An Elementary Treatise on Algebra

To find the Fourth Term of a Proportion.

88. Problem.

to find the fourth.

Given three terms of à proportion,

Solution. The following solution is immediately obtained from the test.

When the required term is an extreme, divide the product of the means by the given extreme, and the quotient is the required extreme.

When the required term is a mean, divide the product of the extremes by the given mean, and the quotient is the required mean.

89. EXAMPLES.

1. Given the three first terms of a proportion respectively

A, B, C; find the fourth.

Ans.

2. Given the three first terms of a proportion respectively 2ab2, 3 a2b, 663; find the fourth.

Ans. 9 a b2.

3. Given the three first terms of a proportion respectively am, an, ap; find the fourth. Ans. an+p-m.

4. Given the first term of a proportion a3 b2, the second 3 a8 63, the fourth 7 a b; find the third. Ans. a-4. 5. Given the first term of a proportion 6 am-2b, the third 15 a3 65, the fourth 40 a-(m-1); find the second.

Ans. 16 a-4b-4.

6. Given the three last terms of a proportion respectively a2 — b2, 2 (a+b), a2+2ab+b2; find the first.

Ans. 2 (a-b).

90. When both the means of a proportion are the same quantity, this common mean is called the mean proportional between the extremes.

Mean Proportional. Continued Proportion.

Thus, when

A: BB: C,

B is a mean proportional between A and C.

91. If the test is applied to the preceding proportion it gives

whence

B2 = A X C;

BAX C ;

that is, the mean proportional between two quantities is the square root of their product.

92. A succession of several equal ratios is called a continued proportion.

Thus,

A: B-C: DE: F, &c.

is a continued proportion.

93. Theorem.

The sum of any number of antecedents in a continued proportion is to the sum of the corresponding consequents, as one antecedent is to its consequent.

Demonstration. Denote the value of each of the ratios in the continued proportion of the preceding article by M, and we have

MA:BC: D= E: F, &c.;

and the sum of these equations is

A+ C+E+ &c. = (B+D+F+ &c.) × M;

Ratio of Sum of Antecedents to Sum of Consequents.

whence

A+C+E+&c.
B+D+F+ &c.

= M =

A C E
B D F

= &c.

94. Corollary. Either antecedent may be repeated any number of times in the above sum, provided its consequent is also repeated the same number of times.

95. Corollary. Either antecedent may be subtracted instead of being added, provided its consequent is also subtracted.

96. Corollary. The application of these results to the proportion

gives

whence

A: B= C: D,

A+ C: B+D=A:B= C: D
A-C: B-DA:B= C: D
mA+nCm B÷nD=A: B = C: D
mA-n C: m B-n D=A:B= C:D;

A+ C: B+D=A-C: B-D A+C: mA+nCm B÷nD=m A-n C: m B-n D; or, transposing the means as in art. 87,

AC: A-C=B+D: B-D

mA+nCmA-n C-mB÷nD: mB-nD;

that is, the sum of the antecedents of a proportion is to the sum of the consequents, as the difference of the antecedents is to the difference of the consequents, o~ as either antecedent is to its consequent.

Likewise, the sum of the antecedents is to their difference, as the sum of the consequents is to their difference.

Ratio of Sum of two first Terms to that of two last.

Moreover, in finding these sums and differences, each antecedent may be multiplied by any number, provided its consequent is multiplied by the same number.

97. Corollary. These rules may also be applied to the proportion

obtained from

A:CB:D

A: BC: D

by transposing its means, and give

and

A+B:C+D=A—B:C-D

=mA+n B:m C÷nD=m A-n B:m C-n D =A:C=B: D;

A+B: A—B=C+D: C - D
mA+nB:mA-n B=mC+n D:m C-nD;

that is, the sum of the first two terms of a proportion is to the sum of the last two, as the difference of the first two terms is to the difference of the last two, or as the first term is to the third, or as the second is to the fourth.

Likewise, the sum of the first two terms is to their difference, as the sum of the last two is to their difference.

Moreover, in finding these sums and differences, both the antecedents may be multiplied by the same number, and both the consequents may be multiplied by any number.