evident from what has been said that it will express one tenth of an unit, or 1 10' if we remove the 1 a place farther to the right, by interposing a cipher between the point and the 1, thus .01, this will 1 express one tenth of or one hundredth part of unity; if two , 10 ciphers be interposed, thus .001, it will express one tenth of 1 100 1 terposed, it will express one tenth of or one thousandth part of unity; if three ciphers be in 1000' sandth part of unity; and so on without end". or one ten thou 215. The point which is employed to mark the place of units is called the decimal mark; it separates between whole numbers and decimals; that part of any number which is to the left of the decimal mark being the whole number, and that to the right the decimal. 216. We have chosen the number 1 on account of its simplicity for the illustration of this doctrine; but the same equally holds true of all the other figures; thus, .2 expresses two tenths, or 3 2 ; .03, three hundredths, or -; .004, four thousandths, or 10 100 217. Hence it appears, that a decimal fraction is expressed by the mumerator only; the denominator, (being understood to consist of an unit with as many ciphers subjoined as the decimal has places,) is always omitted; thus, .5 denotes .0% denotes ', .0009 expresses 9 10000 5 10 7 &c.; by this simple artifice, fractions and 100 whole numbers are exhibited under one and the same form; a whole number and a decimal constituting together but one number, in all respects (the decimal mark excepted) similar to a whole number; so that if a whole number and a decimal be connected, they constitute one uniform scale, the steps of which, beginning at the right hand figure, regularly increase through both, up to the left hand figure in a tenfold proportion at each step, and decrease in the same proportion at every step through both, from the left hand figure to the right. 218. Another advantage follows from the similarity of deci mals to whole numbers, namely, that the modes of operation are P Hence it follows, that every cipher on the left hand in any decimal expres sion decreases the value of the decimal tenfold, and therefore such ciphers when they occur must always be put down, otherwise the value of the decimal will, for every cipher omitted, be increased tenfold, &c. more than its proper value. It appears also that ciphers on the right hand of a decimal do not alter its 5 50 500 value, for .5 = .50 = .500, &c. that is, are equal to each other; 10' 100 1000 1 this is evident, for each is equal to ; and the same is true in general. |