tion of Decimals, the answer is 17. 14s. 11 d., as before. 46. If 2.5lb. of sugar cost 2s. what cost 85.25lb.?

47. If 11 yards of linen cost

bought for $78.75 ?

Ans. 31. 8s. 24d.+ $9, Kow much may be Ans. 96 yd. 1 qr.

The following questions are promiscuously placed. 48. A cent weighs 208 grains Troy; and 7000 grains Troy are equal to 1lb. Avoirdupois. What is the Avoirdupois weight of as many cents as are worth $1000? Cent. cents. grains. grains.

First, As 1 : 100000 :: 208: 20800000.

Secondly, As 7000 grains Troy: 20800000 grains Troy :: 1lb. Avoir.: 2971 lb. Avoir. Ans.

Question 48th is solved by two statements. The Troy weight of 100000 cents, (=$1000,) is found by the first statement, and the Avoirdupois weight by the second.

49. A factor bought a certain quantity of broadcloth and drugget, which, together, cost $270: The quantity of broadcloth was 50 yards, at $3 per yard; and for every 5 yards of broadcloth he had 9 yards of drugget. I demand the quantity of drugget, and what it cost per yard?

As 5 yards of broadcloth: 50 yards of broadcloth :: 9yd. of drugget: 90 yd., the quantity of drugget.

yd. yd. $ $

As 1:50:3: 150, value of the 50 yd. of broadcloth. $270-150-$120, value of the 90 yd. of drugget. $120÷90-$14, value of a yard of drugget.

Ans. 90 yards, at $13 per yard.

50. A and B depart from the same place, and travel the same road; but A goes 5 days before B, at the rate of 15 miles a day-B follows at the rate of 20 miles a day: What distance must B travel to overtake A?

As I day 5 days: 15 miles: 75 miles, the distance A had travelled when B started.

20-15-5, the number of miles B gains of A in travelling 20 miles. Then,

Gained. gained. travelled. travelled.

As 5 miles 75 miles :: 20 miles : 300 miles, Ans.

:

51. A harmless dove was soaring high,
To stretch her wings in space-
At length a hawk did her espy,
And gave the dove a chase:
Just forty chains were then between
The hawk and dove that flew--
While the poor dove flew seventeen,
The hawk just twenty-two:

The hawk pursued with all his strength,
As those who saw did say-
Pray tell the chains he flew in length
Before he caught his prey?

52. Just fifteen pair of ladies' gloves
For sixty dimes had I;

Ans. 176 chains.

How many pair of that same kind
Will forty eagles buy?

Ans. 1000.

53. There are four pieces of cloth; the first contains 21 yards, the second 23, the third 24, and the fourth 27. What is the value of the whole, at $1.43 per yard?

21+23+24+27=95 yards, the whole quantity.

:

Then, as lyd. : 95 yd. :: $1.43 $135.85, Ans. 54. If 5 barrels of flour cost $28.75, how many barrels may be bought for $5750?

Ans. 1000. 55. Bought 4 casks of wine, each containing 65 gallons, at the rate of 2 gallons for $3: what did they cost?

Ans. $391.50 56. After observing a flash of lightning, it was 12 seconds before the thunder was heard: required the distance of the cloud from whence it came? Ans. 24 miles.

Note.--Sound, if not interrupted, moves at the rate of 1142 feet in a second of time, or 1 mile in about 4 seconds. Therefore, the answer to the preceding question is found thus: as 4 sec. : 12 sec. :: 1 mile : 24 miles.

57. Perceiving a man at a distance hewing down a tree with an ax, I remarked that 6 of my pulsations passed between seeing him strike and hearing the report of the blow: what was the distance between us, allowing 70 pulses to a minute? Ans. 5873 feet.

58. In what time would wind move from the pole to the equator, at the rate of 3 miles an hour, the distance being 6228 miles? Ans. 86 da. 12 h.

59. The Earth describes its orbit round the Sun in 365 days, 5 h. 48 min. 48 sec.: through what space does it move each hour, at an average, the circumference of the orbit being 596902655 miles? Ans. 68094 m.+ 60. How many yards, 3 quarters wide, are equal in measure to 24 yards, 5 quarters wide?

As 3qr. 5qr. :: 24yd. : 40 yards, Ans. 61. How many yards of carpeting, which is 3 feet wide, will cover a floor which is 27 feet long and 20 feet wide? Width. width. length. length.

As 3ft.: 20 ft. :: 27 ft. : 180 ft.=60 yards, Ans. 62. How many yards of stuff, 3 quarters wide, will line a cloak that is 3yd. 2qr. in length, and lyd. 3qr. wide? Ans. 8yd. 23 na.

