Demonstration. If four numbers are proportional, we have seen that the product of the means is equal to the product of the extremes; (Art. 498;) therefore the product of the second and third terms must be equal to that of the first and fourth. But if the product of two factors is divided by one of them, the quotient will be the other; (Art. 156;) consequently, when the first three terms of a proportion are given, the product of the second and third terms divided by the first, must give the fourth term or answer.

The object of placing that number, which is of the same kind as the answer, for the third term, instead of the second, as is sometimes done, is twofold: 1st, it avoids the necessity of the Rule of Three Inverse; 2d, the third term, in many cases, has no ratio to the first; consequently it is inconsistent with the principles of proportion to put it for the second term. Thus, in the ninth example, if we put $96 for the second term, it would read, 8 yds.: $96:: 20 yds.: $240, the answer. But a yard can have no ratio to a dollar; for one cannot be said to be greater nor less than the other. (Art. 476. Obs. 2.)

OBS. 1. If the first and second terms are compound numbers, reduce them to the lowest denomination mentioned in either, before the multiplication or division is performed.

When the third term contains different denominations, it must also be reduced to the lowest denomination mentioned in it.

2. The process of arranging the terms of a question for solution, or putting it into the form of a proportion, is called stating the question.

3. Questions in Simple Proportion, we have seen, may be solved by Analysis. After solving the following examples by proportion, it will be an ex:ellent exercise for the student to solve them by analysis. (Art. 462. Obs. 2.)

11. If 16 barrels of flour cost $112, what will 129 barrels cost? 12. If 40 acres of land cost $540, what will 97 acres cost? 13. If 641 sheep cost $1923, what will 75 sheep cost? 14. At the rate of 155 miles in 12 days, how far can a man travel in 60 days?

15. How much hay, at $17.50 per ton, can you buy for $350 ? 16. If $45 buy 63 lbs. of tea, how much will $1540 buy? 17. If 90 lbs. of pepper are worth 72 lbs. of ginger, how many lbs. of ginger are 64 lbs. of pepper worth?

18. A bankrupt compromised with his creditors, at 64 cts. on a dollar; how much will be received on a debt of $2563.50?

19. An emigrant has a draft for £1460 sterling: how much is it worth, allowing $4.84 to a pound?

EST. Obs. If the first and second terms contain different denominati ns, how proceed? When the third term contains different denominations, what is to be one? What is meant by stating a question?

SIMPLE PROPORTION BY CANCELATION.

20. If 72 tons of coal cost $648, how much will 9 tons cost?

Operation.

Tons. Tons. Dolls.

72 9: 648: Ans.
8:1

Now $648÷8=$81. Ans.

=

Having stated the question as before, we perceive the factor 9 is common to the first two terms, and therefore may be canceled. (Art. 151.)

the answer. (Art. 503.)

9X648
72,8

$81, the same as before. Hence,

504. When the first term has factors common to either of the other two terms.

Cancel the factors which are common, then proceed according to the rule above. (Arts. 151, 221.)

PROOF.-Place the answer in the denominator, or on the left of the perpendicular line, as the case may be, and if the factors of the divisor exactly cancel those of the dividend, the work is right.

OBS. 1. The question should be stated, before attempting to cancel the common factors. When the terms are of different denominations, the reduction of them may sometimes be shortened by Cancelation.

2. Instead of points, it may sometimes be more convenient to place a perpendicular line between the first and second terms, as in division of fractions. (Art. 231.) In this case the third term should be placed under the second, with the sign of proportion (::) before it to denote its origin, and its relation to the fourth term or the answer.

3. It will be perceived that cancelation is applicable in Simple Proportion to all those examples, whose first term has one or more factors common to either of the other terms.

21 If 24 yds. of cloth cost $63, what will 32 yds. cost?

Operation.

24 yds. $20 yds., 40

$

:: $68, 21

When arranged in this way, th question is read, 24 yds. is to 320

Ans. |$21×40=$840. yds., as $63 is to the answer required 22. If 20 bu. of oats cost £1, how much will 2 quarts cost

23. If 12 bbls. of flour cost $88, what will 108 barrels cost? 24. If 30 cows cost $480, what will 173 cows cost?

25. If a man can travel 240 miles in 16 days, how far can he travel in 29 days?

26. If 48 men can build a ship in 84 days, how long would it take 16 men to build it?

27. If of a ton of hay costs £4, what will § of a ton cost?

Solution.

ton. ton. £

: : Ans. Now, tקׇ=£2 Ans. Hence,

505. If the terms in a proportion are fractional, the question is stated, and the answer obtained in the same manner as if they were whole numbers.

