fraction differ from multiplying by a fraction? 27. When the divisor is less than a unit, what is the quotient compared with the dividend? 28. What is understood by a common denominator? the least common denominator? 29. How does it appear, that each given denominator must be a factor of the common denominator? 30. How is the common denominator to two or more fractions found? 31. What is understood by a multiple? by a common multiple? by the least common multiple?-What is the process of finding it? 32. How are fractions added and subtracted? 33. How is a fraction of a greater denomination reduced to one of a less? of a less to a greater? 34. How are fractions of a greater denomination reduced to integers of a less ? integers of a less denomination to the fraction of a greater? what part of a day? 4. Add lb. Troy to of an oz. Amount, 6 oz. 11 pwt. 16 gr. 5. How much is less? of? &? 144 -? 14 - 44 ? 6-43? 1981 of 3 of 2? Ans, to the last, 188. 6. From shilling take of a penny. Rem. 11 pwt. 3 grs. h. Rem. 2 d. 22 h. 20 m. of a yard of cloth? T 65. The price of unity, or 1, being given, to find the cost of any quantity, either less or more than unity, multiply the price by the quantity. On the other hand, the cost of any quantity, either less or more than unity, being given, to find the price of unity, or 1, divide the cost by the quantity. Ans. $ 1. If lb. of sugar cost of a shilling, what will 3 of a pound cost ?* This example will require two operations: first, as above, to find the price of 1 lb.; secondly, having found the price of 1 lb., to find the cost of of a pound. s. ÷ 11 (11 of 75 s. ¶ 57) s. the price of 1 lb. Then, s. X 13 (13 of 1 s. ¶ 53) = 7817 s. = 4 d. 34875 q., the Answer. 2912 95 If Or we may reason thus: first to find the price of 1 lb.: 1 lb. costs 75 S. 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. lb. costs s., it is evident lb. will cost of T S., and 13 lb. will cost 13 times as much, that is, the price of 1 lb. Then, 23 of 4 s. = 48 s., the cost of 2 of a pound. 2812 s. This process is called solving = 4 d. 34873 q., as before. 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. ? how much? What for 12 pounds? 2 of how Ans. to the last, $14. of much? 4. If 2 oz. of silver cost $224, what costs = yards? Ans. $4'269. oz. ? Ans. $84. Ans. $1'283. 5. If oz. costs $1, what costs 1 oz.? 6. If lb. less by costs 13 d., what costs 14 lb. less by } of 2 lb. ? Ans. 4£. 9 s. 9 d. 7. If yd. cost $7, what will 40 yds. cost? 8. If of a ship costs $251, what is 9. At 33. per cwt., what will 9 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. 2§ 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 12914 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 1 yard. Then, as many times 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 28 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 124 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 yd. wide will be required to make the same? Ans. 34 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. 24 = tons. If 7 horses consume 1 tons in 6 weeks, 1 horse will consume of of a ton in 6 weeks; and if a horse consume of a ton in 6 weeks, he will consume of = To of a ton in 1 week. 12 horses will consume 12 times 168 18 in 1 week, and in 8 weeks they will consume 8 times = 132 6 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 224 weeks when 3 persons more are added to the family? Ans. 280 gallons. 20. If 9. students spend 107. in 18 days, how much will 20 students spend in 30 days? Ans. 39£. 18 s. 4 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 cailed 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 1 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, 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 $785. 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 3d place. || || || || || || || Hundreds 2d place. 1st place. 1st place. 3d place. 4th place. 5th place. 6th place. 7th place. Tens. or Tenths Hundredths, Thousandths. Ten-Thousandths Hundred-Thousandths. 634 10000% = =25'6 3 6 3 4 0 0 0 0 0 0 0 ...... 634. Σ Numbers. Whole Decimal Parts. |