Q. What part of 10 is 9 or, what is the ratio of 10 to 9? A. 10.

Q. What is the ratio of 17 to 18?

18

A. 19.

A. 17.

Q. What is the ratio of 18 to 17? Q. What part of 3 oz. is 12 oz.? or, what is the ratio of 3 oz. to 12 oz.? A. 24, ratio.

Q. What part of four yards is 9 yds.? or, what is the ratio of 4 to 9? A. =24.

Q. Hence, to find the ratio of one number to another, how do you proceed? A. Make the number which is mentioned last (whether it be the larger or smaller), the numerator of a fraction, and the other number the denominator; that is, always divide the second by the first.

1. What part of $1 is 50 cents? or, what is the ratio of $1 to 50 cents?

A. $1100 cents; then,, the ratio, Ans.

2. What part of 5 s. is 2 s. 6 d.? or, what is the ratio of 5 s. to 2 s. 6 d.?

=

2 s. 6 d. 30 d., and 5 s. =60 d. ; ratio, Ans.

3. What is the ratio of £1 to 15 s.? 4. What is the ratio of 2 to 3? A.

Of 20 to 4? A. . Of 8 to 63? A. 73.

. Of 4 to 20? A. 5. Of 200 to 900 ? A. 44. Of 800 to 900? A. 13. Of 2 quarts to 1 gallon? A. 2. Let us now apply the principle of ratio, which we were in pursuit of, to practical questions.

PROPORTION. 22. If 2 melons cost 8 cents, what will 10

cost?

It is evident, that 10 melons will cost 5 times as much as 2; that is, the ratio of 2 to 10 is 10=5; then, 5 X 8=40, Ans. But, by stating the question as before, we have the following proportions:

Q. When, then, numbers bear such relations to each other, what are the numbers said to form? Ans. A proportion.

Q. How may proportion be defined, then? A. It is an equal ity of ratios.

How many numbers must there be to form a ratio?

A. Two.

Q. How many to form a proportion? A. At least, three. To show that there is a proportion between three or more numbers, we write them thus:

melons. melons. cents.
2 : 10: 8 :

cents.

40,

which is read, 2 is to 10 as 8 is to 40; or, 2 is the same part of 10 that 8 is of 40; or, the ratio of 2 to 10 is the same as that of 8 to 40.

Q. What is the meaning of antecedent? A. Going before. Q. What is the meaning of consequent? A. Following. Q. What is the meaning of couplet? A. Two, or a pair. Q. What may both terms of a ratio be called? A. A couplet. Q. What may each term of a couplet be called, as 3 to 4? A. The 3, being first, may be called the antecedent'; and the 4, being after the 3, the consequent.

Q. In the following proportion, viz. 2: 10 :: 8: 40, which are the antecedents, and which are the consequents? A. 2 and 8 are the antecedents, and 10 and 40 the consequents. Q. What are the ratios in 2 10: 8:

40?

In the last proportion, 2 and 40, being the first and last terms, are called extremes; and 10 and 8, being in the middle, are called the means. Also, in the same proportion, we know that the extremes 2 and 40, multiplied together, are equal to the product of the means, 10 and 8, multiplied together, thus; 2 X 40=80, and 10 X 880. Let us try to explain the reason of this. In the foregoing proposition, the first ratio, 10, (= 5), being equal to the second ratio, 40, (= 5), that is, the fractional ratios being equal, it follows, that reducing these fractions to a common denominator will make their numerators alike; thus, 40 and 40 become & and ; in doing which, we multiply the numerator 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. Q. When, then, any four numbers are proportional, what may we learn respecting the product of the extremes and means? A. That the duct of the extremes will always 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:

pro

Multiplying together 8 and 10, the two means, makes 80; then 80 ÷ 40, the known extreme, gives 2, the other extreme required, or first term. Ans. 2.

Again, let us suppose the 10 absent, the remaining terms

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 exemplify 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 proportion thus:

OPERATION.

horses. horses. bushels.

10 : 40 :: 30

40

1|0)120|0

120 bushels, Ans.

By multiplying together 40 and 30, the two means, we have 1200, which, divided by the known extreme, 10, gives 120; that is, 120 bushels, for the other extreme, or 4th term, that was required. Let us apply the principle of ratio in

finding the 4th term in this example. The ratio of 10 to 40 is 48=4, that is, 40 horses will consume 4 times as many bushels as 10; then 4 X 30 bu. = 120 bushels, the 4th term, or extreme, as before.

