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Now, since the product of the extremes is equal to that of the means, 4 times x equals 6 times 12, or, according to the second arrangement, 19 times 6. Hence, if 12 times 6, or 72, be divided by 4, the first extremne, the quotient, 18, is evidently the other extreme, or the value of x.
196. 3. If 4 men can do a piece of work in 6 days, in how many days can 8 men do it!
By analyzing the example, we find that 4 men 6 days=1 man 24 days, and 1 man 24 days-8 men' 3 days. 8 then is the answer. Moreover is is obvious, that if 4 men can do a piece of work in 6 days, twice the number men will do it in half the time, or 3 days; and generally the greater the number of men, the less the time, and the reverse ; and also, the longer the time, the less the number of men, and the reverse. In the above example, the ratio of the men, 4 to 83, but the ratio of the times, 6 to 33. Now, if we invert the first ratio, it becomes, 8 to 43; me we have two equal ratios, and consequently a proportion : i. e. 8:7::6:3, or 8:6:: 4 :3. By the question, the proportion would stand, 8:6::4:6; then 8 4X6, and r33. Ans. Where more requires less or less requires more, that is, when one of the ratios is inverted, as explained in this article it is denominated inverse proportion ; otherwise it is called direct propor lion.
I. Single Rule of Three. 197. When three terms of a proportion are given, the operation by which-the fourth is found, is called the Single Rule of Three. Ah questicns, whicli can be solved by the single rule of three, must contain three given numbers, two of which are of the same kind, and the other of the kisd. of the required answer; and from an examination of the preceding analysis, it will be seen that the giverr number, which is of the same kind as the answer, may always be one of the means in the proportion; and, since the proportion is not altered by changing the places of the means, (195) it may always be regarded as the first mean, or ilie middle one of the three given
Now if the conditions of the question require the answer to be greater than the given number of the same kind, or first mean, the other inean inust obviously be greater than the first extreme; but if the answer be required to be less, the second mean must be less than the first extrema Hence we have the following general
RULE. 198. Write down the given number, which of the same kind as the answer, or number sought; for the second term. Consider whether the answer ought to be greater, or less, than this number; and if greater, write the greater of the other two given nuinbers for the third term, and the less for the first term; but if less, write the least of the other two given numbers for the third term, and the greater for the first. Multiply the second and third terms together, and divide the product by the first, the quotient will be the answer.
Notx.-Before stating the question, the first and third terms must be reduced to the same denomination, if they are not already so, and the uniddle term to the lowest denomination mentioned in it. The answer will be in the game denomination as the second torm, and may be brought w higtaer by reluction, if necessary.
QUESTIONS FOR PRACTICE. 4. If 15 bushels of corn cost 8. If 81b. 4oz. of tobacoo $7.50, what will 25 bushels cost 58. 6d., what will 41b. cost?
12oz, cost? bu. % cts. bu.
s. d. 15 : 7.50 :: 25
84 5 6 24 12 25
16 12 16
132oz. 66d. 156 1500
24 $ cts. 15) 187.50 ( 12.50 Ans.
396oz. By analysis.--If 15 bushels of
oz. cost $7.50, One bushel will 132 : 66 :: 396 cost ($7.50-15=) $0.50, and if
66 one bushel cost $0.50, 25 bushels will cost ($0.50X253) $12.50. The pupil should be required to
2376 solve the following questions by
2376 analysis as well as by the rule of three.
132 ) 26136 ( 198d. 5. If $7.50 buy 15 bushels
16s. 6d. Ans. of corn, what will $12.50 buy?
Here the several terms are re$ cts.bu. $ cts.
duced to the lowest denominations 7.50: 15 : : 12.50
mentioned, before stating the question.
9. If 8 acres produce 176 6250
bushels of wheat, what will 34 1250
acres produce ? bu.
Ans. 748 bushels. 7.50 ) 187.50 ( 25 Ans. This is the reverse of the pre
10. A borrowed of B 250 ceding example, and therefore dollars for 7 months; after
wards B borrowed of A 300 6. If a family of 12 persons dollars; how long must he spend 5 bushels of wheat in 4 keep it to balance the former weeks, how much will last favor ? Ans. 5mo. 25d. them a year, allowing 52 weeks 11. A goldsmith sold a tankto a year?
ard weighing 39oz. 15pwt., for w. bu.
£10 125.; what was it per oz ? 4:5 :: 52 Ans. 65 bash.
02. pwt. £ s. 7. If 9lb, of sugar cost 6s.
39 15 : 10 12::1 Ans. 58. 4d. what will 25lb. cost?
12. If the interest of $100 Ans. 16s, 8d. When there is a remainder after be the interest of 336 dollars
for 1 year be 6 dolls., what will. dividing the product of the second and third terms by the first, reduce for the same time? it to the next lower denomination,
$ and divide as beforç.
100: 6:: 336 Ans. $20.10
13. If 100 men can do a 22. If a staff 4ft. 6in. in length piece of work in 12 days, how cast 4 shadow 6 feet, what many men can do the same in is the height of a tree whose 3 days?
