Art. 167.-The number of terms must be at least four, for the equality is between the ratios of the couplets; and each couplet must have an antecedent and consequent. There may be, however, a proportion between three quantities; for one of the quantities may be repeated, so as to form the two terms. Thus, 6 12:12:24.

Art. 168.-If four numbers are in geometrical proportion, the product of the extremes is equal to the product of the means. Thus, 128:15: 10, for 12 × 10=8×15.

Art. 169.-By multiplying the extremes and means together, a proportion is reduced to an equation. When the product of any two numbers is equal to the product of any other two, the numbers may be formed into a proportion by taking the factors on one side of the equation for the extremes, and those on the other for the means. Thus, 4×3=6×2. Making 4 and 3 constitute the extremes, and 6 and 2 the means, we have the following proportion; 4:2::6:3. Form proportions of the following equations :

6 × 8= 4 × 12
3×12= 4X 9
4× 7=14 X 2

8X 9-12 × 6

Art. 170.-In compounding proportion, equal factors may be rejected from antecedents and consequents. Thus:

12: 4::9: 3

4: 8:3: 6
8:20:6:15

12:20:9:15

Art. 171.-If the corresponding terms of two or more ranks of proportiona' quantities be multiplied together, the products will be proportional. Thus:

12: 4:: 6:2

10: 5:: 8:4

120:20:48:8

Art. 172.-If the terms in one rank of proportionals be divided by the corresponding terms in another rank, the quotients will be proportional.

Art. 173.-If to or from the terms of any proportion, there be added or subtracted the corresponding terms of any other proportion, having the same ratio, their sums or remainders will be proportional. Thus,

14:7:16:

16: 8

4:2:: 6: 3

18:9:22:11

10:5:10: 5

Art. 174.-A factor may be transferred from one mean to the other, or from one extreme to the other, without altering the ratio, 16:8::12:6.

The scholar may be exercised upon the foregoing proposition, in the following manner:

Teacher. What are the factors of the antecedent of the first couplet? Scholar. 4 and 4.

T. Transfer one of these to the consequent, and illustrate.

4

S. 4:12:6. To divide antecedent and consequent by the same. quantity, does not affect the ratio.

7. How does it appear that the antecedent has been divided?

S. We have removed from it the factor, 4. Removing a factor from any quantity, divides by that factor.

7. What are the factors of the antecedent of the second couplet?

S. 4 and 3.

T. Transfer the 4 to the consequent, and illustrate.

S. 4:3: 6

4

7. Remove the denominators from the consequents.

S. 4:8:: 3 : 6.

7. What effect upon the consequents has removing the denominators? T. Is there a proportion between the four following numbers? 4:2::6 3. Illustrate.

S. 4X3-2×6.

T. Remove the factor, 4, from the left of the equation, and illustrate. To divide both members of the equation by the same quantity, does not affect the equation.

S. 3=

2X6
4

T. Do the four following numbers, 16:8::12:6, constitute a proportion?

S. They do.

7. How do you know?

S. The ratios between the couplets are equal.

7. Divide the consequents by 2, and will there then be a proportion ?

S. There will.

7. Are not the ratios affected?

S. They are.

7. Why then is not the proportion destroyed?

S. The ratios are still equal.

7. In what, therefore, does proportion consist?

S. In equality of ratios.

7. How do you ascertain when the ratios are equal?

S. By dividing the antecedents by the consequents, or by dividing consequents by antecedents, or by multiplying the extremes together and the means together. ·

T. How do you reduce a proportion to an equation? How do you form a proportion from an equation? Reduce to an equation the following proportion; 5:10::4:8. Add 2 to each member of the equation. Is the equation affected? Why not? Add 2 to one member, and 3 to the other. Is the equation now affected? Repeat the axiom.

Let the teacher multiply exercises of this kind.

1

3

Art. 175.-Inverse, or reciprocal proportion, is an equality between a direct and reciprocal ratio. Thus, 4:2:: That is, 4 is to 2 as 3 is to 6 reciprocally. Sometimes the crder of the terms is inverted, without writing them in the form of a fraction. Thus, 4:2 Thus, 4:2:: 3:6, inversely. In this case the first is to the second as the fourth is to the third.

We have seen that a factor may be removed from antecedent to consequent, and the reverse, and the proportion still be preserved.

