1. If lb. of sugar cost of a shilling, what will of a pound cost ?* This example will require two operations: first, as above, to find the price of 1 lb.; secondly, having of 1 lb., to find the cost of 3 of a pound. ofs. I 57) = s. the price of 1 lb. (2 of s. π 53) = 231 s. Answer. 65 found the price 5 s. ÷ 11 (11 Then, s. X 4 d. 3483 q., the 971 165 Or we may reason thus: first to find the price of 1 lb. : lb. costs $. If we knew what lb. would cost, we might repeat this 13 times, and the result would be the price of 1 lb. is 11 parts. If 13 lb. costs 7 s., it is evident lb. will cost of T3 S., and 13 lb. will cost 13 times as much, that is, the price of 1 lb. Then, 2 of s. = 8 s., the cost of of a pound. 3 s. = 4 d. 34873 q., as before. This process is called solving the question by analysis. s. After the same manner let the pupil solve the following questions: 2. If 7 lb. of sugar cost of a dollar, what is that a pound?of= how much? What is it for 4 lb. ? of how much? What for 12 pounds? much? = of how Ans. to the last, $12. yards? Ans. $4'269. 3. If 6 yds. of cloth cost $3, what cost 9 4. If 2 oz. of silver cost $2'24, what costs oz. ? Ans. $84. 5. If oz. costs $11, what costs 1 oz. ? Ans. $1'283. 6. If lb. less by costs 13 d., what costs 14 lb. less by * of 2 lb. ? Ans. 4£. 9 s. 923 d. 7. If yd. cost $3, what will 40 yds. cost? 8. If of a ship costs $251, what is 9. At 3§£. per cwt., what will 93 lb. Ans. $59'062 +. of her worth? Ans. $53'785 +. cost? Ans. 6 s. 3 d. sold of his share Ans. $1794'375. 10. A merchant, owning of a vessel, for $957; what was the vessel worth? 11. If yds. cost £., what will of an ell Eng. cost? Ans. 17 s. 1 d. 24 q. This and the following are examples usually referred to the rule Proportion. or Rule of Three. See ¶ 95 ex. 35. 12. A merchant bought a number of bales of velvet, each containing 12917 yards, at the rate of $7 for 5 yards, and sold them out at the rate of $11 for 7 yards, and gained $200 by the bargain; how many bales were there? First find for what he sold 5 yards; then what he gained on 5 yards-what he gained on I yard. Then, as many times 1 as the sum gained on 1 yd. is contained in $200, so many yards there must have been. Having found the number of yards, reduce them to bales. Ans. 9 bales. 13. If a staff, 53 ft. in length, cast a shadow of 6 feet, how high is that steeple whose shadow measures 153 feet? Ans. 144 feet. 14. If 16 men finish a piece of work in 283 days, how *long will it take 12 men to do the same work? First find how long it would take 1 man to do it; then 12 men will do it in of that time. Ans. 377 days. 15. How many pieces of merchandise, at 20 s. apiece, must be given for 240 pieces, at 12 s. apiece? Ans. 149. 16. How many yards of bocking that is 14 yd. wide will be sufficient to line 20 yds. of camlet that is of a yard wide? First find the contents of the camlet in square measure; then it will be easy to find how many yards in length of bocking that is 14 yd. wide it will take to make the same quantity. Ans. 12 yards of camlet. 17. If 14 yd. in breadth require 20 yds. in length to make a cloak, what in length that is 2 yd. wide will be required to make the same? Ans. 344 yds. 18. If 7 horses consume 2 tons of hay in 6 weeks, how many tons will 12 horses consume in 8 weeks? If we knew how much 1 horse consumed in 1 week, it would be easy to find how much 12 horses would consume in 8 weeks. 23 tons. If 7 horses consume 1 tons in 6 weeks, 1 horse will consume of of a ton in 6 weeks; and weeks, he will consume 12 horses will consume if a horse consume of a ton in 6 of of a ton in 1 week. 12 times 18 in 1 week, and in 8 weeks they will consume 8 times 183264 tons, Ans. 19. A man with his family, which in all were 5 persons, did usually drink 74 gallons of cider in 1 week; how much will they drink in 22 weeks when 3 persons more are added to the family? Ans. 2804 gallons. X 20. If 9 students spend 107£. in 18 days, how much will 20 students spend in 30 days? Ans. 39£. 18 s. 43o d. DECIMAL FRACTIONS. 