Ans. 400 men. shadow measures 108 feet? 14. If 100 dollars gain 6 dol
Ans. 81 feet lars in one year, in what time 23. If the earth revolve On will a sum of money double at its axis 366 times in 365 days, that rate, simple interest ? in what time does it perform 1 $ yr.
revolution? 6:1:; 100 Ans. 16 yrs.
366 : 365 ::1 15. If $100 gain $6 in 12
Ans. 23h. 56m. 4s. nearly.* months, in how many months wul a sum of money double at 24. Bought 4 bales of cloth, that rate, simple interest? cach containing 6 pieces, and $ mo.
each piece containing 27 yds. 6:12:: 100 Ans. 200 mo. at £16 4s. per picce; what is 16, If $100 gain 6 dollars in
the value of the whole, and the 365 days, in how many days price per yard ! will a sum of inaney double at
Ans. £388 10s. and 12s. per that rate, simple interest ? yard.
Ans. 6083} days. 25. If a hogshead of rum 17. A owes B £296 17s., but cost $75.60, how much water becoming a bankrupt, can pay
must be added to it to reduce only 7s. 6d. on the pound; the price to $1 per gallon ? how much will B receive ?
Aus, 127 gal Ans. £111 6s. 4d. 2qrs. 26. If a board be 9 inches 18. If 1 dozen of eggs cast wid-, how much in length will 10 cents, what will 250 eggs make a square foot ? Ans. $2.187.
Ans. 16in. 19, If a penny loaf weigh 27. How many yards of pa9oz. when wheat is bs. 3d. per per 3 quarters of a yard wide, bushel, what ought it to weigh will paper a room that is 24 when wheat is 8s. 23d. per yards round, and 4 yds. high? wushel? Ans. 6oz. 13drs.
Ans. 128 yards, 20. How many yards of 28. If a man spend 75 cents Aannel 5qrs. wide, will line 20 per day, what does he spend yards of cloth 3qrs. wide ?
29. A garrison of 500 men 21. If a person at the equa- has provisions for six months ; tor be carried by the diurnal | how many must depart that motion of the earth, 25000 there may le provisions for miles in 24 hours, how far is those who remain 8 months? he carried in a minute ?
* This is called a sideral day:
30. The salary of the Presi- 39. The earth is 3600 in dent of the United States is circumference, and revolvos $25000 a year; what is that on its axis in 24 hours ; how
Ans. $68.493. far does a place move in one 31. If a field will feed 6 minute in lat. 44°, a degree in cows 91 days, how long will it that latitude being about 50 feed 21 cows?
miles? Ans. 124m. nearly.
h. Ans. 26 days.
24X60 : 360X50 :: 1. 32. A lends B 66 dollars for year; how much ought B to 40. If the earth perform its lend A for 7 months, to bal- diurnal revolution in 24 hours, ance the favor?
in what time does a place on Ans. $113.142. its surface move through one 33. At $1.25 per week, how degree? Ans, 4 minutes. many weeks' board can I have
360°: 24::10 for $100 ? Ans. 80 weeks.
41. There is a cistern which 34. If my watch and seal be has a pipe that will empty it in worth $48, and my watch be 6 hours; how many such pipes worth 5 times as much as my
will be required to empty it in seal, what is the value of the 20 minutes ? watch? Ans. $10.
Ans. 18 pipes. 6 : 48 : : 5
42. What is the value of 35. A cistern containing 642 dollars against an estate 230 gallons, has 2 pipes; by which can pay only 69 cents one it receives 50 gallons per on the dollar ? hour, and by the other dis
Ans. $442.98. charges 35 gallons per hour ;
43. If 6352 stones of 3 feet in what time will it be filled ?
long complete a certain quanAns. 15h. 20m.
tity of walling, how many 36. What will 39 weeks” stones of 2 feet long will raise board come to at $1.17 per
a like quantity ? Ans. 9528. week?
Ans. $15,63. 37. If 40 rods in length and
44. Suppose 450 men have 4 in breadth make 1 acre, how provisions for 5 months, how many rods in breadth, that is many must depart, that the 16 rods long will make 1 acre ? provisions may serve those. Ans. 10 rods. who remain 9 months ?
Ans. 200 men. 38. How many men must be employed to finish in 9 days, 45. A person's annual in what 15 would do in 30 days? come being £146, how much
Ans, 50 men. is that per day? Ans. Ss.
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?
1. If a person can travel 96 miles in 4 days, he can travel (96-543) 24 miles in 1 day, and if he can travel 24 in a day, which is 8 hours long; he can travel (27:8=) 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, (12X3=) 36 miles in 1 day, or (36X2=172 miles in 2 days, which is the answer.
11. 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 (12x2= 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 propor Lions together, we have
:X, by which it appears that 96 is to be multiplied by 2 and 12, or (2X123 21, and divided by 4 and 8, or (4x8=) 32, which is the same as the second method of solving the question.
200. 2. It 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 :: 18 : x, 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 tho 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 Ords.
:: Ords. 8h.
10h. by which it appears that 9X4X8 : 12 :: 18X6X 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 example,