Art. 176.-The terms of the proportion may also be changed, provided that the equality of the ratios be not affected. Thus,

In all these changes the product of the extremes will be found equal to the product of the means. If, therefore, we have the product of the extremes, and one of the means, it is easy to find the other. We can, therefore, find any one term of the proportion when we know the other three, for the term sought must be one of the extremes, or one of the means. The operation by which, three terms being given, a fourth proportional is found, is called the "Rule of Three," or "Rule of Proportion." There must always be three terms or numbers given, two of which are of the same kind, and the other of the kind of the answer required.

SIMPLE PROPORTION, OR RULE OF THREE.

Art. 177.-Proportion is of two kinds, Direct and Inverse. Proportion is direct, when the ratios are in the order in which the question is proposed; Inverse, when one of the ratios is inverted. A question is known to belong to Direct Proportion when more requires more, or less requires less. More requires more, when the second term is greater than the first, and requires that the fourth be greater than the third. Less requires less, when the second term is less than the first, and requires that the fourth be less than the third.

1. If 3 men build 12 rods of wall in a given time, how many rods will 6 men build in the same time?

In this question the ratios are in the order in which the question is reposed, 3 : 12 :: 6 : the answer. More requires more: for, vidently, 6 men will perform more labor in the

same time than 3 men.

We may employ the same numbers in the proposal of a different question, and the ratios will be inverted. ·

2. If 3 men perform a certain amount of labor in 12 days, in how many days would 6 men perform the same?

In this question more requires less: for 6 men would require less time to perform the same amount of labor than 3 men: 3 is to 6 reciprocally as 12 is to the answer, }} 12: the answer.

1 1 3 6

Of the three terms given in Proportion, two are called the terms of condition, and one the term of demand. Thus, 3 men, 12 days, are the terms of condition, and 6 men the term of demand.

It may be observed that, in Proportional questions, the term of demand is the only term which presents any difficulty. The two other terms simply state a fact, or the condition upon which the conclusion rests, and are to be employed as the means of solving the difficulty. The answer to the question proposed, in Direct Proportion, depends upon the ratio of the term of demand to that term of the condition, which is of the

QUESTIONS.-5. What is Proportion? 6. What are the first and last terms called? 7 Having the extremes and one of the means given, how may the other mean be found? 8. What is the Rule of Three? 9. How are the ratios? 10. What is meant by the order in which the question is proposed? 11. How is a question known to belong to the Rule of Three Direct? Illustrate. 12. Rule for stating the question?

same name or kind. In Inverse Proportion, the answer depends not upon the ratio of demand to condition, but of condition to demand. In either case, this ratio multiplied into that term, which is of the same name or kind as the answer required, gives the desired result.

Solution of Question 1st.

×12=24 Ans.

The ratio of 6 men to 3 men expresses how much more labor

6

3

Solution of Question 2d.

labor.

3 X 12 6 Ans.

The ratio of 3 men to 6 men, expresses how much less time 6 6 men can perform in a given men would require than 3 men, time, than 3 men, 2. They to perform the same amount of would perform twice as much. The first step in solving a question is to find the ratio; the second, to multiply. If 3 men build 12 rods, then 6 men will build of 12-24 rods.

3. If 5 tons of hay cost 10 dollars, how many dollars will 20

tons cost?

Does this question belong to Direct or Inverse Proportion? How do you know?

|

If 3 men will perform a certain amount of labor in 12 days, 6 men will perform the same in 3 of 12-6 days. Also, if it takes 3 men 12 days to perform a certain amount of labor, it will take 1 man 3 times as long, 3X12= 36, and 6 men one-sixth as long as 1 man, 36÷÷6=6 days.

4. If 7 men reap a field of grain in 14 days, how many men can reap the same in 21 days?

Does this question belong to Direct or Inverse Proportion? How do you know?

What is meant by more re

What is meant by more requiring more? Illustrate by ex-quiring less? ample 3d.

Which is the term of demand? Which are the terms of condition? Stat the question according to the example given.

What is the first step in the solution? What is the second? Which term presents the difficulty?

How are the other terms to be employed?

How do you know when more requires less?

Are the ratios direct or recip

rocal?

What do you mean by a reciprocal ratio?

Is the ratio that of the demand to the condition, or of the condition to the demand? State the question, and illustrate.

Art. 178.-It has been stated that, in Proportion, we have either the two extremes and one of the means, or the two means and one of the extremes given, to find the other.