8. What is the net weight of 18hhd. of tobacco, each weighing gross 8cwl 3qr. 14lb.; tare 167b. to the cwt.? Ans. 67. 16cwt. 3qr. 20lb.

9. In 4 T. 3cwt. 3qr. gross, tare 20lb. to the cwt., what is the net weight? Ans. 3 T. 8cwt. 3qr. 5lb. 10. What is the net weight and value of 80 kegs of figs, gross weight 7 T. 11cwt. 3qr., tare 147b. per cwt., at $2,31 per cwt.? 6 T. 12cwt. 3qr. 3lb. 8oz.

Ans. {Value $306,724 4+.

DUODECIMALS.

§ 170. 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 divided 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

1' inch or prime

of a foot.

1"third is 12 of 12 of 12=1728 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

1. In 185', how many feet?

2. In 250, how many feet and inches?

3. In 4367"", how many feet?

Ans. 15ft. 5'.

Ans. 1ft. 8' 10". Ans. 2ft. 63" 11"".

Q. 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 subtracted? How many of one denomination make 1 of the next greater?

EXAMPLES IN ADDITION AND SUBTRACTION.

1. What is the sum of 3 ft. 6' 3"

2. What is the sum of 8ft. 9' 7"

and 2ft. 1' 10" 11";

Ans. 5ft. 8' 2" 1"".

and 6ft. 7 3" 4""?
Ans. 15ft 4' 10" 4"".

3. What is the difference between 9ft. 3' 5'' 6'' and 7ft. 31 6 7 ?

Ans.

4. What is the difference between 40ft. 6' 6" and 29ft. 7!!!? Ans. 11ft. 6' 5" 5"".

MULTIPLICATION OF DUODECIMALS.

§ 171. It has been shown (§ 64) that feet multiplied by feet give square feet in the product.

EXAMPLES.

1. Multiply 6ft. 6′ 6′′ by 2ft. 7'. Set down the multiplier under the multiplicand, so that feet shall fall under feet, inches under inches, &c. It is generally most convenient to begin with the highest denomination of the multiplier, and then multiply first the lower denominations of the multiplicand.

6

of

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

Then mulitply by 7'. The product of 7' by 6"=42′′" : for, 7=11⁄2 of a foot, and 6 of a foot: hence 7'6" =12×14=1238=42""=3" 6′′." Then

22=

42", and 3" to carry make 45"-3′ 9′′: set down 9". Then by 6=42', and 3' to carry make 45'=3ft. 9', which are set down in their proper places.

Hence, we see,

1st, That feet multiplied by feet give square feet in the product.

2nd, That feet multiplied by inches give inches in the product.

3rd, That inches multiplied by inches give seconds, or twelfths of inches in the product.

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

2. Multiply 9ft. 4in. by 8ft. 3in. Beginning with the 8 feet, we say 8 times 4 are 32', which is equal to 2 feet 8': set down the 8'. Then say 8 times 9 are 72 and 2 to carry are 74 feet: then multiplying by 3', we say, 3 times 4' are 12", equal to 1 inch: set down 0 in the second's place: then 1 to carry make 28', equal to 2ft. 4'. product is equal to 77ft.

3

9

OPERATION.

4'

8 3'

74 8'

2 4' 0"

77 0' 0" Ans.

times 9 are 27 and Therefore the entire

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

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

5. Multiply 24ft. 10in. by 6ft. 8in. 6. Multiply 70ft. 9in. by 12ft. 3in.

Ans. 87 ft. 1'. Ans.

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 feet.

NOTE. It must be recollected that 16 solid feet make one cord foot § 65.

Q. In multiplication how do you set down the multiplier? Where do you begin to multiply? How do you carry from one denomination to another? Repeat the four principles.

ALLIGATION MEDIAL.

§ 172. A merchant mixes 876. of tea worth 75cts. per pound, with 167b. 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 Alligation Medial. Hence,

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

In the example above, the simples 876. and 167b., and also their prices per pound, 75cts. and $1,02, are known. 87b. of tea at 75cts. per lb.

$1,02 per lb.

24 sum of simples.

Now if the entire cost of the mixture, which is $22,32, be divided by 24 the number of pounds, or sum of the simples, the quotient 93cts. will be the price per pound. Hence, we have the following

RULE.

6,00

16,32

Total cost $22,32

OPERATION.

24)$22,32(93cts

216

72

72

Divide the entire cost of the whole mixture by the sum of the simples the quotient will be the price of the mixture.

EXAMPLES.

1. A farmer mixes 30 bushels of wheat worth 5s per bushel, with 72 bushels of rye at 3s per bushel, and with 60 bushels of barley worth 2s per bushel: what is the value of a bushel of the mixture?

2. A wine merchant mixes 15 gallons of wine at $1 per gallon with 25 gallons of brandy worth 75 cents per gallon what is the value of a gallon of the compound? Ans. 84cts.+

3. A grocer mixes 40 gallons of whiskey worth 31cts. per gallon with 3 gallons of water, which costs nothing: what is the value of a gallon of the mixture? Ans. 2839cts.

4. A goldsmith melts together 2lb. of gold of 22 carats fine, 6oz. of 20 carats fine, and 6oz. of 16 carats fine: what is the fineness of the mixture? Ans. 203 carats.

5. On a certain day the mercury in the thermometer was observed to average the following heights: from 6 in the morning to 9, 64°; from 9 to 12, 74°; from 12 to 3, 84°; and from 3 to 6, 70°: what was the mean temperature of the day? Ans. 73°.

Q. What is Alligation Medial? How do you find the price of the mixture?

ALLIGATION ALTERNATE.

§ 173. A farmer would mix oats worth 3s per bushel with wheat worth 9s per bushel, so that the mixture shall be worth 5s per bushel: what proportion must be taken of each sort?

The method of finding how much of each sort must be taken, is called Alligation Alternate. Hence,