twice as many sheep as calves. buy?

How many of each did he

13. Divide the number 336 into four such parts that the second shall be twice the first, the third 2 times as much as the first and second, and the fourth as much as the other three.

NOTE. The parts are as the numbers 2, 4, 15 and 21. Why?

14. The greater of two numbers is 7 times the less, and the sum of the numbers is 204. What are the numbers? 15. A spendthrift, after spending of his money and of the remainder, had $500 left. How much had he at first? 16. A cistern has 3 pipes; the first can fill it in an hour, the second in of an hour, and the third in of an hour. How long will it take to fill it if they are all left open together?

17. Three fifths of a certain number exceed of it by 270. What is the number?

18. If it takes a man 3 days to perform a piece of work, working 9 hours per day, how long will it take him to perform it if he works 10 hours per day?

19. If 9 men mow 10 acres of grass in a day, how much will 15 men mow in 3 days?

20. Bought 978 pounds of sugar, at 8 cents per lb.; for of which paid in potatoes at 45 cents, and the rest in cash. How much cash and how many potatoes did I pay?

136. MISCELLANEOUS EXERCISES IN SECTION XII.

1. What is the ratio of 83 to 57? Of 3% to 754? 2. What is the ratio of 8.3 to 9.75? Of 5.47 to 3.3? 3. Express the ratio of .87 to .087; of .00015 to .19, and reduce the ratios to their lowest terms.

4. Express in a simple form the ratio compounded of 8 : 7, and 15: 4.

5. Express in a simple form the ratio compounded of 83: 7, and 15: 46.

6. What is the fourth term of this proportion -81:71:: 25: -?

7. If 1000 bricks will build a wall 9 inches broad, 26 feet long, 4 feet high, how many will build a wall 18 inches broad, 130 feet long, and 6 feet high?

8. If an iron bar 2 feet long, 3 inches broad, and 1 inch thick, weighs 18 lb., what will be the weight of another bar of iron, which is 7 feet long, 6 inches broad, and 34 thick?

9. How many men will build a wall 240 yards long, 6 feet high, and 3 feet thick, in 8 days, when 7 men can build another wall 40 yards long, 4 feet high, and 2 feet thick, in 32 days?

10. A person engaged to complete a portion of railway 490 yards long in 38 days, and for that purpose hired 60 men; but at the end of 22 days, he finds no more than 210 yards finished. How many additional men must he employ, to complete the work in the stipulated time?

11. A, B, and C contract to build a railroad for $9150. A employs 20 men 60 days, B 30 men 50 days, and C 60 men 55 days, and he is to receive $150 for superintending the work. How much should each man receive?

12. Three men purchase a vessel. A pays of the cost, B, and C the remainder, which was $1150. What was C's share, and how much did A and B pay?

13. If 10 barrels of flour can be bought for 45 bushels of wheat, and 15 bushels of wheat for 25 bushels of corn, and 24 bushels of corn for 20 bushels of rye, and 18 bushels of rye for $12.75, how many barrels of flour can be bought for $100?

14. If 25 lb. at New York are equal to 22 lb. at Nuremburg, and 88 lb. at Nuremburg are equal to 92 lb. at Hamburg, and 46 lb. at Hamburg are equal to 49 lb. at Bourdeaux, how many pounds at New York are equal to 98 lb. at Bourdeaux?

15. Six merchants trade after this manner. A puts in $250 for 6 months, and $300 for 4 months; B puts in $450 for 8 months; C puts in $800 for 5 months, and $500 for 4 months; D puts in $200 for 7 months, and $500 for 5 months; E puts in $1500 for 10 months, $500 for 2 months, and $300 for 5 months. Their whole gain is $1000. What part of it shall each have?

16. A commences trade Jan. 1st, with a capital of $1000. April 1, he admits B as a partner, with a capital of $800. Their profits at the end of the year are $650. What is each person's share of the gain, after paying to A a salary of $350, and deducting interest at 6 per cent. on each person's capital? How much of the whole profits must each receive?

NOTE. A's interest is $60; his salary $350. B's interest is $34. Deducting these sums from $650, there remain $206, to be divided in

the ratio of to §, (134,) which gives for A's net profit $128.75; for B's, $77.25. A's net profit, $128.75, + his salary, $350, +his interest, $60,=$538.75. B's net profit, $77.25, + his interest, $34, = $111.25.

=

17. A, B, and C enter into partnership. A at first contributes $500, and in 3 months afterward $250 more; B contributes at first $650, but at the end of 6 months he withdraws $200; Cat first puts in $375, and at the end of 7 months $400 more. At the end of the year they find they have gained $875; $275 of which belongs to A for transacting the business. How much of the whole gain must each receive?

18. Four men own a saw-mill. A paid $1000, B $1600, C $1750, and D $2250. What is their yearly income from the investment, if the mill rents for $600, the taxes and other expenses being $150.50? What per cent. do they realize per annum on the money invested?

137. ALLIGATION is of two kinds, MEDIAL and ALTER

NATE.

ALLIGATION MEDIAL is the process by which we find the average or mean value of a mixture composed of several different ingredients, when the value and quantity of each are given.