63. Supposing I have 200 yards of cloth, which cost me 90 cents per yard, but some damage having happened to it, I am willing to lose $20 by the whole; at what rate must I sell it per yard? Ans. 80 cents.

64. Bought a pipe of wine for $84, and afterwards found it had leaked out 12 gallons; I sold the remaining 114 gallons at 20 cents a quart: what did I gain or lose?

Ans. I gained $7.20

65. If the Legislature of a State grant a tax of 8 mills on the dollar, how much must that man pay whose list amounts to $1084.75? Ans. $8.678 66. If 30 bushels of corn, at 50 cents a bushel, will pay a certain debt, how many bushels at 75 cents a bushel would pay the same debt? Ans. 20.

67. If 50 gallons of water, in one hour, fall into a cistern that will hold 230 gallons, and by a pipe in the cistern 35 gallons run out in an hour; in what time will the cistern be filled? Ans. 15 hours, 20 min. 68. Bought 4 pieces of cloth, each containing 24 English ells, for $96: how much is that per yard?

Ans. 80 cents. 69. Bought 126 gallons of rum for $110: how much water must be added to reduce the cost to 75 cents per gallon? Ans. 20 gal. 70. The moon moves through 13 degrees, 10 min. 35 sec. of the zodiac in a day: in what time does it move through 360 degrees, or perform an entire revolution?

Ans. 27da. 7h. 43 min.+

CONTRACTIONS IN THE RULE OF THREE.

There are several methods of contraction, which may be used in particular cases, in working out the proportions. The following are some of them.

Note. When the first and second terms of the given analogy are of different denominations, they must be reduced to the same denomination before any of the following methods of contraction can be used.

CONTRACTION I.

When the first term of the proportion is 1, multiply the second and third terms together, and the product will be the answer: And when either the second or the third term is 1, divide the other by the first term, and the quotient will be the answer, in the same denomination as the third term.* Ex. 1. What cost 80 lb. of cheese, at 4 cents a lb.?

As 1lb. 80 lb. :: $.045: the 4th term.

Ans. 57. 15s.

Then, $.045×80-$3.60, Ans. 2. What is the value of 7 acres of land, if each acre be worth 8l. 5s.? 3. If8lb. of butter cost $1.28, what cost 1lb.? As 8lb. 1 lb. :: $1.28: the 4th term. Then, $1.28÷8=$.16=16 cents, Ans. 4. If 1 yard of broadcloth cost $5, how many yards may be bought for $70? Ans. 14yd.

CONTRACTION II.

The work may often be much abbreviated by dividing either, the first and second, or the first and third terms, (but never the second and third,) by any number which will measure or divide them, and then using the quotients instead of the numbers so divided.f

* If any number be either multiplied or divided by 1, the product, or quotient, will evidently be the same as the given number, and hence, in such cases, the operation of multiplying or dividing may be omitted.

The ratio of any two numbers, is the same as the ratio of the products or quotients obtained by multiplying or dividing both the given numbers by any one number: Or, which amounts to the same thing, if the divisor and dividend be both multiplied or divided by the same number, the quotient is not changed; and hence the reason of the 2d method of contraction is evident.

Note. It will be best to take, for a divisor, one of the two terms which you would divide, when the other can be divided by it; for then one of the quotients will be 1, and the answer may be found by Contraction 1st.

Ex. 1. If 15 yards of cloth cost $17.43, what will 55 yards cost, at the same rate?

2. If 16 men eat 25 loaves of bread in a week, how many loaves will 24 men eat in the same time? Ans. 37.

3. Find a fourth proportional to 100, 256, and 700. As 100 256 :: 700: the 4th term.

Or, as 1 256 :: 7 : 1792, Ans.

Here I divide the first and third terms by 100; or, which is the same thing, I reject two ciphers from the right hand of each of these terms; and then I find the answer by Contraction 1st.

4. If 8 yards of broadcloth cost $32.80, what will 15 yards cost? Ans. $61.50

CONTRACTION III.

If by adding to, or subtracting from the first term, any part of itself, the sum or the remainder will be equal to the second term; then add to the third term, or subtract from it, (as the case shall require,) the like part of itself, and the result will be the fourth term, or answer.

Ex. 1. If 50 bushels of wheat be worth $60.25, what is the value of 75 bushels?

As 50 bush.: 75 bush. :: $60.25: the 4th term.

Here, of the first term added to the first, is equal to the second term; and therefore, of the third term added to the third, gives the answer.