66. We have seen, that an individual thing or number may be divided into any number of equal parts, and that these parts will be called halves, thirds, fourths, fifths, sixths, &c., according to the number of parts into which the thing or number may be divided; and that each of these parts may be again divided into any other number of equal parts, and so on. Such are called common, or vulgar fractions. Their denominators are not uniform, but vary with every varying division of a unit. It is this circumstance which occasions the chief difficulty in the operations to be performed on them; for when numbers are divided into different kinds or parts, they cannot be so easily compared. This difficulty led to the invention of decimal fractions, in which an individual thing, or number, is supposed to be divided first into ten equal parts, which will be tenths; and each of these parts to be again divided into ten other equal parts, which will be hundredths; and each of these parts to be still further divided into ten other equal parts, which will be thousandths; and so on. Such are called decimal fractions, (from the Latin word decem, which signifies ten,) because they increase and decrease, in a tenfold proportion, in the same manner as whole numbers. ¶ 67. In this way of dividing a unit, it is evident, that the denominator to a decimal fraction will always be 10, 100, 1000, or i with a number of ciphers annexed; consequently, the denominator to a decimal fraction need not be expressed, for the numerator only, written with a point before it () called the separatrix, is sufficient of itself to express the true value. Thus, fo are written '6. The denominator to a decimal fraction, although not expressed, is always understood, and is 1 with as many ciphers annexed as there are places in the numerator. Thus, 3765 is a decimal consisting of four places; consequently, 1 with four ciphers annexed (10000) is its proper denominator. Any decimal may be expressed in the form of a com mon fraction by writing under it its proper denominator. Thus, '3765 expressed in the form of a common fraction, is 378. 3765 0000 When whole numbers and decimals are expressed together, in the same number, it is called a mixed number. Thus, 25'63 is a mixed number, 25', or all the figures on the left hand of the decimal point, being whole numbers, and '63, or all the figures on the right hand of the decimal point, being decimals. The names of the places to ten-millionths, and, generally, how to read or write decimal fractions, may be seen from the following TABLE, 18853 1880 τόσ 71008000 = 3d place. || || || || || || Hundreds. 2d place. ∞ Tens. or Tenths read 5 Tenths. 6 3 4 0 0 0 0 0 0 0........ 634. Hundredths. Ten-Thousandths Hundred-Thousandths. Ten-Millionths. Numbers. Whole Decimal Parts. From the table it appears, that the first figure on the right hand of the decimal point signifies so many tenth parts of a unit; the second figure, so many hundredth parts of a unit; the third figure, so many thousandth parts of a unit, &c. It takes 10 thousandths to make 1 hundredth, 10 hundredths to make 1 tenth, and 10 tenths to make 1 unit, in the same manner as it takes 10 units to make 1 ten, 10 tens to make 1 hundred, &c. Consequently, we may regard unity as a starting point, from whence whole numbers proceed, continually increasing in a tenfold proportion towards the left hand, and decimals continually decreasing, in the same proportion, towards the right hand. But as decimals decrease towards the right hand, it follows of course, that they increase towards the left hand, in the same manner as whole numbers. 68. The value of every figure is determined by its place from units. Consequently, ciphers placed at the right hand of decimals do not alter their value, since every significant figure continues to possess the same place from unity. Thus, 5, 50, 500 are all of the same value, each being equal to or But every cipher, placed at the left hand of decimal fractions, diminishes them tenfold, by removing the significant figures further from unity, and consequently making each part ten times as small. Thus, '5, '05, '005, are of different value, 5 being equal to, or 1; '05 being equal to τẳʊ, or 2; and '005 being equal to Tobo, or goo. Decimal fractions, having different denominators, are readily reduced to a common denominator, by annexing ciphers until they are equal in number of places. Thus, '5, '06, '234 may be reduced to '500, ‘060, 234, each of which has 1000 for a common denominator. 69. Decimals are read in the same manner as whole numbers, giving the name of the lowest denomination, or right hand figure, to the whole. Thus, '6853 (the lowest denomination, or right hand figure, being ten-thousandths) is read, 6853 ten-thousandths. Any whole number may evidently be reduced to decimal parts, that is, to tenths, hundredths, thousandths, &c. by annexing ciphers. Thus, 25 is 250 tenths, 2500 hundredths, 25000 thousandths, &c. Consequently, any mixed number |