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(183.) If four quantities are proportional, the product of the two extremes is equal to the product of the

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and, by clearing the equation of fractions, we have

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(184.) Conversely, if the product of two quantities is equal to the product of two others, the first two quantities may be made the extremes, and the other two the means of a proportion.

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dividing each of these equals by bd the expression becomes

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(185.) The preceding proposition is called the test of proportions, and any change may be made in the form of a proportion which is consistent with the ap

QUEST.-If four quantities are proportional, by what property may they be distinguished? How may every equation be converted into proportion

plication of this test. In order, then, to decide whether four quantities are proportional, we must compare the product of the extremes with the product of the

means.

Thus, to determine whether the numbers 5, 6, 7, 8 are proportional, we multiply 5 by 8, and obtain 40. Multiplying 6 by 7, we obtain 42. As these two products are not equal, we conclude that the numbers 5, 6, 7, 8 are not proportional.

Again: take the numbers 5, 6, 10, 12. The product of 5 by 12 is 60, and the product of 6 by 10 is also 60. Hence these numbers are proportional; that is, 5:6:10:12.

(186.) If three quantities are in continued proportion, the product of the extremes is equal to the square of the mean.

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(187.) Conversely, if the product of two quantities is equal to the square of a third, the last quantity is a mean proportional between the other two.

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QUEST.-HOW may we determine whether four quantities are proportional? When three quantities are in continued proportion, by what property are they distinguished? How is a mean proportiona between two quantities found?

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then 6 is a mean proportional between 4 and 9.

Examples.

1. Given the first three terms of a proportion, 24, 15, and 40, to find the fourth term.

2. Given the first three terms of a proportion, 3ab', 4a'b', and 9ab, to find the fourth term.

3. Given the last three terms of a proportion, 4a3b3, 3a'b', and 2a'b, to find the first term.

4. Given the first, second, and fourth terms of a proportion, 5y', 7x'y', and 21x'y, to find the third term. 5. Given the first, third, and fourth terms of a proportion, 22, 72, and 252, to find the second term. 6. Are the quantities 25, 70, 78, and 218 proportional?

7. Resolve the equation 22x105=33×70 into a proportion.

(188.) Ratios that are equal to the same ratio are equal to each other.

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QUEST.-Compare two ratios that are equal to the same ratio.

Thus, if

and

then

2:7::18:63,

2:7::22:77,

18:63:22:77.

(189.) If four quantities are proportional, they will be proportional by alternation; that is, the first will have the same ratio to the third that the econd has to the fourth.

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(190.) If four quantities are proportional, they will be proportional by inversion; that is, the second will have to the first he same ratio that the fourth has to the third

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QUEST.-Explain the principle of alternation. Explain the principle

of inversion.

(191.) If four quantities are proportional, they will be proportional by composition; that is, the sum of the first and second will have to the second the same ratio that the sum of the third and fourth has to the fourth. Let

then will

For, since

we have

a:b::c:d;
a+b:b::c+d:d.
a:b::c:d,

a C

b d

Add unity to each of these equals, and we have

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(192.) If four quantities are proportional, they will be proportional by division; that is, the difference of the first and second will have to the second the same ratio that the difference of the third and fourth has to the fourth.

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Subtract unity from each of these equals, and wo

have

QUEST.-Explain the principle of composition. Explain the princi. ple of division.

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