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

Q. How, then, may it be defined? A. It is the process of finding, by the help of three given terms, a fourth term, be tween which and the third term there is the same ratio or proportion as between the second and first terms.

It will sometimes be necessary to change the order of the terms; but this may be determined very easily by the nature of the question, as will appear by the following example:24. If 8 yards of cloth cost $4, what will 2 yards cost?

In this example, since 2 yards will cost a less sum than 8 yards, we write 2 yards for one mean, which thus becomes the multiplier, and 8 yards, the known extreme. for the divisor; for the less the multiplier, and the greater the divisor, the less will be the quotient; then, 2 X4=88=$1, Ans. But multiplying by the ratio will be much easier, thus; the ratio of 8 to 2 is s; then, 4X = $1, Ans., as before.

From these illustrations we derive the following

RULE.

I. Which of the three given terms do you write for a third term? A. That which is of the same kind with the answer.

II. How do you write the other two numbers, when the answer ought to be greater than the third term? A. I write the greater for a second term, and the less for a first term.

III. How do you write them, when the answer ought to be less than the third term? A. The less for a second term, and the

greater for a first term.

IV. What do you do, when the first and second terms are not of the same denomination? A. Bring them to the same by Reduction Ascending or Descending.

V. What is to be done, when the third term consists of more than one denomination? A. Reduce it to the lowest denomination mentioned, by Reduction.

VI. How do you proceed in the operation? A. Multiply the second and third terms together, and divide their product by the first term; the quotient will be the fourth term, or answer, in the same denomination with the third term.

VII. How may this process of multiplying and dividing be, in most cases, materially shortened? A. By multiplying the third term by the ratio of the first and second, expressed either as a fraction in its lowest terms, or as a whole number.

VIII. If the result, or fourth term, be not in the denomination required, what is to be done? A. It may be brought to it by Reduction.

IX. If there be a remainder in dividing by the first term, or multiplying by the ratio, what is to be done with it? A. Reduce it to the next lower denomination, and divide again, and so on, till it can be reduced no more.*

As this rule is commonly divided into direct and inverse, it may not be amiss, for the benefit of some teachers, to explain how they may be distinguished; also, to give the rule for each.

The Rule of Three Direct is when more requires more, or less requires less. It may be known thus; more requires more, when the third term is more than the first, and requires the fourth term, or answer, to be more than the second; and less requires less, when the third term is less than the first, and requires the fourth term, or answer, to be less than the second.

RULE 1. State the question, that is, place the numbers so that the first and third terms may be of the same name, and the second term of the same name with the answer, or thing sought.

2. Bring the first and third terms to the same denomination, and reduce the second term to the lowest denomination mentioned in it.

3. Divide the product of the second and third terms by the first term; the quotient will be the answer to the question, in the same denomination with the second term which may be brought into any other denomination required.

More Exercises for the Slate.

25. If 600 bushels of wheat cost $1200, what will 3600 bushels 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, 600

=

36.00-6 X 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 bushels cost? A. $1200.

Ratio,

23. If board for 1 year, or 52 weeks, amount to $182, what will 39 weeks come to? A. $136,50. Ratio, & × 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 bush els of potatoes? A. 150 bushels rye. Ratio, 5.

30. If 4 cwt. 1 qr. of sugar cost $45,20, what will 21 cwt. qr. cost? (Bring 4 cwt. 1 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?

The Rule of Three Inverse is, when more requires less, or less requires more, and may be known thus; more requires less, when the third term is more than the first, and requires the fourth term, or answer, to be less than the second; and less requires more, when the third term is less than the first, and requires the fourth term to be more than the second.

RULE. State and reduce the terms as in the Rule of Three Direct; then multiply the first and second terms together, and divide their product by the third term; the quotient will be the answer, in the same denomination with the middle

term.

Note. Although the distinction of direct and inverse is frequently made, still it is totally useless. Besides, this mode of arranging the proportional nuinbers is very erroneous, and evidently calculated to conceal from the view of the pupil the true principles of ratio, and, consequently, of proportion, on which the solution proceeds. The following example will render the absurdity more apparent.

A certain rich farmer gave 20 sheep for a sideboard; how many sideboards may be bought for 100 sheep?

20 sheep 1 sideboard: 100 sheep: 5 sideboards, the 4th term, or Ans. It must appear evident, to every rational mind, that there can be no analogy between 20 sheep and 1 sideboard, or 100 sheep and 5 sideboards. With the same propriety it may be asked, what ratio or analogy there is between such heterogeneous quantities as 2 monkeys and 5 merino shawis; or between 7 lobsters and 4 bars of music; the one is equally as correct as the other.