9. I purchase flour at $4.75 per barrel. What must I sell it for to gain 12 per cent.?

10. A merchant bought a ship for 11475 dollars, and sold her for $13680. What did he gain per cent.?

What

11. The population of the state of New York, in 1810, was 959949. In 1820, it was 1372812. was the gain per cent. in that term of 10 years? 12. In 1830, it was 1918604. What was the gain per cent. from 1820 to 1830?

13. In 1840, it was 2428921. What was the gain per cent. from 1830 to 1840?

14. The population of Ohio, in 1810, was 230760. In 1820, it was 581434. What was the gain per cent.?

15. The population of Ohio, in 1830, was 937903. What was the gain per cent. from 1820 to 1830? 16. In 1840, it was 1519467. What was the gain per cent. from 1830 to 1840?

17. Massachusetts had, in 1810, 472040 inhabitants. In 1820, it had 523287. What was the gain per cent. in 10 years?

18. Massachusetts had, in 1830, 610408 inhabitants. What was her gain per cent. from 1820 to 1830?

19. In 1840, Massachusetts had 737699 inhabitants. What was the gain per cent. from 1830 to 1840? 20. An agent sells 12000 dollars' worth of cloth for a factory, charging 2 per cent. commission. What will be his remuneration?

21. If I buy for a merchant, at a commission of 4 per cent., 500 barrels of flour, at $4.40 per barrel, what am I entitled to for my commission?

22. What is 3 per cent. on $674.54?

23. What is 2 per cent. on $781.50?

24. What is the value of five 100 dollar shares in a bank, at 4 per cent. advance?

25. What is the value of seven 100 dollar shares, at 6 per cent. discount?

26. What is the value of 18 shares bank stock, 60 dollars a share, at 4 per cent. discount?

27. What is the duty on a quantity of broadcloth, whose value is 1735 dollars, at 15 per cent. ?

28. What is the duty on a quantity of iron, whose value is 3456 dollars, at 18 per cent.?

29. What is the commission on the sale of 1246 dollars' worth of cloth, at 3 per cent.?

30. A man bought a lot of hay for 13 dollars a ton. He sold it for $14.25 a ton. What did he gain per

cent.?

31. Bought tea for 46 cents a pound. What must I sell it for a pound to gain 12 per cent.?

32. What is the worth of 750 dollars, bank stock, at 7 per cent. advance?

33. What is the worth of 8500 dollars, bank stock, at 9 per cent. discount?

34. I sell flour at $5.32 per gain 12 per cent. on my outlay.

cost?

barrel, and thereby What did the flour

Every $1.00 laid out in the purchase has brought me a return of $1.12. The number of dollars I paid out on a barrel must therefore equal the number of times $1.12 will go in $5.32.

35. A merchant sells a ship for 13680 dollars, gaining thereby 14 per cent. on what she cost him. What did the ship cost?

36. 300 dollars is 2 per cent. on what sum?

37. $15.63 is 2 per cent. on what sum?

38. Bought 12 barrels of flour, each containing 196 pounds, at $5.42 per barrel, and sold it at 26 cents for 7 pounds. How much gain in the whole, and how much gain per cent.?

39. Bought 43 dozen pairs of shoes, at dozen, and sold them at 62 cents per pair. in all? What gain per cent.?

$4.30 per What gain

40. Bought 20 barrels of apples, each containing 2 bushels, at $2.10 per barrel, and sold them at $1.25 per bushel. What gain in all? What gain

per cent.?

41. Bought 375 barrels of flour, at $5.20 per barrel, and sold 200 barrels at $6.10; the remainder at $6.42 per barrel. What gain in all? What gain per cent.?

42. Bought 34 acres of land, at 41 dollars per acre. Sold it for $1700.00. How much gain in all? What gain per cent.?

SECTION XXXIII.

ALLIGATION.*

The operations under this rule show the method of finding the value of a mixture, when the price and quantity of each of its ingredients are given; also, to find the quantity of each ingredient, when its price is given, and it is required to unite them so as to form a mixture of a given value.

CASE 1.-To find the value of the mixture, when the quantity and price of each of the ingredients are given.

1. Mix 15 bushels of oats, at 40 cents per bushel; 12 bushels of barley, at 60 cents; and 24 bushels of

* The word alligation signifies a tying together, and has reference to a particular way of linking numbers together, by means of which operations of this kind have been performed. The name is retained as a matter of convenience; but I have thought it best for the progress of the pupil, that he should pursue a strictly analytical method in all the operations.

corn, at 83 cents. What will the mixture be worth per bushel?

It is evident that, if you find the value of the whole, and divide the sum by the number of bushels, the quotient will be the value per bushel.

2. Mix 20 pounds of tea, at 43 cents per pound; 18 lbs. at 61 cents; and 11 lbs. at 74 cents per pound. What will the mixture be worth?

3. If 41 lbs. of coffee, at 13 cents per lb., be mixed with 45 lbs. at 94 cents, and 27 lbs. at 15 cents, what will the mixture be worth per pound?

CASE 2. To find the quantity of each ingredient, when its price and that of the required mixture are given.

4. If I mix oats, worth 2s. per bushel, with rye, worth 5s., so as to make the mixture worth 3s. per bushel, in what proportion must I mix them?

It is evident, that, if I put in 1 bushel of oats, I gain 1 shilling. Now, I must put in rye enough with this bushel of oats to lose 1 shilling. On every bushel of rye put in, I lose 2 shillings; therefore, in order to lose 1 shilling, I must put in a bushel. I must therefore put in 1 bushel of oats to a bushel of rye. It is evident that, if I double the quantity thus found of each ingredient, the value of the mixture will be the same; or I may take any equal multiples of the quantities, as 4 bushels of oats and 2 bushels of rye, 6 bushels of oats and 3 bushels of rye, 20 bushels of oats and 10 bushels of rye, &c.

5. If I mix oats, worth 2s. per bushel, with rye, worth 6s., so as to make the mixture worth 3 s. per bushel, in what proportion must they be mixed ?

6. Mix oats, worth 3s. per bushel, with wheat, worth 7 s., so as to make the mixture worth 5 s. per bushel. In what proportion must they be mixed?

In

7. Mix the same ingredients, at the same price, so as to make the mixture worth 6s. per bushel. what proportion must they be mixed?

8. In what proportion must oats, worth 2s., and wheat, worth 8s., be mixed, to make the mixture worth 4s. per bushel?

9. How can you mix corn, worth 80 cents per bushel, and rye, worth 85 cents, with barley, worth 46 cents, so as to make a mixture worth 60 cents per bushel?

Here you have three ingredients. First, mix barley with one of the dearer ingredients, so as to make a mixture of the required value. Then mix barley with the other ingredient, and see how much you have taken of each.

10. Mix 3 sorts of tea, at 25 cents, 33 cents, and 40 cents, per pound, so as to make a mixture worth 30 cents per pound.

11. Mix tea at 20 cents, with tea at 45 cents, and tea at 54 cents, per pound, so as to make a mixture worth 38 cents per pound.

12. If you mix sugar, at 6 cents, 8 cents, 10 cents, and 11 cents, per pound, in what quantities may they be taken so as to make a mixture worth 9 cents per pound?

First, take two of the ingredients, one cheaper and one dearer than the mixture. Form a mixture of these. Then take the two remaining ingredients in the same way.

13. If three sorts of spirit, worth 60 cents, 75 cents, and 80 cents, per gallon, are mixed with water, costing nothing, what must be the proportion to make a mixture worth 70 cents per gallon?

It is immaterial in what way you select the pairs of ingredients, provided, in each pair, one of the ingredients be cheaper and the other dearer than the