RULE. Take such a part of the given quantity as the price is part of one dollar. NOTE. Since the shilling in most of the different currencies is some aliquot part of the dollar, this rule is of much practical use in making out bills and accounts where the prices of the items are given in State Currency, and the amounts are required in United States Money. EXAMPLES FOR PRACTICE. 1. What cost 568 pounds of butter at 25 cents a pound? Ans. $142. 2. A merchant sold 51 yards of prints at 163 cents per yard, 8 pieces of sheeting, each piece containing 33 yards, at 6 cents per yard, and received in payment 18 bushels of oats at 333 cents per bushel, and the balance in money; how much money did he receive? Ans. $19. 3. Required the cost of 28 dozen candles, at 1 shilling per dozen, New York currency. Ans. $3.50. 4. What cost 576 lbs. of beef at 10d. per pound, Pennsylvania currency? Ans. $64. 5. If a grocer in New York gain $7.875 on a hogshead of molasses containing 63 gallons, how much will he gain on 576 gallons at the same rate? Ans. $72. CASE X. 406. To find the cost of a quantity, when the quantity is a compound number, some part or all of which is an aliquot part of the unit of price. 1. What cost 5 bu. 3 pk. 4 qt. of cloverseed, at $3.50 per bu.? the price by 8, or the cost of 2 pk. by 4, or the cost of 1 pk. by 2, we 2. At £6 7s. 5d. Sterling per hhd., how much will 4 hhd. 9 gal. 3 qt. of West India Molasses cost? Dividing the cost of 3 qt. And the sum 9 gal. by 12, we have the cost of 11⁄2 of 36 qt. RULE. I. Multiply the price by the number of units of the denomination corresponding to the price. II. For the lower denominations, take aliquot parts of the price; the sum of the several results will be the entire cost. NOTE. This method is applicable in certain cases of multiplication, where one compound number is taken as many times as there are units and parts of a unit of a certain kind, in another compound number. This will be seen in the first example below. OPERATION. in 1 da. EXAMPLES FOR PRACTICE. 1. A chemist filtered 18 gal. 3 qt. 1 pt. of rain-water in 1 day; at the same rate how much could he filter in 4 da. 6 h. 30 min.? "4 1 “ 3 gi. “ 6 h. 7 66 ΤΣ "30 min. ANALYSIS. Multiplying the quantity filtered in 1 day by 4, we have the quantity filtered in 4 days. Dividing the quantity filtered in 1 day by 4, we have the quantity filtered in da. = = 6 30 min. And the sum of these several results h. Dividing the quantity filtered in 6 hours by 12, we have the quantity filtered in h. is the entire result required. 2. What will be the cost of 3 lb. 10 oz. 8 pwt. 5 gr. of gold at $15.46 per oz.? Ans. $717.52. 3. A man bought 5 cwt. 90 lb. of hay at $.56 per cwt.; what was the cost? Ans. $3.304. 4. What must be given for 3 bu. 1 pk. 3 qt. of cloverseed, at $4.48 per bushel? Ans. $14.98. 5. A gallon of distilled water weighs 8 lb. 5 oz. 6.74 dr.; required the weight of 5 gal. 3 qt. 1 pt. 3 gi. Ans. 49 lb. 12 oz. 5.73- dr. 6. At $17.50 an acre, what will 3 A. 1 R. 35.4 P. of land cost? 7. If an ounce of English standard gold be worth £3 17s. 101⁄2d., what will be the value of an ingot weighing 7 oz. 16 pwt. 18 gr.? Ans. £30 10s. 4.14375d. 8. If a comet move through an arc of 4° 36′ 40′′ in 1 day, how far will it move in 5 da. 15 h. 32 min. 55 sec.? 9. What will be the cost of 7 gal. 1 qt. 1 pt. 3 gi. of burning fluid, at 4s. 8d. per gallon, N. Y. currency? Ans. $4.35+. 10. What must be paid for 12 days' labor, at 5s. 3d. per day, New England currency? FOR DIVISION. CASE I. 407. When the divisor is an aliquot part of some higher unit. 1. Divide 260 by 31, and 1950 by 25. OPERATION. 26/0 19/50 78 78 ANALYSIS. Since 31 is † of 10, the next higher unit, we divide 260 by 10; and having used 3 times our true divisor, we obtain only of our true quotient. Multiplying the result, 26, by 3, we have 78, the true quotient. Again, since 25 is of 100, the next higher unit, we divide 1950 by 100; and having used 4 times our true divisor, the result, 19.5, is only of our true quotient. Multiplying 19.5 by 4, we have 78, the true quotient. Hence the RULE. I. Divide the given dividend by a unit of the order next higher than the divisor, by cutting off figures from the right. II. Take as many times this quotient as the divisor is contained times in the next higher unit. 408. When the right hand figure or figures of the divisor are an aliquot part of 10, 100, 1000, etc. OPERATION. 1875) 601387 4 4 7500) 2405548 4 4 310000) 96212192 3201387, Ans. ANALYSIS. Since 33 is of 100, we multiply both dividend and divisor by 3, (117, III), and we obtain a divisor the component factors of which are 100 and 37. We then divide after the manner of contracted division, (112). ANALYSIS. Multiplying both dividend and divisor by 4, we obtain a new divisor, 7500, having 2 ciphers on the right of it. Multiplying again by 4, we obtain a new divisor, 30000, having 4 ciphers on the right. Then dividing the new dividend by the new divisor, we ob tain 320 for a quotient, and 22192 for a remainder. As this remainder is a part of the new dividend, it must be 4 x 4 = 16 times the true remainder; we therefore divide it by 16, and write the result over the given divisor, 1875, and annes the fraction thus formed to the integers of the quotient. From these illustrations we derive the following RULE. I. Multiply both dividend and divisor by a number or numbers that will produce for a new divisor a number ending in a cipher or ciphers. II. Divide the new dividend by the new divisor. NOTE. If the divisor be a whole number, or a finite decimal, the multiplier will be 2, 4, 5, or 8, or some multiple of one of these numbers. 409. Ratio is the relation of two like numbers with respect o comparative value. NOTE. There are two methods of comparing numbers with respect to value; st, by subtracting one from the other; 2d, by dividing one by the other. The elation expressed by the difference is sometimes called Arithmetical Ratio, and The relation expressed by the quotient, Geometrical Ratio. 410. When one number is compared with another, as 4 with 2, by means of division, thus, 12÷43, the quotient, 3, shows he relative value of the dividend when the divisor is considered s a unit or standard. The ratio in this case shows that 12 is 3 imes 4; that is, if 4 be regarded as a unit, 12 will be 3 units, or he relation of 4 to 12 is that of 1 to 3. 411. Ratio is indicated in two ways: 1st. By placing two points between the two numbers compared, riting the divisor before and the dividend after the points. hus, the ratio of 8 to 24 is written 8: 24; the ratio of 7 to 5 is ritten 7 : 5. |