2. John has $.50 and George has $5. George's money is how many times John's? In the four classes there are ten types of problems; show that the examples given illustrate the types. 208. Method of Reasoning.-In arithmetic, as in all other subjects, we reason from the known to the related unknown; but there are other principles especially applicable in this subject. Since unity is the basis of all numbers, in analysis the following principles are fundamental: 1. Reason from one to many. 2. Reason from many to one. 209. These principles may now be illustrated by EXAMPLES 1. If 1 top costs 8 cents, what will 5 tops cost? SOLUTION : 1. The cost of 1 top = 8 cents. 2. The cost of 5 tops 5 x 8 cents = 40 cents. .. if 1 top cost 8 cents, 5 tops will cost 40 cents. In this we reason from one to many. = 2. If 8 books cost $12, what will 1 book cost? SOLUTION: 1. The cost of 8 books = $12. 2. The cost of 1 book 3 of $12 = $1.50. .. if 8 books cost $12, 1 book will cost $1.50. In this we reason from many to one. 3. If 15 pencils cost 90 cents, what will 8 pencils cost? SOLUTION: 1. The cost of 15 pencils = 90 cents. 2. The cost of 1 pencil = 15 of 90 cents = 6 cents. 3. The cost of 8 pencils 8 x 6 cents : 48 cents. .. if 15 pencils cost 90 cents, 8 pencils will cost 48 cents. In this the two principles are combined. = Exercise XXIV Using analysis, solve the following: 2. A merchant, owning / of a ship, sells of his share for $16800: at this rate what is the value of the whole ship? 3. A man pays $350 a year for house rent, which is 1 of his income: what is his income? 4. A school enrolls 208 boys, and is of the pupils are girls: how many pupils in the school? 5. If { in. on a map corresponds to 7 mi. of a country, what distance on the map represents 20 mi.? 6. A certain sum of money gains of itself, the total amount then being $728; what is the sum gained? 7. A man agreed to work 20 days for $3 a day and his board, and to pay $1 a day for board when idle; at the end of the time he received $44. How many days was he idle? 8. If a box 7 ft. long, 5 ft. wide, and 4 ft. deep holds 112 bushels, how deep is another box which is 20 ft. long and 9 ft. wide, and holds 864 bushels ? NOTE. —See second solution of problem 1 under 186. 9. A and B could have done a work in 15 days, but after working together 6 days, B was left to finish it, which he did in 30 days: in what time could A have finished it, if B had left at the end of the 6 days? 10. A ship has water in it and water is running in through a leak at a uniform rate. If 60 sailors can bail out the water in 8 hours, and 90 sailors can do it in 5 hours, in what time can 50 sailors do the work? 66 450 SOLUTION: If 1 sailor in 1 hr. can do 1 unit of work, then 60 sailors 8 480 units and 90 5 The amount of water running in during 8 – 5, or 3 hours, = 480 – 450, or 30 units of work. Therefore, the flow in 1 hour 10 units of work, or the work of 10 sailors. During the 8 hours that the 60 sailors worked, 10 sailors were keeping out the flow, while the remaining 50 were emptying the ship. Hence, the amount of water in the ship when the work begins is represented by 50 ~ 8, or 400 units of work. Of the 50 sailors, 10 will keep out the flow, and the number of hours required for the remaining 40 to empty the ship will be the number of times that 40 units of work is contained in 400 units of work ; that is, 10 hours. 11. There is coal now on the dock, and coal is running on also from a shoot, at a uniform rate. Six men can clear the dock in one hour, but 11 men can clear it in 20 minutes: how long will it take 4 men?—R. N. H., p. 406. 12. If } of a number is 35, how much is of it? 13. Three boys had 169 apples, which they shared in the ratio of 1, }, and 1. How many did each boy receive? 14. The 9th term of a geometric series is 137781, and the 13th term 11160261: what is the 4th term? 15. Express as common fractions: .963; .378; .2045. 16. Change 52364, to the decimal scale. 17. Express in the quaternary scale, 5439. 18. A room is 21 ft. long, 16 ft. wide, and 12 ft. high. How far is it from an upper corner to the opposite lower corner? 19. At what time between 3 and 4 o'clock will the minute hand and the hour hand be together? PRINCIPLES USED IN TIME PROBLEMS 1. The distance moved by the minute hand (min. h.) = 12 times the distance moved by the hour hand (hr. h.). 2. The distance gained by the min. h. 11 times the distance moved by the hr. h. Therefore, 3. Every 12 spaces moved by the min. h. = 11 spaces gained. 4. 1 space gained by min. h. spaces moved by it. SOLUTION: 1. Since 1 space gained tř spaces moved, 2. .. 15 spaces gained 15 x it spaces moved moved. .. the hands will be together at 164 min. past 3 o'clock. 16 f spaces 20. At what time between 2 and 3 o'clock will the hands form a right angle? SOLUTION: In this problem the min. h. must gain 25 spaces to form a right angle with the hr. h. 1. Since 1 space gained = ff spaces moved, 27 i spaces moved. ., the hands will form a right angle at 27 ( min. past 2 o'clock. 21. At what time between 3 and 4 o'clock will the min. h. be opposite the hr. h.? 22. At what time between 3 and 4 o'clock is 3 midway between the two hands? SUGGESTION.—The sum of the distances moved by the two hands 15 spaces. The min. h. dist. t% of 15 spaces 131} spaces. 23. At what time between 6 and 7 o'clock will 6 be midway between the two hands? 24. At what time between 5 and 6 o'clock are the two hands perpendicular to each other? 25. What time is it when šof the time past noon equals of the time till midnight? 26. A cistern containing 480 gallons can be emptied by two pipes in 4 and 5 minutes, respectively. If both pipes are left open, in what time will they empty the cistern? 27. A reservoir has 3 pipes; the first can fill it in 10 days, the second in 16 days, and the third can empty it in 20 days. In what time will the reservoir be filled if they are all allowed to run at the same time? 28. The rate of the current of a river is 4 miles an hour. How far up the river can a boat go and return in 12 hours, if the boat's rate of travel in still water is 8 miles an hour? SOLUTION: 1. Boat's own rate of travel up stream = 8 mi. an hr. 2. Distance driven back by the current = 4 3. Therefore the rate of boat up stream 4 4. Boat's own rate of travel down stream 8 5. Distance carried by the current 4 6. Therefore the rate of boat down stream 12 |