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RULE.

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

1. If 19 bushels of wheat at 75c. per bushel, 40 bushels of rye at 50c. per bushel, and 12 bushels of barley at 37.5c. per bushel, be mixed together, what is a bushel of the mixture worth?

19 x 75 =14.25D.
40 x 50 =20.00
12 x 371= 4.50

71

71)38.75(54c. 6m. Ans. 2. If 4 ounces of silver, worth 62.5c. per ounce, be melted with 8 ounces at 50c. per ounce, what is an ounce of this mixture worth?

Ans. 54c. 3. A goldsmith melted together 8 ounces of gold, 22 carats fine ; 1 lb. 8 oz. of 21 carats fine; and 10 oz. of 18 carats fine ; what is the quality of fineness of this composition ?

Ans. 201, carats fine. 4. In buying tea, I pay 90c. per lb. for 12 lbs. and 11.20 per lb. for 16 lbs.; what is the mixture worth per lb.? Ans. D1.07.1+.

5. A wine merchant mixed 12 gallons of wine, at 75c. per gallon, with 24 gallons at 90c. per gallon, and 16 gallons at D1.10; what is l gallon worth?

Ans. 92c. 6ın. 6. 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 12,740; from 12 to 3, 840; and from 3 to 6, 700; what was the mean temperature of the day? Ans. 73o.

ALLIGATION ALTERNATE

Is the method of finding what quantity of each of the ingredients whose rates are given, will compose a mixture of a given rate; so that it is the reverse of Alligation Medial, and may be proved by it.

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.

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

2. 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 mixtures is linked with one or more that is greater, and every one that is greater with one or more that is less.

3. 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 at that price.

1. If you would mix wines worth 16c., 18c., and 22c. per quart in such a way that the mixtures be worth 20c. quart, how much must be taken of each sort ? Thus :

2 at 16c. 2 qts. at 16c. given price 20c.

18
2 at 18c. 2

18c.
22: 4+2=6 at 22c. 6

22c. 2. How much corn at 42c., 60c., 67c., and 78c., per bushel, must be mixed together that the compound may be worth 64c.

16

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per bushel ?

Ans. 14 bushels at 42c., 3 bushels at 60c., 4 bushels at 67c., and 22 bushels at 78c.

3. What quantity of oats at 50c. per bushel, and 30c. per bushel, must be mixed together, that the compound may be worth 40c. per bushel ? Ans. an equal quantity of each sort.

4. How much rye at 50c. per bushel, barley at 371c. per bushel, and oats at 25c. per bushel, will make a mixture worth 31c. per bushel ?

Ans. 6 bushels at 50c., 6. at 37°c., aud 251 at 25c. Note.--If all the given prices are greater or less than the mean or given price, they must be linked to a cipher. A variety of answers may be obtained, according to the method of linking.

RULE II.

1. Find the proportional quantities of the simples as in rule 1st. 2. Then say as the number opposite the simples whose quantity is given, is to the given quantity, so is either proportional quantity to the part of its simple to be taken.

5. What quantity of cossee at 20c. and at 16c. must be mixed with 35 lbs. at 14c. per lb. to make a mixture worth i

18c.
per

Ib.!
2:35 :: 2 : 35lbs. at 16c.
18 16.

14

)

2:35 :: 6:105lbs. at 20c. 20 4+26 For 35 X 2=-2= 35 at 16c.?

Ans. and 35 X 6-2=105 at 20c.

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6. A farmer would mix 14 bushels of wheat at D1.20 per bushel, with rye at 72c., barley at 48c., and oats at 36c.; how much must be taken of each sort to make the mixture worth 64c. per bushel ?

Ans. 14 bushels of wheat, 8 bushels of rye, 4 bushels of barley, and 28 bushels of oats.

7. A person wishes to mix 10 bushels of wheat at 70c. per bushel, with rye at 48c., corn at 36c., and barley at 30c. per bushel, so that a bushel of this mixture may be worth 38c.; what quantity of each must be taken ?

Ans. 2} bushels of rye, 124 bushels of corn ; 40 bushels of barley.

8. How much water-must be mixed with 100 gallons of wine worth 90c. per gallon, to reduce it to 75c. per gallon.

Ans. 20 gallons.

When the quantity of the compound is given as well as the price.

RULE III.

1. Find the proportional quantities as in rule 1st.

2. Then say, as the sum of the proportional quantities is to the given quantity, so is each proportional quantity to the part to be taken of each.

