154. How many miles an hour must William travel for 20 hours, in order to overtake John, who is 40 miles ahead, and is travelling at the rate of 5 miles an hour?

155. A company sets out for California by land, and travels at the rate of 30 miles a day. Three days afterwards a man undertook to overtake them, and accomplished his purpose in 10 days. At what rate per day did the man travel?

156. A man bound for California by land, on his arrival at the usual starting point, learns that a company was 3 days ahead of him, with the intention of travelling 30 miles a day. He set out after them, and travels at the rate of 39 miles a day. In how many days will he overtake them?

157. The velocity of sound through the air is found to be 1142 feet per second of time; and, in a healthy person, the number of pulsations is, say 70 in a minute. Now, between the time of observing a flash of lightning and hearing the explosion of the thunder I counted 20 pulsations. Required the number of feet which sound moves in a minute; the number in one pulsation; and the distance of the cloud in miles.

158. A gentleman counted 20 pulsations between the sight of a flash of lightning and the arrival of the sound. By calculation he found the distance of the cloud to be 3 miles, 226 rods, and 84 feet; allowing the pulsations to be at the rate of 70 in a minute, how many feet did the sound travel in a second?

159. A lunar month, in which the moon makes a complete apparent revolution round the earth, is, say 291 days. Supposing the moon's motion to be uniform, what is the apparent distance of the sun and moon in degrees, &c., in one day after the change? in 12 days 6 hours after the change? in 19 days? [Take notice that the moon again begins to approach the sun. after 14 days. Why?] Ans. to the last question, 128°.

160. The earth, being 360° in circumference, turns on its axis, say in 24 hours. How many miles are the inhabitants at the equator carried by that movement in one minute, a degree there being 69 miles? Ans. 17 miles.

161. If an ingot of gold, weighing 9 lb. 9 oz. 12 dwt., be worth $1128 96, what is it worth per grain?

162. If gold be worth 2 cents per grain, what is the value of 9 lb. 9 oz. 12 dwt.?

163. A saves of his income; but B, who has the same

salary, by living at double the expense, runs
year. How much has each per annum?
164. Required the sum of which 1, 1, and

behind $120 a
Ans. $360.
make $94.
Ans. $120.

B's age equals A's and half of
What are their several ages?

165. A is 20 years of age. C's; and C's equals them both. If C's age is considered (1); B's will be ()+20; and A's 20; but C's equals A's and B's; how much, then, is the other half of C's age equal to ? Prove by trial.

166. The head of a fish is 9 inches long, and its tail is as long as its head and half the body, and the length of the body equals those of the head and tail. What is its whole length? Prove by trial.

167. A laborer hired for 42 days upon this condition, that he should receive 80 cents for every day he worked, and forfeit 40 cents for every day he was idle. On settlement it was found that nothing was due him. How many days did he work? Perform this mentally by division. Ans. 14 days. 168. Two foot travellers sat down to dinner, one having 5 small loaves of bread, the other 3 of the same size. A third traveller coming along proposed to join them on condition of paying for his share of the bread. The whole was eaten, and the third traveller paid 8 cents. How should this money be shared between the two owners of the bread ?

Ans. 7 to the first, 1 to the second. 169. Tell the difference between 365 times 421 and 375 times the same number, by a mere glance of the eye, and prove by multiplication and subtraction.

170. Tell the difference between 235 times 364 and 240 times the same number, by halving a certain number mentally, and prove as in last exercise.

171. Tell the difference between 328 times 465 and 333 times the same number, by a similar process, and prove.

172. Tell the difference between 425 times 326 and 475 times the same number, by halving a certain number mentally, and prove.

173. Tell the difference between 354 times 248 and 379 times the same number, by dividing a certain number mentally by, and prove.

174. Tell the difference between 432 times 368 and 4571 times the same number, by a similar mental process, and prove.

175. Tell the difference between 216 times 344 and 241 times the same number, by a mental process, and prove.

176. Tell the difference between 284 times 275 and 334 times the same number, by a mental process, and prove.

177. Tell the difference between 341 times 432 and 367 times the same number, by a mental process, and prove.

178. Tell the difference between 286 times 76 and 310 times the same number, by a mental process, and prove.

179. Tell the difference between 525 times 340 and 552 times the same number, by a mental process, and prove.

180. Find, by inspection, the product of 244 by 24'9, by division and subtraction, and prove by multiplication on the slate.

Suggestive Questions.- If 244 be taken 25 times in place. of 24.9 times, how many times is it taken too much? What is of 2.44?

181. Find, by inspection, the product of 5 by 23'84, by division, and prove by multiplication.

Suggestive Question.-Why are there only two decimal places in the product, when there are three in the two factors? 182. Find, by inspection, the square of 49 by division and subtraction, and prove by multiplication.

183. Find, by inspection, the square of 26, by division and addition, and prove by multiplication.

184. Find, by inspection, the squares of 24, 16, 14, by processes similar to those in the last two problems, and prove by multiplication.

185. Find, by inspection, the square of 75, by division and subtraction of the quotient, and prove by multiplication.

186. Find, by inspection, the square of 3'1, by multiplication and addition, and prove by multiplication only.

187. Find, by inspection, the product of 324 by 49, by division and subtraction, and prove by multiplication.

188. Find, by inspection, the product of 64 by 43, by division and subtraction, and prove by multiplication. 189. Find, by inspection, the product of 72 by 51, by division and addition, and prove by multiplication.

190. Find the product of 16 by 12 by division.

Solution. 8)1600=16×100.

Suggestive Questions.-How many times 12 in 25? How many times 25 in 100? Then how many times 12 in 100? If, then, we multiply by 100 in place of 12, how many times. will the product be too large? Why, then, should we divide by 8?

191. Find the product of 24 by 37, by division and sub

192. Find the product of 64 by 621 (=50+12) by division and addition.

Solution. 2)6400=64×100

4)3200-64X 50
800-64× 12

4000 64× 621

193. Find the product of the same numbers by division and multiplication.

Solution. 8)6400=64X100

800-64×121

5

4000-64×62

194. Find the product of 328 by 874 by division and subtraction.

Solution. 8)32800=328×100

4100=328X 121

28700-328 871

195. Find the product of 176 by 1121 by division and addition.

196. Find the product of 3724 by 125, by division and addition.

199. Find the above 9 products by inspection, without the aid of pen or pencil.

200. Find the following products severally by inspection, and prove each by calculation: 4572 by 121; 3968 by 25; 8128 by 37; 1437 by 50; 2456 by 621; 7415 by 75; 9284 by 87; 667 8 by 112; 1529 by 125; 7924 by137 ; 6414 by 162; 2256 by 175; 6328 by 187; 4428 by 212; 7732 by 225; 3648 by 487.

201. The property of John and Thomas was estimated by the assessors at $18,600, and John was twice as rich as Thomas. What was the capital of each?

Ans. John's $12,400, Thomas' $6200. Suggestive Questions.-If Thomas's property was considered one share, how many shares had John? Into how many parts, then, should the property be divided?

202. The sum of $2500 is to be divided between two persons, the one to have $4 as often as the other $1. How much would each receive? Ans. $2000 and $500.

203. A man sold a farm for $3960, and received payment in bank notes and gold, there being 4 times as much in paper as in gold. How much was there of each?

Ans. $720 in paper, $3240 in gold. Suggestion.-Get rid of the fraction by multiplication.

204. Divide 1394 into two such parts that the one may be contained in the other 3 times. Ans. 328 and 1066.