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sum of the payments : the quotient will be the meant time.

2. B owes A $600: $200 is to be paid in two months, $200 in four months, and $200 in six months: what is the mean time for the payment of the whole ?

200x2= 40C
200 X4= 800
200 X 6=1200
600 )24 00

4 Ans. 4 months. 3. A merchant owes $600, of which $100 is to be paid in 4 months, $200 in 10 months, and the remainder in 16 months : if he pay the whole at once, at what time must he make the payment?

Ans. 12 months. 4. A owes B $600; one third is to be paid in 6 months, one fourth in 8 months, and the remainder in 12 months : what is the mean time of payment ?

Ans. 9 months. 6. A merchant has due him $300 to be paid in 60 days, $500 to be paid in 120 days, and $750 to be paid in 180 days: what is the equated time for the payment of the whole ?

Ans. 1371 days. 6. A merchant has due him $1500; one sixth is to be paid in 2 months ; one third in 3 months ; and the rest in 6 months : what is the equated time for the payment of the whole ?

Ans. 4 months.

QUESTIONS. $ 199, What is Fellowship? What is the gain or loss cali. "? What is the rule for finding each one's share ?

10. What is Double Fellowship? What two circum

İstances determine each one's share of the profits ? What is the rule for finding each one's share ?

$201. What is Equation of Payments ? What is the rule for finding the mean time of payment of several sums due at different times ?

DUODECIMALS. Ø 202. Duodecimals are denominate fractions in which 1 foot is the unit that is divided.

The unit 1 foot is first supposed to be divided into 12 equal parts, called inches or primes, and marked'.

Each of these parts is supposed to be again divi. ded into 12 equal parts, called seconds, and marked":

Each second is divided in like manner, into 12 "equal parts called thirds, and marked "'.

This division of the foot, gives l' inch or prime

= of a foot 1" second is=ia of iz

Fita of a foot. 1"" third is='s of te of i

Fitzg of a foot. Duodecimals are added and subtracted like other denominate numbers, 12 of a lesser denomination making one of a greater, as in the following

TABLE.

12"

make 1" second 12

l' inch or prime 12

1 foot.
Ex. 1. In 185, how

many
feet?

Ans. 15ft. 5. 2. In 250', how many feet and inches ?

Ans. ift. 8 10". 3. In 4367'", how many feet?

Ans. 2ft. 6' 3" 11".

EXAMPLES IN ADDITION AND SUBTRACTION.

1. What is the sum of 3ft. 6' 3" 2" and 2ft. 1' 10" 11" ?

Ans. 5ft. 8 2" 1". 2. What is the sum of 8ft. 9' 7" and 6ft. 7' 3'' 4" ?

Ans. 15ft. 4' 10" 4". 3. What is the difference between 9ft. 3 5" 6"" and 7ft. 3 6" 7'"'? , Ans. ift. 11' 10" 11"".

4. What is the difference between 40ft. 6' 6" and 29ft. 7"" ?

Ans. 11ft. 6' 5" 5'". MULTIPLICATION OF DUODECIMALS. Ø 203. It has been shown Ø 84, that feet multiplied by feet gives square feet in the product.

Ex. 1. Multiply 6ft. 6' 6" by 2ft. 7'.

Set down the multiplier under ft. the multiplicand, so that feet shall 6 6 6" fall under feet, inches under in 2 7 ches, &c. It is generally most 13 1' convenient to begin with the high 3 9' 9" 6"" est denomination of the multiplier,

16 10' 9" 6" and first multiply the lower denominations of the multiplicand.

The 6" of the multiplicand, are is of an inch, or Tft of a foot. Therefore when we multiply it by 2 feet, the product is 12", equal to 1 inch. Multiply. ing 6', or is of a foot, by 2 feet the product is 12, to which add 1 inch from the last product, making 13'. Set down l' under the column of inches and carry 1 foot to the product of the 6 by 2, making 13 feet.

Then multiply by 7'. The product of 7' by 6"= 42'" : for 7=it of a foot, and 6"=tlt of a foot : 'hence 7' X 6"=11&t=14*5=42"=3" 6'". Then 5x9==42" and 3” to carry make 45"=3' 9": set down 9". Then I by 6=42' and 3' to carry make 15'=3ft. 9' which are set down in their proper places.

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204. Hence we see 1st. That feet multiplied by feet gives square feet in the product.

2d. That feet multiplied by inches gives inches in the product.

3d. That inches multiplied by inches gives seconds, or twelfths of inches in the product.

4th. That inches multiplied by seconds gives thirds in the product.

2. Multiply 9ft. 4in. by 8ft. 3in.

Beginning with the 8 feet, 9 4
we say 8 times 4 are 32', 8 3
which are equal to 2 feet 8': 74 8
sent down the 8'. Then say 2 4' 0"
8 times 9 are 72 and 2 to car-

77 0' 0" Ans.
ry are 74 feet : then multi-
plying by 3', we say, 3 times 4' are 12", equal to 1
inch: set down 0 in the seconds place: then 3 times
9 are 27 and 1 to carry make 28, equal to 2ft. 4'.
Therefore the entire product is equal to 77ft.
3. How

many

solid feet in a stick of timber which is 25ft. 6in. long, 2ft. 7in. broad, and 3ft. 3 inches thick ?

ft.

25 6' length
It is shown in $ 85 that 2 breadth
the number of solid or cu 51 0
bic feet in a solid, is equal 14 10'6"
to the length, breadth, and

65 10' 6"
thickness multiplied to-

3 3' thickness gether.

197 7'6"

16 5' 7" 6" Ans. 214 1' 1" 6'

4. Multiply 9ft. 2in. by 9ft. 6in.

Ans. 87ft. 1'. 5. Multiply 24ft. 10in. by 6ft. 8in.

Ans. 165ft. 6' 8". 6. Multiply 70ft. Sin. by 12ft. 3in.

Ans. 866ft. 8 3". 7. How many cords and cord feet in a pile of wood 24 feet long, 4 feet wide, and 3ft. 6in. high?

Ans. 2 cords and 5 cord ft. Note. It must be recollected that 16 solid feet make one cord foot Ø 84.

QUESTIONS.

8202. In Duodecimals what is the unit that is divided ? How is it divided ? How are these parts again divided ? What are the parts called ? How are duodecimals added and sub. tracted ? How many of one denomination make 1 of the next greater ?

$ 203. In multiplication how do you set down the multi. plier ? Where do you begin to multiply? How do you carry from one denomination to another ?

$ 204. Repeat the four principles.

ALLEGATION MEDIAL.

$ 205. A merchant mixes 8Zb. of tea worth 75cts. per pound, with 1616. worth $1,02 per pound : what is the value of the mixture per pound ?

The manner of finding the price of this mixture is called Allegation Medial. Hence,

ALLEGATION MEDIAL teaches the method of finding the price of a mixture when the simples of which it is nosed, and their prices, are known.

he example above, the simples 8lb. and 1616.,

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