105. How many gills in 16 casks, each holding 354 gallons ? 106. If 3cwt. 3qr. 3lb. of pork cost $19.50, what will 193lbs. cost? 107. A boy being asked how many marbles he had, replied, “If I had as many more, and 1 as many more, I should have 45.” How many had he? 108. What is the cost of a load of wood that is 9ft. long, 4ft. wide, and 4.3ft. high, at 6.62, per cord ? 1109. A boy having some peaches, found, after giving away 2 more than }, that 1 more than 1 remained. How many had he at first? 110. Add 16%, 4, 171, 491, and 2176 111. i, j, 1, 1, 1 and 4 of a certain number, is equal to z of g of į of of 168588. What is the number? 112. Add 5.793cwt., 4 jocwt., 3cwt. 3qr. 15lb. and lqr. glb., and reduce the result to ounces. 113. 916.4*.0015-363.18=? 114. .0091 x .00037-1950000=? 115. Reduce to their lowest terms 112 2 32 236 138 828• 116. What is the value of a farm 189.5 rods long, and 150 rods wide, at $37.50 per acre ? 117. How many times will a half-seconds' pendulum tick in i of a year? 118. What is the value of £63.795 at į cent per far. thing? 119. What part of a solar year is 11 days 5 hours ? 120. What part of a rod is ift. 7.9in. ? 121. What part of a yard is 3qr. Ina. 1.5in. ? X 122. What part of £4 11s. 3d. is £1 58. 23d.? * 123. What part of 515 73 33 is 2 1 43 17 ? 124. What part of 4gal. 3qt. is 2qt. lpt. 3gi. ? 125. What part of 47. 3cwt. 1gr. is 2}T.?X 126. What part of 25cwt. is 1831b. ? 127. At $1.75 per square yard, how much must I give for carpeting a floor 15 3' long, and 12ft. wide ? 128. What will s of 153 yards of broadcloth cost, at of $7.25 per yard ? 129. If A. can do a piece of work in 6 days, B. can do it in 9 days, and C. can do it in 12 days, in what time can it be done if they all work together ? 130. What number is that, which, increased by 1, j, and of itself, makes 48 ? 131. In a certain orchard the trees bear apples, 1 bear peaches, } bear cherries, and the remaining 30 bear pears. How many trees are there in the orchard, and how many of each sort ? 132. What is the value of 5 of a ton ? 133. How many inches in of a mile? 134. Reduce 11oz. 7dwt. 3gr. to the fraction of a tb. 135. Reduce 4.790%, to the fraction of a pound Avoirdupois. 136. Reduce 38.75616. to the fraction of a ton. 137. Reduce 4.763in. to the fraction of a mile. 138. Reduce z of a qr. to the decimal of a ton. 139. Reduce 43 pence to the decimal of a pound. 140. Reduce 3.725 shillings to the fraction of a pound. 141. Reduce 59.63 minutes to the fraction of a day. 142. Reduce of a day to hours, minutes, &c. 143. If 25 pounds of sugar cost 119 cents, what will 54 pounds cost? 144. If 9 men can reap a field in 3.1 days, how long will it take two men to reap it? 145. How much rice, at 44 cents a pound, can be bought for $1.50 ? 146. If 291 gallons run from a cistern in an hour, how many pints will run out in 165 hours ? 147. What is the price of 33cut. of rice, at 4 cts. per pound? 148. At 122 cents a yard, what will be the cost of 13y. 3qr. 3na. of sheeting? 149. If of a bushel of wheat cost $0.75, what will be the price of 3 pecks? 150. How far will a pigeon fly in 3h. 45m. 45sec., at the rate of 25 miles an hour ? 151. How many cents would reach from Philadelphia to Washington, supposing the distance to be 132, miles, and the diameter of a cent 1.15 inches? 