merator 40 (one extreme) by the denominator 2 (the other extreme), also the numerator 10 (one mean) by the denominator, 8, (the other mean); hence the reason of this equality. When, then, any four num bers are proportional, what may we learn respecting the product of the extremes and means?

A. That the product of the extremes will al ays be equal to the product of the means.

Hence, with any three terms of a proportion being given, the fourth or absent term may easily be found. Let us take the last example:

Melons. Melons. Cents. Cents.

: 10 :: 8: 40

Multiplying together 8 and 10, the two means, makes 80; then 80 + 43 known extreme, gives 2, the other extreine required, or first term.

Ans. 2.

Again, let us suppose the 10 absent; the remaining terms are Melons. Melons. Cents. Cents.

By multiplying together 40 and 2, the extremes, we have 80; which, divided by 8, the known mean, gives 10, the 2d term, or mean, required. Let us ap emplify this principle more fully by a practical example.

23. If 10 horses consume 30 bushels of oats in a week, how many bushels will serve 40 horses the same time?

In this example, knowing that the number of bushels eaten are in proportion to the number of horses, we write the propo tion thus:

this example. The ratio of 10 to 40 is 84, that is, 40 horses will con sume 4 times as many bushels as 10; then 4 × 30 bu. = 120 bushels, the 4th term, or extreme, as before.

Q. When any three terms of a proportion are given, what process of finding the fourth term called?

A. The Rule of Three.

Q. How, then, may it be defined !

4. It is the process of finding, by the help of three given terms, a fourth term, between which and the third term there is the same ratio or pro portion as between the second and first terms.

More Exercises for the Slate.

25. If 600 bushels of wheat cost $1200, what will 3600 bu els cost? and what is the ratio of the 1st and 2d terms?

Perform the foregoing example, and the following, first, with out finding the ratio; then, by finding the ratio, and multiply ing by it.

A. $7200. The ratio, 3602-6X 1200-$7200, the same. 26. How many bushels of wheat may be bought for $7200, if 600 bushels cost $1200? A. 3600 bushels. Ratio, 6; then, 6 x 600 3600 bushels.

27. If $7200 buy 3600 bushels of wheat, what will 600 bush els cost? A. $1200. Ratio, .

28. If board for 1 year, or 52 weeks, amount to $182, what will 39 weeks come to? A. $136,50. Ratio, X 182=$136,50,

the same.

29. If 30 bushels of rye may be bought for 120 bushels of potatoes, how many bushels of rye may be bought for 600 bushels of potatoes? A. 150 bushels rye. Ratio, 5.

30. If 4 cwt. 1 qr. of sugar cost $45,20, what will 21 cwt. 1 gr. cost? (Bring 4 cwt. I qr., and 21 cwt. 1 qr. into quarters first.) A. $226. Ratio, 5.

31. If I buy 60 yards of cloth for $120, what is the cost per yard?-2. What is the cost per ell Flemish ?-150. What per

ell English ?-250. What per ell French-3. A. $9.

32. Bought 4 tuns of wine for $322,56; what did 1 pipe cost?4032. What did 1 hhd. ?-2016. What did 1 tierce ?-1344. What did 1 bbl.?-1008. A. $84. What did 2 quarts cost?16. What did 3 pints cost?-12. What did 4 gills cost?-4.

A. $,32.

33. Bought 6 tuns of wine for $500,50; what did 1 pipe cost t A. $41,708+.

34. When a merchant compounds with his creditors for 40 cents on the dollar, how much is A's part, to whom he owes $2500? How much is B's part, to whom he owes $1600?

A. A's, $1000; "B's, $640.

35. A, failing in trade, owes the following sums, viz. to B, $1600,60; to C, $500; to D, $750,20; to E, $1000; to F, $230; and his property, which is worth no more than $1020,20, he gives up to nis creditors; how much does he pay on the dollar? and what is the amount of loss sustained by all?

A. 25 cents on $1; and the amount of loss is $3060,60.

It must appear evident, to every rationa; mind, that there can be no analogy between 20 sheep and 1 sideboard, or 100 sheep and 5 sideboaros. With the same propriety it may be asked, what ratio or analogy there is between such heterogeneous quantities as 2 monkeys and 5 merino shawls ; or between 7 lob pters and 4 bars of music; the one is equally as correct as the other

35. Bought 4 tierces of rice, each weighing 7 cwt. 2 qrs. 10 fos.; what do they come to at $9,35 per cwt. ?-285842. As $2,50 per qr.?-305714. At 10 cents per pound ?-34240.

A. $933,956+

37. Bought, by estimation, 300 yards of cloth, for $450,60; but, by actual measurement, there were no more than 275 yds 2 qrs.; for how much must I sell the measured yards per yd., so as to neither make nor lose?-1635. How much must I sell 20 yds. for, so as to lose nothing?-32711. How much 25 yds. e qrs. ?-41707. How much 30 yds. 1 qr. 3 na. ?-49782.

A. $125,835+. 38. If a staff 6 feet long cast a shade on level ground 9 feet, what is the height of that steeple whose shade measures at the same time 198 feet? A. 132 feet in height. Ratio, 22.

39. If 3 cwt. 2 qrs. 16 lbs. of sugar cost $51, what will 10 ewt. 3 qrs. 20 lbs. cost?-Ratio, 3 X 51: = 153. What will 21 ewt. 3 qrs. 12 lbs. ?-6, ratio, x 51306. What will 43 cwt. 2 qrs. 24 lbs. ?-Ratio, 12; then, 12 × 51=612. A. $1071.

Note.-The following examples may be performed either by analysis, or by finding the ratio, or by the common rule. Perhaps it would be well to let the pupil take his choice. The one by ratio is recommended.

40. If $100 gain $6 in a year, how much will $20 gain in the same time?-120. Ratio, . How much will $10 gain?60. Ratio, How much will $50 gain?-3. Ratio,. How much will $75 gain?-450. Ratio, . What will $200 gain 212. Ratio, 2. How much will $300 gain?-18. Ratio, 3. How much will $500 gain?-30. Ratio, 5. How much will $800 gain?-48. Ratio, 8. How much will $1000 gain?-60. Ratio, 10. How much will $1250 gain?-75. Ratio, 12. How much will $2000 gain ?-120. Ratio, 20. A. $372,30.

41. If 12 men can build a wall in 20 days, how many can do the same in 5 days? A. 48 men. Ratio, 4.

42. If 60 men can build a wall in 4 days, how many men can do the same in 20 days? A. 12 men. Ratio, .

43. If 4 men can build a wall in 120 days, in how many days will 12 men do the same ?-40. Will 16 men ?-30. Will 20 men?-24. Will 24 men ?-20. A. 114 days.

44. If a man perform a journey in 30 days, by travelling 6 hours each day, in how many days will he perform it by travelling 10 hours each day? (10 hours will require a less number of days than 6 hours; that is, the multiplying or 2d term must be the smallest.) A. 18 days. Ratio, x 3018, the same. 45. If a field will keep 2 cows 20 days, how long will it keep 5 cows- Will it keep 8 cows?-5. Will it keep 10 cows?

ག

Will keep 20 cows-2. 4. 19 days. 46. 1 bushels of grain, at $1 per bushel, will pay a debt, how many bushels, at $1,50 a bushel, will pay the same ?-40.