1. A grocer mixes 12 lb. of with 15 lb. worth 10 cts. per lb. worth?

sugar worth 8 cents per lb., What is 1 lb. of the mixture

RULE. Divide THE TOTAL VALUE of the quantities by THE SUM of the quantities.

2. If 15 bushels of oats worth 45 cts. per bushel, 10 bushels of rye worth 60 cts., and 12 bushels of barley worth 50 cts. per bushel, be mixed together, what will 1 bushel of the mixture be worth?

3. A goldsmith melted 3 oz. of gold of 18 carats fine, with

5 oz. 20 carats fine, and 4 oz. 21 carats fine, with 2 oz. of alloy. What was the fineness of the mixture?

NOTE. A carat is

of any quantity of gold. Gold 18 carats fine is pure gold; the rest is of some baser metal, which is regarded as of no value.

4. What is the average length of 22 pieces of cloth, of which 5 pieces measure 20 yards each, 8 pieces 20 yards each, 6 pieces 21 yards each, and 3 pieces 214 yards each?

5. A composition is made of 18 lb. of tea at 66 cents per lb., with 20 lb. at 75 cents, and 16 lb. at 78 cents per lb. What is the worth of 3 lb. of this mixture?

138. ALLIGATION ALTERNATE is the process by which we find what quantity of each of several ingredients, whose values are given, will compose a mixture of a given rate.

1. In what proportions must I mix barley worth 50 cts. and oats worth 40 cts. per bushel, that the mixture may be worth 47 cents a bushel?

NOTE. I must mix them so as to gain just as much on the oats as I lose on the barley. I gain 7 cts. for every bushel of oats I use, and as I lose but 3 cents on 1 bushel of barley, I must use as many bushels of barley as there are times 3 in 7, or 2. I must therefore use 24 bushels of barley to 1 of oats, or 7 of barley to 3 of oats; for the ratio of 2 to 1= the ratio of 7 to 3. (126.)

Analyze the following examples in the same manner.
Prove your answers to be correct, by Alligation Medial.

2. In what proportions must a grocer mix sugars worth 8 and 12 cts. together, to make a mixture worth 11 cts.? 10 cts.? 9 cts.?

In making a mixture worth 11 cts., he will gain 3 cents on 1 lb. at 8 cents; but he must use 3 lb. at 12 cts., to lose 3 cents. Ans. 1 lb. at 8 cents to 3 lb. at 11 cents.

3. In what proportions must wine that costs 80 cents per gallon, and water, be mixed together, to reduce the price of the wine to 75 cts. per gallon? 70 cts.? 50 cts.? 60 cts? 85 cts.?

4. In what proportions must a grocer mix wines at 50 cts.

and 60 cts. per gallon, with water, that the value of the mixture may be 45 cts. per gallon? 40 cts.? 35 cts.? 37 cts.?

NOTE. To make a mixture at 45 cents per gallon, by using 1 gallon of water, he gains 45 cents; on 1 gallon of wine at 50 cents, he will lose 5 cents; and on 1 gallon at 60 cents, he will lose 15 cents. He may therefore use 1 gallon at 50 cents, and 23 at 60 cents, to 1 gallon of water; or 1 at 60 and 6 at 50 cents, to 1 gallon of water. The proportions may, therefore, be either 1 at 50, 23 at 60, and 1 of water; or, 6 at 50 cents, 1 at 60, and 1 of water; or, 2 at 50, 2 at 60, and 1 of water; or, 3 at 50, 2 at 60, and 1 of water; or, 4 at 50, 1 at 60, and 1 of water. The proportions may thus be varied indefinitely.

By this method of solving the question, the grocer may use a limited quantity of one or more of the ingredients, or a larger or smaller proportion of either, as may suit his convenience.

5. A grocer has 8 gallons of wine worth 50 cts. per gallon, which he would mix with 3 gallons of wine at 60 cts., and with water, to make a mixture worth 45 cts. per gallon. How much water must he use? How much water, if he use 5 gallons at 50 cts. and 6 at 60 cts.?

6. Mix teas at 30, 36, 40, and 50 cts. per lb., so as to make a mixture worth 42 cents per lb.; 37 cents per lb.; 45 cts. per lb.

7. How much port wine of American manufacture, at $1.75, temperance wine, at $1.25 per gallon, and water, may be mixed together, to make a mixture of 500 gallons that may be sold at $1 per gallon?

NOTE. Find the proportions as above, and then find the quantities, as in Art. 133, quest. 6 to 10.

8. A grocer has two kinds of sugar, worth 8 and 11 cents per lb. How much of each must he take to answer an order for 300 pounds at 9 cents per lb. ?

per

9. I have 2 kinds of cloves, one of which cost me 15 cents lb. and the other 20 cents; I wish to fill an order for 800 pounds at 20 cents. Of how many pounds of each kind shall the mixture be made, that I may gain 20 per cent. on the cost?

10. There is a mixture made of wheat at 4s. per bushel, rye at 3s., barley at 2s., with 12 bushels of oats at 18d. per bushel. How much has been taken of each sort, when the mixture is worth 3s. 6d. ?