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49. If a staff 4 feet long, cast a shade (on level ground). 7 feet long, what is the height of a steeple, whose shade at the same time is 198 feet?
Ans. 1134 feet. Note-The operation may frequently be contracted, by dividing the dividing term, and either of the other two, one by the other, or by any number that will divide them both without a remainder, using their quotients in their stead, as in the following examples :
50. If 24 yards of muslin cost 60 shillings, how much will' 4 yards cost? yd8. 8. yds (12)yds. s. yds. (12)yds. 8. yds. ZA 60 4 ZA 60
24 $0 $ Ø 10s. Ans. 2 5
Z 5 2
2 Thus 24 ; 45 And 60; 6 = 10 220
10s. Ans. 51. If 36 yards of linen cost 90s. what will 12 yards cost?
Ans. 30s. 52. If 24 yards of check cost 60s. what 'will 8 yards cost?
Ans. 20s. 53. What will 12 gallons of oil come to, if 6 gallons cost 18s.?
Ans. 36s. 54. If 8 yards of velvet cost $ 3.20, what will 96 yards come to?
Ans. $ 38.40. 55. If 9 lb. of sugar cost 9s. 4d. what is the value of 27 lb.?
Ans. IL. 8s. 56. How much will 60 bushels of apples come to, if 4 bushels cost 1 L. 4s.?
Inverse proportion is that wherein the third term is, greater than the first, and requires the fourth to be less than the second; or the third term less than the first, and requires the fourth to be greater than the second.
RULE. Multiply the first and second terms together, and divide the product by the third, the quotient will be the answer in the same denomination as the second.
EXAMPLES. 1. If 12 men can build a house in 48 days, in what time could 36 men build it?
Ans. 16 days. days men
days 12 48 36
36 : 16
12 A8 36 12 576
Proof 48 days 2. If 48 men can build a wall in 24 days ; how many men can do it in 192 days ?
Ans. 6 men. 3. If 100 men can finish a piece of work in 12 days, how many can do it in 3 days?
Ans. 400 men. 4. How many labourers must be employed to finish a piece of work in 15 days, which 5 can do in 24 days?
Ans. 8. 5. If 6 reapers can reap a field of wheat in 12 days, in what time could 24 do it?
Ans. 3 days. 6. If 100 dollars in 12 months bring 6 dollars interest, what sum will bring the same in 8 months ? Ans. $ 150.
7. If a footman perform a journey in 3 days, when the days are 16 houis long; how many days will he require of 12 hours long, to perform the same in?
Ans. 4. 8. How many yards of matting, 2 feet 6 inches broad, will cover a floor that is 27 feet long and 20 feet broad?
Ans. 72 yds. 9. What quantity of shalloon, that is 3 quarters of a yard wide, will line 71 yards of cloth, that is 14 yard wide?
Ans. 15 yds
10. How many yards of carpeting, that is 3 quarters of a yard wide, are sufficient to cover a floor that is 18 feet wide, and 60 feet long?
Ans. 160 yds. 11. If a board be 9 inches broad, how long must it be to measure 12 square feet ?
Ans. 16 feet. 12. How much in length that is 44 inches broad will make a square foot ?
Ans. 32 inches.
PROMISCUOUS EXAMPLES IN DIRECT AND INVERSE PROPORTIONS. Note. It will be of use to the scholar, in working the following examples, to use the rule referred to under the rule for stating, at page 74.
1. Calculate the value of 26 yards of linen, at 5s. 6d. a yard?
Facit 7L. 58. 9d. 2. Purchased 156 lb. of soap for 15 dollars 60 cents ; what was the price per pound?
Ans. 10 cts. 3. How many yards of cloth, 3 quarters of a yard wide,
3 are equal in measure to 30 yards, of 5 quarters wide ?
Ans. 50 yds. 4. Bought 274 yards of musling at 63. 9 d., per yard, what does it amount to?
Ans. 9 L. 58. 01d.+2. 5. In what time will 600 dollars gain 50 dollars interest, when 80 dollars would gain it in 15 years! Ans. 2 years.
