be balanced by what quantity at 7? How can this fraction be got rid of ? Why must there be 1 lb. of the 6 cent coffee ? Ans. Because it is balanced by of - ? Why 3 lbs. of 10 cent coffee ? Ans. Because they are balanced by of Why should there be 3 of the 7 cent ? Why 2 of the 12 ? 114. How many pounds did the mixture of the last example contain? If a bag, holding 36 lbs., then, were required to be filled of such a mixture, how many times would the mixing be repeated? Then how many pounds would be required of each kind of coffee? Prove as in last example. 115. If a merchant, wishing a mixture like that named in Ex. 113, should put in 16 lbs. of the 6 cent coffee, how many lbs. would be required of each of the others ? Prove. 116. Again : if he wished to use 16 lbs. of the 12 cent coffee, how many times must the proportions of the others be repeated to make a similar mixture ? How many pounds would such a mixture contain ? Prove. 117. A farmer had several kinds of provender which he wished to mix for his cattle and horses, so as to form a compound worth 50 cts. a bushel, namely, corn meal, worth 75 cts. a bushel ; rye, worth 55; oats, 40; shorts, 25 ; and wheat bran, 15 cents a bushel. What proportions of each sort should be used, and how many bushels would the smallest quantity of the mixture contain, excluding fractions of a bushel ? Ans. to the last question, 13 bushels. Suggestive Questions.—In what proportions may we mix the meal at 75 cts. and that at 25 cts. to form a compound worth 50 cts. ? Mark the quantities 1 each, then. In what proportions can those of 55 and 40 be so mixed ? Enter them, then. In what proportions those of 55 and 15? Enter them, then, separating the two numbers at 55 by the sign of addition, + Why? Prove as in example 113 above. 118. If 208 bushels of the above compound were wanted, how many times would the proportion of each sort be repeated, and how many bushels would there be of each ? Prove. Prove. 119. If 5} bushels of the corn meal were used, how much would be required of each of the others ? Prove. 120. A grocer has four sorts of sugar, worth 4 cts., 5 cts., 7 cts., and 8 cts. a pound. He would make a mixture of 200 lbs., worth 6 cts. a pound? What quantity must be taken of each sort? 121. A goldsmith has four sorts of gold, namely, of 22 carats fine, of 20 carats fine, of 18 carats fine, and of 15 carats fine. He would make a mixture of 48 oz. of 17 carats fine. How many oz. of each sort must he take? Prove. 122. Afterwards, of the same material, he wished to make a mixture of the same fineness, containing 4 oz. of 20 carats fine. How many ounces must he take of each of the other sorts ? Prove. 123. A rectangular field was 16 rods long and 12 wide. How many square rods did it contain ? 124. What is the width of a rectangular field containing 192 square rods, whose length is 16 rods? 125. There are 192 rods in a rectangular field, whose width is 12 rods. What is its length ? 126. There is a square field whose sides are 16 rods long. How many square rods does it contain ? 127. What is the length of a square field containing 256 square rods? 128. The sides of one of the square fields of a farm is 40 rods long, and those of another 80. How many times is the one larger than the other ? 129. There are two square fields in a farm, one of which is 40 rods long; the other is 4 times the size. What is the length of its sides? 130. The inside of a box is 2 feet every way. How many cubical feet does it contain ? 131. The contents of a box with equal sides are 8 cubical feet. What are its length, width, and depth inside ? 132. A farmer erected a stable 50 feet long by 25 feet wide. The height of the gable was 8 feet. The eaves projected a foot over each side of the building, and the roof was 2 feet longer than the frame, so as to project a foot over each gable. How many thousand shingles would be required for the roof, if one thousand shingles cover 10 feet square ? Ans. 18 nearly. 133. One man exchanged with a broker £4 10s. 10d. sterling for 11 crowns and 7 dollars; and another man, at the same rate, £1 15s. for 4 crowns and 3 dollars. How much were the crown and dollar severally valued at ? Prove by trial. Suggestive Questions.—What is 3 times the amount of the first exchange ? 7 times the amount of the 2d? What is the difference between the exchanges thus increased ? What, then, is the value of a crown? Of a dollar ? In what respects does this operation differ from bringing fractions to the same denomination ? 134. Required two such numbers that if of the first be added to of the second, the sum shall be 66; and if á of the first be added to of the second, the sum shall be 60. Prove by trial. 135. If the greater of two numbers be divided by the less, the quotient is 6, and the sum of the two numbers is 252. What are the numbers ? Prove by trial. 