13. If 100 men can do a piece of work in 12 days, how many men can do the same in 3 days? Ans. 400 men. 14. If 100 dollars gain 6 dollars in one year, in what time will a sum of money double at that rate, simple interest? $ yr. 6:1:100 Ans. 163 yrs. 15. If $100 gain $6 in 12 months, in how many months will a sum of money double at that rate, simple interest? $ mo. 6:12:: 100 Ans. 200 mo. 16. If $100 gain 6 dollars in 365 days, in how many days will a sum of money double at that rate, simple interest? Ans. 6083 days. 17. A owes B £296 17s., but becoming a bankrupt, can pay only 7s. 6d. on the pound; how much will B receive? Ans. £111 6s. 4d. 2qrs. 18. If 1 dozen of eggs cost 10 cents, what will 250 eggs Ans. $2.187. cost? 27. How many yards of pa per 3 quarters of a yard wide, will paper a room that is 24 yards round, and 4 yds. high? Ans. 128 yards. 28. If a man spend 75 cents per day, what does he spend per annum? Ans. $273.75. 29. A garrison of 500 men has provisions for six months; how many must depart that there may be provisions for those who remain 8 months? Ans. 125. * This is called a sideral day. 35. A cistern containing 230 gallons, has 2 pipes; by one it receives 50 gallons per hour, and by the other discharges 35 gallons per hour; in what time will it be filled? Ans. 15h. 20m. 36. What will 39 weeks' board come to at $1.17 per week? Ans. $45.63. 37. If 40 rods in length and 4 in breadth make 1 acre, how many rods in breadth, that is 16 rods long will make 1 acre? Ans. 10 rods. 38. How many men must be employed to finish in 9 days, what 15 would do in 30 days? Ans. 50 men. 2. Compound Proportion. ANALYSIS. 199. 1. If a person can travel 96 miles in 4 days, when the days are 8 hours long, how far can he travel in 2 days, when the days are 12 hours long? I. If a person can travel 96 miles in 4 days, he can travel (964) 24 miles in 1 day, and if he can travel 24 in a day, which is 8 hours long, he can travel (248) 3 miles in 1 hour, and if he can travel 3 miles in an hour, he can travel, when the days are 12 hours long, (12×3) 36 miles in 1 day, or (36X2=) 72 miles in 2 days, which is the answer. II. It must be evident that the distances travelled by a person going all the time at the same rate will be in proportion to the times in which they are travelled. In this case, 4 days, which are 8 hours long, are equal to (8X4) 32 hours, and 2d. 12 hours long equal (12×2) 24h. and hence we have this proportion, 32h.:: 96m.:: 24h.: x, or the distance travelled in the 2 days, which we find to be 72 miles as before. III. It will be obvious, in the above question, that the distance travelled depends upon two circumstances, viz. the number of days and the length of the days. Now, supposing the days had all been of the same length, we should have had this proportion, viz. 4d. : 96m.:: 2d. x, or the distance travelled in 2 days; or, supposing the number of days had been the same in both cases, the proportion would stand, 8h.: 96m.:: 12h. : x, or the distance travelled when the days are 12 hours long. Uniting these proportions together, we have 4d. } by which it appears that 96 is to be multiplied by 2 and 12, or (2×12) 24, and divided by 4 and 8, or (4×8=) 32, which is the same as the second method of solving the question. 200. 2. If 12 men can make 9 rods of fence in 6 days, when the days are 10 hours long, how many men will be required to make 18 rods of fence in 4 days, when the days are 8 hours long? In this question, the number of days and their length being supposed to be the same in both cases, we should have this proportion, 9rds.: 12 men 18x, or the number of men required to build the 18 rods-supposing the number of rods to be the same in both cases, and the days to be of equal length, we should have this proportion, 4d. : 12 men; : 6d. : x, or the number required to build the fence in 4 days, and supposing the number of rods and also the number of days to be the same in both cases, we should have this proportion, 8 hours: 12 men :: 10h. x, or the number required, when the days are 8 hours long. These three proportions combined, we have 9rds. 4d. 8h. 18rds. :: 6rds. 12 10h. : x, by which it appears that 9X4X8:12:: 18×6×10: x, and multiplying the product of the third terms by the second, and dividing by the product of the first terms, we find the value of x to be 45 men, which is the answer. DOUBLE RULE OF THREE. 201. A proportion which is formed by the combination of two, or more, simple proportions, as in the preceding examples, is called a Compound Proportion. The rule by which the fourth term of a compound proportion is found, is called the Double Rule of Three, and may be understood from the preceding analysis. RULE. 202. Make that number, which is of the same kind as the required answer, the second term. Take any two of the remaining terms which are of the same kind, and place one for a first, and the other for a third term, as directed in the Single Rule of Three (198); then take any other two of the same kind, and place them in the same way, and so on till all are used. Multiply the product of the third terms by the second term, and divide the result by the product of the first terms: the quotient will be the required answer. QUESTIONS FOR PRACTICE. 7. If the interest of $45 for 6 months be $1.80, what is the rate per annum? Ans. 8 per cent. 8. If 8 men spend 48 dolls. in 24 weeks, how much will 40 men spend in 48 weeks, at the same rate? Ans. $480. 9. If the freight of 5 tierces of salt, each weighing 53 cwt. 80 miles, cost $28, what will be the freight of 75 sacks of salt, each weighing 24 cwt., 150 miles? |