105. How many gills in 16 casks, each holding 354 gallons?

106. If 3cwt. 3qr. 3lb. of pork cost $19.50, what will 19 lbs. cost?

107. A boy being asked how many marbles he had, replied, "If I had as many more, and 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 41ft. high, at 6.621 per cord?

109. A boy having some peaches, found, after giving away 2 more than 1, that 1 more than many had he at first?

remained. How

110. Add 165, 43, 171, 491, and 21.

111. 1 1

1 1

2' 3' 4' 9

to of of of

and of a certain number, is equal of 168583. What is the number? 112. Add 5.793cwt., 47cwt., 3cwt. 3qr. 15lb. and 1qr. 9lb., and reduce the result to ounces.

113. 916.4 x .0015÷363.18=?

114. .0091 x .00037÷1950000=?

646 112. 232

115. Reduce to their lowest terms 38; 32; 343;

138

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 of a year?

118. What is the value of £63.795 at cent per farthing?

119. What part of a solar year is 11 days 5 hours? 120. What part of a rod is 1ft. 7.9in.?

121. What part of a yard is 3qr. 1na. 1.5in.?

X 122. What part of £4 11s. 3d. is £1 5s. 23d.? × 123. What part of 5 73 33 is 2 43 19 ?

124. What part of 4gal. 3qt. is 2qt. 1pt. 3gi.? 125. What part of 4T. 3cwt. 1qr. is 23T.?

126. What part of 25cwt. is 183lb.?

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 of 15 yards of broadcloth cost, at å 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, 3, and of itself, makes 48 ?

131. In a certain orchard the trees bear apples, 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?

16

134. Reduce 11oz. 7dwt. 3gr. to the fraction of a .

135. Reduce 4.79oz. to the fraction of a pound Avoirdupois.

136. Reduce 38.756lb. to the fraction of a ton.

137. Reduce 4.763in. to the fraction of a mile.

138. Reduce of a qr. to the decimal of a ton. 139. Reduce 47 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 118 cents, what will 51 pounds cost?

144. If 9 men can reap a field in 3 days, how long I will it take two men to reap it?

145. How much rice, at 43 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 33cwt. of rice, at 41⁄2cts. per pound?

148. At 12 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?

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:

21

3

Divd.-Rem.=3216

Final rem. 2 = 2 Div. Rem. Quot. Rem.

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 number 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 countingroom 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.

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.

Similar tables may be made to any required extent, but these are sufficient to show their application.

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 11cwt. 3qr. 17lb. of sugar, at £1 3s. 6d. per cwt.?

What is the interest of $187.50 for 3yr. 9mo. 19d. at