OBS. When the first and second terms are fractions, we may reduce them to a common denominator, and then employ the numerators only; for the ratio of two fractions which have a common denominator, is the same as the ratio of their numerators. (Art. 487. Obs. 2.)

28. If of a cord of wood cost $1.35, what will of a cord cost?

29. If of a yard of berége cost 6 shillings, what will of a yard cost?

30. If of a yard of sarcenet cost of a dollar, what will 3 yds. cost?

31. If of a pound of chocolate cost of a dollar, what will 25 pounds cost?

32. What will 165 melons cost, at of a dollar for 5 melons? 33. A man had 420 acres of land which he wished to divide among his three sons A, B, and C, in proportion to the numbers 7, 5, and 3: how much land would each receive ?

Solution. Since the several parts are to be proportional to the numbers 7, 5, and 3, the sum of which is 15, it is evident that the sum of all the given numbers is to any one of them, as the whole quantity to be divided to the part corresponding to the number used as the second term.

That is 157:: 420A. to A's share, which is 196 acres;

PROOF.-196+140+84-420A. the given number. (Ax. 11.)

506. Hence, to divide a given number or quantity into parts which shall be proportional to any given numbers.

Place the whole number or quantity to be divided for the third term, the sum of the given numbers for the first term, and each of the given numbers respectively for the second; then multiply and divide as before. (Art. 503.)

34. A farmer wishes to mix 100 bushels of provender of oats and corn in the ratio of 3 to 7: how many bushels of each must he put in?

35. Bell metal is composed of 3 parts of copper, and 1 of tin: how much of each ingredient will be used in making a bell which weighs 2567 pounds?

36. Gunpowder is composed of 76 parts of nitre, 14 of charcoal, and 10 of sulphur: how much of each of these ingredients will it take to make a ton of powder ?

37. If 40.12 lbs. of sugar are worth $5.13, how much can be bought for $125.375?

38. The Vice-President's salary is $5000 a year: if his daily expenses are $10, how much can he lay up?

39. If lb. of snuff cost £4, what will 150 lbs. cost?

40. If of of of a sloop cost $1500, what will the whole cost? 41. If of of an acre of land on Broadway is worth $8200, how much is of of an acre worth?

42. A man bought of a vessel and sold of what he bought for $8240, which was just the cost of it: what was the whole vessel worth?

43. How many times will the fore wheel of a carriage which is 7 ft. 6 in. in circumference turn round in going 100 miles?

44. How many times will the hind wheel of a carriage 9 ft. 2 in. in circumference, turn round in going the same distance? 45. There are two numbers which are to each other as 12 to 34, the smaller of which is 75: what is the larger?

46. What two numbers are those which are to each other as 5 to 6, the greater of which is 240?

47. If two numbers are as 8 to 12, and the less is 320, what is the greater?

48. There are two flocks of sheep which are to each other as 15 to 20, and the greater contains 500: how many does the less contain?

49. An express traveling 60 miles a day had been dispatched 5 days, when a second was sent after him traveling 75 miles a day: how long will it take the latter to overtake 、ne former ?

50. A fox has 150 rods the start of a hound, but the hound runs & rods while the fox runs 5 rods: how far must the hound un before he catches the fox?

51. A stack of hay will keep a cow 20 weeks, and a horse 15 weeks how long will it keep them both?

52. A traveler divided 80s. among 4 beggars in such a manner, that as often as the first received 10s., the second received s., the third 8s., and the fourth 7s.: what did each receive?

53. Pure water is composed of oxygen and hydrogen in the ratic of 8 to 1 by weight: what is the weight of each in a cubic foot of water, or 1000 ounces avoirdupois ?

COMPOUND PROPORTION.

507 COMPOUND PROPORTION is an equality between a compound ratio and a simple one. (Arts. 479, 480.)

Thus, 6 3 123, is a compound proportion.

Into 4:2)

That is, 6×4 : 3×2 :: 12:3; for, 6×4X3=3×2×12.

OBS. Compound proportion is chiefly applied to the solution of examples which would require two or more statements in simple proportion. It is sometimes called Double Rule of Three.

Ex. 1. If 8 men can reap 32 acres in 6 days, how many acres can 12 men reap in 15 days?

Suggestion. When stated in the form of a compound proportion, the question will stand thus:

That is, the product of the antecedents 8X6, has the same

QUEST.-507. What is compound proportion? Obs. To what is it chiefl #pplied?

What is it sometimes called?