9. A grocer has currants at 4c. 6c., 9c., and llc., per pound, and he wishes to make a mixture of 240lb., worth 8c. per pound; what quantity of each kind must be taken ? -3

10 : 3 :: 240 : 72lbs. at 4c.
1

10 : 1 :: 240 : 24lbs. at 6c.
8
Then

• Ans.
9
-2

10 : 2 :: 240 : 48lbs. at 9c.
10 : 4 :: 240 : 96lbs. at llc.

10

10. How much water, at 0 per gallon, must be mixed with wine at 80c. per gallon, so as to fill a vessel of 90 gallons, which may be offered at 50c. per gallon?

Ans. 56 gallons of wine, 339 gallons of water. 11. A goldsmith has several sorts of gold, namely: of 15, 17, 20, and 22 carats fine, and would melt together, of all these sorts, so much as may make a mass of 4002., 18 carats fine; how much of each sort is required ? Ans. 16oz. of 15 carats fine, 8oz. of 17, 4oz. of 20, 12oz. of 22.

12. How much barley at 40c. per bushel, rye at 60c., and wheat at 80c., must be mixed together that the compound may be worth 62 c. per bush. ? Ans. 17} bu. barley, 17} rye, 25 wheat

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13. A composition was made of 5 lbs. of tea at Dit per lb., 9 lbs. at D1.80 per lb., and 17 lbs. at Di} per lb. ; what is a pound of it worth?

Ans. D1.54.6. 14. A grocer would mix different quantities of sugar, namely: one at 20c., one at 23c., and one at 26c. ; what quantity ol each sort must be taken to make a mixture worth 22c. per lb. ?

Ans. 5 lbs. at 20c., 2 at 23c., and 2 at 26c. Demonstration.-By connecting the less rate with the greater, and placing the difference between them and the mean rate alternately, or one after the other, in turn; the quantities are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss, upon the whole, are equal, and exactly the proposed rate. In like manner, let the number of simples be what it may, and with how many soever each one is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss or gain between every two, and consequently an equal balance on the whole. The rule is founded on the principles of proportion.

What is Alligation ? What is Alligation Medial? How do you find the price of the mixture ? What is the rule ? What is Alligation Alternate ? How can you prove it? How do you find the proportional parts when the price only is given ? Repeat rule 1st.

What is the rule when a given quantity of one of the simples is to be taken?

What is the rule when the quantity of the compound as well as the price is given ? What more can you say or Alligation ?

15. Bought a pipe of wine, containing 120 gallons, at D1.30 a gallon ; how much water must be mixed with it to reduce the first price to D1.00 a gallon ?

Ans. 21 i gallons.

REVIEW.

ARITHMETICAL PROGRESSION.

Any series of numbers more than two, increasing or decreasing by an even ratio or common difference, is in Arithmetical Progression. When the numbers are formed by continual addition of the ratio or common difference, they form an ascending series ; but when formed by continual subtraction of the common difference, it is a descending series. Thus

$ 0, 2, 4, 6, 8, 10, &c., is an ascending arithmetical series.

1, 2, 4, 8, 16, 32, is an ascending geometrical series. 10, 8, 6, 4, 2, 0, is a descending arithmetical series.

is a descending geometrical series.

And { 32, 16,8, 4, 2, 1,

The numbers which form the series are called the terms of the progression.

The first and last terms of a progression are called the extremes, and the other terms the means. Any three of the fol lowing things being given, the other two may be easily found 1st, the first term; 2d, the last term ; 3d, the number of terms; 4th, the common difference ; 5th, the sum of all the terms. The first term, the last term, and the number of terms being given,

to find the common difference.

RULE I.

Divide the difference of the extremes by the number of terms, less 1, and the quotient will be the common difference sought.

1. The extremes are 3 and 39, and the number of terms is 19; what is the common dfference ?

Extremes.

39—3=36-19-1=18, or 18)36(2 Ans. 2. A man had 10 sons whose several ages differed alike; the youngest was 3 years old, and the eldest 48; what was the common difference of their ages ?

Ans. 5 years. The first term, the last term, and the number of terms, being given,

to find the sum of all the terms.

RULE II.

Multiply the sum of the extremes by the number of terms, and half the product will be the answer.

3. A lady purchased 19 yards of riband, for which she gave 1c. for the first yard, 3c. for the second, and 5c. for the third yard, increasing 2c. per yard ; required the cost ? 19 less l=18 then 1+37=38

a com. diff.

x19 no. of terms,

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2x

36

2)722 1st term, +1

half product, D3.61 Ans. last term, 37 4. If 100 stones were laid two yards distant from each other, in a right line, and a basket placed two yards from the first stone ; what distance would a person travel, to gather them singly into a basket ?

Ans. 11 m., 3 fur. 180 yds. NOTE.—In this question, there being 1760 yards in a mile, and the man returning with each stone to the basket, his travel will be doubled.

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