Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

42", and 3" to carry make 45"=-3' 9": set down 9". Then 7 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?

[blocks in formation]

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

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

[blocks in formation]

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. 81b. of tea at 75cts. per lb.

167b.

- $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?

[merged small][merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

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. 283cts.

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. 20 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,

ALLIGATION ALTERNATE teaches the method of finding what proportion must be taken of several simples, whose prices are known, to form a compound of a given price. Alligation Alternate is the reverse of Alligation Medial, and may be proved by it.

For a first example, let us take the one before stated. If oats worth 3s per bushel be mixed with wheat worth 98, how much must be taken of each sort that the compound may be worth 5s per bushel?

5

3

4 Oats.

9.

12 Wheat.

If the price of the mixture were 6s, half the sum of the prices of the simples, it is plain that it would be necessary to take just as much oats as wheat.

But since the price of the mixture is nearer to the price of the oats than to that of the wheat, less wheat will be required in the mixture than oats.

Having set down the prices of the simples under each other, and linked them together, we next set 5s, the price of the mixture, on the left. We then take the difference between 9 and 5 and place it opposite 3, the price of the oats, and also the difference between 5 and 3, and place it opposite 9, the price of the wheat. The difference standing opposite each kind shows how much of that kind is to be taken. In the present example, the mixture will consist of 4 bushels of oats and 2 of wheat; and any other quantities, bearing the same proportion to each other, such as 8 and 4, 20 and 10, &c., will give a mixture of the same value.

PROOF BY ALLIGATION MEDIAL.

4 bushels of oats at 3s
2 bushels of wheat at 9s.

в

[ocr errors]
[ocr errors]

12s.

[ocr errors]

18s.

6)30

Ans. 5s.

Q. What is Alligation Alternate? How do you prove Alligation Alternate?

CASE 1.

§ 174. To find the proportion in which several simples of given prices must be mixed together, that the compound may be worth a given price.

RULE.

1. Set down the prices of the simples under each other, in the order of their values, beginning with the lowest.

II. Link the least price with the greatest, and the next least with the next greatest, and so on, until the price of each simple which is less than the price of the mixture is linked with one or more that is greater; and every one that is greater with one or more that is less.

III. Write the difference between the price of the mixture and that of each of the simples opposite that price with which the particular simple is linked; then the difference standing opposite any one price, or the sum of the differences when there is more than one, will express the quantity to be taken of that price.

EXAMPLES.

1. A merchant would mix wines worth 16s, 18s and 22s per gallon in such a way that the mixture be worth 20s per gallon: how much must be taken of each sort?

[blocks in formation]

Ans. {2gal. at 168, 2 at 188, and 6 at 228: or any other

quantities bearing the proportion of 2,2 and 6.

2. What proportions of coffee at 16cts., 20cts., and 28cts. per lb. must be mixed together so that the compound shall be worth 24cts. per lb. ?

In the proportion of 476. at 16cts., Ans. {4lb. at 20cts., and 1216. at 28cts.

3. A goldsmith has gold of 16, of 18, of 23 and of 24 carats fine: what part must be taken of each so that the mixture shall be 21 carats fine?

Ans. 3 of 16, 2 of 18, 3 of 23, and 5 of 24.

4. What portion of brandy at 14s per gallon, of old Madeira at 24s per gallon, of new Madeira at 21s per gallon, and of brandy at 10s per gallon, must be mixed together so that the mixture shall be worth 18s per gallon?

Ans. 6 gal. at 10s, 3 at 14s, 4 at 21s, and 8g al. at 24s.

« ΠροηγούμενηΣυνέχεια »