136. A gentleman gave $4350 for a house-lot, the land being valued at $2 per foot. If it had been 6 feet wider, it would have cost $5394. What were the length and breadth of the lot? Prove by trial. 137. A boy bought at one time 5 apples, 6 pears, and 4 oranges, for 48 cents; at another time, 3 apples, 4 pears, and 5 oranges, for 43 cents; and again, 2 apples, 3 pears, and 6 oranges, for 43 cents, all at the same rate. What did he pay for each kind of fruit? Prove by trial. 138. There were 5 Sundays in the month of February in 1852. In what year will this occur again; that is, when will the first day of February fall on a Sunday in a bissextile or leap year? Ans. In 1880. Suggestive Questions.—How many days of the week does the advance from one bissextile to another? What is the smallest number of fives exactly divisible by 7 ? Then how many bissextiles must elapse till 5 Sundays again occur in February ? year Equation of Payments. 139. A man bought a farm for $2000, one half of which was to be paid in two years, and the remainder in 4 years; that is, the purchaser was to have the use of $1000 of the purchase money for 2 years, and the use of the remaining $1000 for 4 years. At what time may the whole be paid at once without loss to either of the parties ? years Use of 1000 for 2 1000 " 4 =2000 for one year. =4000 for one year. paid 2000 =$6000 for one year. Suggestive Questions.—How long must he keep the $2000 so as to balance the use of $6000 for one year? By what process can this be ascertained ? By addition, subtraction, multiplication, or division ? 140. A man owed his neighbor $300, which he engaged to pay as follows : $50 in 2 months, 100 in 4 months, and $150 in 6 months. When may the whole be paid at once without loss to either party? $300 =$1400 for 1 Suggestive Question.-$1400 for 1 month=$300 for how many months ? 141. A friend lent me $400 for three months. How long should I lend him $100 to balance the favor ? Ans. 12 months. 142. A man bought a piece of property for $600, and agreed to pay $100 in 2 months, 200 in 5 months, and the rest in 8 months ? What would be the proper time to make one payment of the whole ? Ans. 6 months. 143. What is the mean time for the settlement of a debt of $800, contracted to be paid as follows : $200 in 3 months, of the remainder in 4 months, į of what then remains in months, and the rest in 6 months ? Ans. 4 months. 144. One merchant owes another $800, payable in 6 months, but wishes to pay him $200 of the debt in 2 months. How long should the time of payment of the remainder be suspended to balance the favor ? Ans. 11 months. 145. A country merchant makes purchases from a merchant in Boston, on a credit of 6 months, as follows: $1500 on May 1, $400 on June 1, $500 on July 1, and $300 on Aug. 1. \ What is the mean time for the whole from Aug. 1 ? Ans. 4 months. 146. What is the mean length of the following pieces of cloth : No. 1, 30 yds.; No. 2, 28 yds.; No. 3, 27 yds.; No. 4, 29 yds. ; No. 5, 32 yds.; No. 6, 25 yds.; No. 7, 25 yds. Ans. 28 yards. 147. A man on horseback ravelled the following distances : the first day, 30 miles; the second day, 34 miles ; the third day, 36 miles; and the fourth day, 42 miles. How many miles did he average a day? Ans. 35) miles. 148. Four men are engaged in building a wall measuring 820 cubic feet. The first can build 9 cubic feet in 4 days; the second, 10 cubic feet in 4 days; the third, 8 cubic feet in 6 days; and the fourth, 7 cubic feet in 3 days. How many days will be necessary to complete the whole wall, when all work together? 149. A merchant had 32 tons of plaster for sale. On examining his sale books at the end of a week, he found that there remained 8 tons more than he had sold. How many tons were sold ? Prove. 150. Three farmers bought a pasture jointly, consisting of 140 acres. On dividing it, it was agreed that A's share should be to B's as 6 to 11, and that C should have 4 acres more than A and B together. What is the share of each ? Prove. 151. A man being asked how much money he had in his pocket, answered that } and } of it amounted to $3-20. How much had he? Prove. 152. A traveller, being asked what o'clock it was, replied that it was between 3 and 4. But a more particular answer being requested, said that the hour and minute hands were exactly together. What was the time? Ans. 164 min. past 3. Suggestive Questions. How far are the hands apart at 3 o'clock? In what time will the minute overtake the hour hand ? 153. John sets out on a journey, and travels at the rate of 5 miles an hour. He travels 8 hours the first day, and the next morning a friend sets out after him at the rate of 7 miles an hour. If both start at the same hour in the morning, and travel the same number of hours in a day, how far must the friend travel before he overtakes John? |