6. What quantity of wine, at 6s. per gallon, may be bought with 18 L. 18s.?
Ans. 63 gals. 7. If i hundred weight of sugar cost 13 dollars-50 cents, what must be paid for 17 Cwt. 3 qrs. 14 lb. ?
Ans. & 241,313 cts. 8. A wall which is to be built to the height of 27 feet, has been raised 9 feet in 6 days, by 12 men: how many men must be employed to finish the work in 4 days?
Ans. 36 men. 9. A cistern has a cock which will empty it in 10 hours : how many cocks of the same capacity will empty it in 30 minutes ?.
Ans. 20. 10. How many yards of paper, 21 feet wide, will be required to cover a wall, which is 12 feet in length, and 9 in breadth ?
Ans. 14 yds. I ft. 2 in.
11. If 1 oz. of spice cost 64d. what will 34 oz. cost at the same rate ?
Ans. Is. lid.+ 12. What is the value of a piece of cloth containing 52 English ells, 3 quarters, at 1 dollar 76 cents per yard?
101b. Ş 12 men
DOUBLE RULE OF THREE. The Double Rule of Three is that, wherein five numbers or terms are given, to find a sixth ; three of which are a supposition, and two a demand.
RULE FOR STATING. Set the two terms of supposition which are of the same name or kind as those of the demand, one under the other, in the first place; that of the same kind as the answer in the second, and those of the demand in the third, with the two correspondent terms of the supposition and demand opposite to each other, and of the same denominations as in the following examples.
1. If 6 men in 8 days eat 10 lb. of bread, how much will 12 men eat in 24 days?
24 days 2. If 6 men eat 10 lb. of bread in 8 days, in how many days will 12 men eat 60 lb. 6 men
12 men 10 lb.
60 lb When a question is stated, consider the two upperterms with the middle one, as a stating in the Single Rule of Three, and also the two under terms, with the middle one, as a staling in the same rule; if, in both instances, the proportion be direct, the question is in direct proportion; but if in either of them the proportion be inverse, the question is in inverse proportion. Thus the former of the above examples is in direct, and the latter in inversa proportion.
RULE. Multiply the two terms in the third place together, and multiply the product by the middle term; divide the last
n} 8 days }
product by the product of the terms in the first place, and the quotient will be the answer, in the same denomination as the middie term.
PROOF. By two statings in the Single Rule of Three. Note. If either of the two first terms, or both, will divide, or can be divided by any of the three last, or by any other number with out a remainder, the operation may be contracted by using their quotients in their stead.
EXAMPLES 1. If 6 men in 8 days eat 10 lb. of bread, how much will 12 men eat in 24 days?
Ans. 60 lb, 6 men Tolb.
Contracted. $ 12 men 8 days 24 days 6 men
12 men 2 s
224 days 3 43
10 43)2880(60 Ans. 288
s day's }
0 2. If 3 men in 4 days eat 5 lb. of bread, how much will suffice 6 men for 12 days?
Ans. 30 lb. 3. Suppose 4 men in 12 days mow 48 acres, how many acres can 8 men mow in 16 days?
Ans. 128 A. 4. If 10 bushels of oats be sufficient for 18 horses 20 days, how many bushels will serve 60 horses 36 days, at that rate ?
Ans. 60 bu. 5. If 7 quarters of malt are sufficient for a family of 7 persons 4 months, how many quarters will 46 persons use in 10 months ?
Ans. 115. 6. Suppose the wages of 6 persons for 21 weeks be 288 dollars, what must 14 persons receive for 46 weeks?
Ans. 1472 dols. 7. If 8 reapers have 3 L. 4s. for 4 days work, how much will 48 men have for 16 days work?.. Ans. 76 L. 16s.
8. If 100 L. in 12 months gain 6 I. interest, how muchi will 75 1.. gain in 9 months ?
Ans. 3 L. 7. 6d. 9. If 100 L. in 52 weeks gain 6 L. interest, how much will 200 L. gain in. 26 weeks?