7 PROOF BY CASTING OUT 9's. The figure 9 has the curious property of exactly dividing any number, when the sum of its digits is divisible by 9, and on this property is founded the mode of proof known as casting out the nines. This is done by adding the figures which compose any number, and rejecting 9 from the sum as often as possible. Thus, if we wish to cast out the 9's from 7683217, we say 7 and 6 are 13 less 9 are 4 and 8 are 12 less 9 are 3 and 3 are 6 and 2 are 8 and 1 are 9 less 9 are 0 and 7 are 7. We then know that if 7683217 is divided by 9, there will be a remainder, 7: To prove any operation, we must reject the nines from each of the original numbers, perform the operation with the remainder, and reject the nines from the result, and also from the original result. If the work is right, the final remainders will be equal. The following examples will show the application of the rule: Addition. Rem. Subtraction. 168412 4 23849047 1+9 49037 5 9108736 7 684673 7 14740311 3 Final rem. 3-3 Division. 268 ) 3462 ( 12 8 268 233 251607 56 782 251607 536 167738 246 Rem. 19541477 Divd.-Rem.=3216 3 Final rem. 2 Div. Rem.X Quot. Rem.= 21 3 In the example of subtraction, the remainder of the minuend being less than that of the subtrahend, we increase the upper number by one of the rejected nines. The proof is not infallible, as, if any error is 9 or some multiple of 9, it will not be detected. There is, however, great advantage in the facility thus afforded for discovering errors arising from transposition TRANSPOSITION. The difference between any number, and the same num. ber transposed, is divisible by 9. Thus, 723—327, 1680 –0681, 231-123, are each exactly divisible by 9. Therefore, if we find in comparing the books of a counting. room or banking-house, that they do not agree, and the amount of their disagreement is divisible by 9, we know that it may have arisen from a transposition. We shall thus frequently be enabled to discover an error readily, which would otherwise have required a long and tedious examination. CHAPTER VIII. PRACTICE. cts. =1 6 33}: In PRACTICE, many questions arise that can be solved more readily than by adopting either of the foregoing rules. Most of the operations of business, in which compound numbers are concerned, may be abbreviated by first finding values for the highest denomination, and considering the lower denominations as aliquot parts of the higher. TABLES OF ALIQUOT PARTS. Of a dol. Of a £. Of a shi. Of a ton. Of a cwt. Of a year. s. d. £ d. s. cwt. gr. ton. gr. lb. cwt. mo. d. y. 50 10 10 = 2 8 5 1 = 4 25 5 3 4 16= 3 20 2 14= 2 12 16 83 2 1 15: 10 45 1 10= 1 2= 1 10 8 1 4 4 24= 8= 12 1 205 in 65 4= 16 9 18 2 24= 4 155 2= 44= 7 3 12= 7 15: Similar tables may be made to any required extent, but these are sufficient to show their application. = 12 20 1 60 1 80 50 120 RULE. Assume the price at some unit higher than the given price, and take aliquot parts of the assumed price for the answer. TO FIND THE VALUE OF A QUANTITY OF SEVERAL DENOMINATIONS.—Multiply the price by the integers of the highest denomination, and take aliquot parts for the lower denominations. EXAMPLES FOR THE BOARD. What is the value of 1lcwt. 3qr. 1776. of sugar, at £1 3s. 6d. per cut. ? £ d. 2qr.= cwt. 1 3 6 price of lcwt. 11 167b.= cut. 66 12 6 18 6 price of 11cwt. lqr.= 2qr. 10 9 price of 2qr. 101 price of lqr. ilb.= 116lb. 3 43 price of 161b. 2zł price of 116. 12 6 39 66 price of 11cwt. 3qr.17lb. What is the interest of $187.50 for 3yr. 9mo. 19d. at .07 per year? 187.50 .07 3mo. 15d. 3d. 1d. 39.375 int. for 3yr. 6.5621 int. for 6mo. 3.2811 int. for 3mo. ,5467 int. for 15d. .1093 int. for 3d. .03611 int, for id. 49.911,1 int. for 3yr. 9mo. 19d. |