Continued Proportion

392. When quantities are in continued proportion, all the ratios are equal. (Art. 368.) If
            a:b = b:c = c:d = d:e,
the ratio of a:b is the same, as that of b:c, of c:d, or of d:e. The ratio of the first of these quantities to the last, is equal to the product of all the intervening ratios; (Art. 348,) that is, the ratio of a:e is equal to $\frac{a}{b}\frac{b}{c}.\frac{c}{d}.\frac{d}{e} $

But as the intervening ratios are all equal, instead of multiplying them into each other, we may multiply any one of them into itself; observing to make the number of factors equal to the number of intervening ratios. Thus the ratio of a:e, in the example just given, is equal to
            $\frac{a}{b}.\frac{a}{b}.\frac{a}{b}.\frac{a}{b}=\frac{a^4}{b^4} $
When several quantities are in continued proportion, the number of couplets, and of course the number of ratios, is one less than the number of quantities. Thus the five proportional quantities a, b, c, d, e, form four couplets containing four ratios; and the ratio of a:e is equal to the ratio of a4:b4, that is, the ratio of the fourth power of the first quantity, to the fourth power of the second. Hence,

393. If three quantities are proportional, the first is to the third, as the square of the first, to the square of the second; or as the square of the second, to the square of the third. In other words, the first has to the third, a duplicate ratio of the first to the second. And conversely, if the first of the three quantities is to the third, as the square of the first to the square of the second, the three quantities are proportional.
      If a:b = b:c, then a:c = a2:b2. Universally,

394. If several quantities are in continued proportion, the ratio of the first to the last is equal to one of the intervening ratios raised to a power whose index is one less than the number of quantities.
If there are four proportionals a, b, c, d, then a:d = a3:b3
If there are five a, b, c, d, e; a:e = a4:b4, etc.

395. If several quantities are in continued proportion, they will be proportional when the order of the whole is inverted. This has already been proved with respect to four proportional quantities. (Art. 376. cor.) It may be extended to any number of quantities.
      Between the numbers,      64, 32, 16, 8, 4,
      The ratios are        2, 2, 2, 2,
      Between the same inverted       4, 8, 16, 32, 64,
      The ratios are        ½, ½, ½, ½.
So if the order of any proportional quantities be inverted, the ratios in one series will be the reciprocals of those in the other. For by the inversion, each antecedent becomes a consequent, and v. v. and the ratio of a consequent to its antecedent is the reciprocal of the ratio of the antecedent to the consequent. (Art. 346.) That the reciprocals of equal quantities are themselves equal, is evident from Ax. 4.

396. Harmonical or Musical Proportion may be considered as a species of geometrical proportion. It consists in an equality of geometrical ratios; but one or more of the terms is the difference between two quantities.

Three or four quantities are said to be in harmonical proportion, when the fifst is to the last, as the difference between the two first, to the difference between the two last.
If the three quantities a, b, and c, are in harmonical proportion, then a:c = a-b:b-c.
If the/our quantities a, b, c, and d are in harmonical proportion, then a:d = a-b:c-d.
Thus the three numbers 12, 8, 6, are in harmonical proportion.
And the four numbers 20, 16, 12, 10, are in harmomcal proportion.

397. If, of four quantities in harmonical proportion, any three be given, the other may be found. For from the proportion
            a:d = a-b:c-d,
by taking the product of the extremes and the means, we have ac - ad = ad - bd.
And this equation may be reduced, so as to give the value of either of the four letters.
Thus by transposing -ad, and dividing by a,
            $c=\frac{2ad-bd}{a}$.

Examples, in which the principles of proportion are applied to the solution of problems.

1. Divide the number 49 into two such parts, that the greater increased by 6, may be to the less diminished by 11; as 9 to 2.
      Let x= the greater, and 49-x = the less.
By the conditions proposed,      x+6:38-x = 9:2
Adding terms, (Art. 385, 2.)      x+6:44 = 9:11
Dividing the consequents, (Art. 378, 8.)    x+6:4 = 9:1
Multiplying the extremes and means, x+6 = 36. And x = 30.

2. What number is that, to which if 1, 5, and 13, be severally added, the first sum shall be to the second, as the second to the third?
            Let x= the number required.
By the conditions, x+1:x+5 = a+5:x+13
Subtracting terms, (Art. 385,2.) x+1:4 = x+5:8
Therefore 8x+8 = 4x+20. And x = 3.

3. Divide the number 18 into two such parts, that the squares of those parts may be in the ratio of 25 to 16.
      Let x= the greater part, and 18 - x= the less.
By the conditions, x2:(18-x)2 = 25:16
Extracting, (Art. 387,) x:18-x = 5:4
Adding terms, x:18 = 5:9
Dividing terms, x:2 = 5:1
Therefore, x = 10.

4. If the number 20 be divided into two parts, which are to each other in the duplicate ratio of 3 to 1, what number is a mean proportional between those parts?
      Let x= the greater part, and 20-x= the less.
By the conditions, x:20-x = 32:12 = 9:6
Adding terms, x:20 = 9:10
Therefore, x= 18. And 20-x = 2
A mean propor. between 18 and 2(Art. 372.) =√2.18 = 6.

5. There are two numbers whose product is 135, and the difference of their squares, is to the square of their difference, as 4 to 1. What are the numbers?
            Ans. 15 and 9.

6. What two numbers are those, whose difference, sum, and product, are as the numbers 2, 3, and 5, respectively?
              Ans. 10 and 2.

7. Divide the number 24 into two such parts, that their product shall be to the sum of their squares, as 3 to 10.
              Ans. 18 and 6.

8. In a mixture of rum and brandy, the difference between the quantities of each, is to the quantity of brandy, as 100 is to the number of gallons of rum; and the same difference is to the quantity of rum, as 4 to the number of gallons of brandy. How many gallons are there of each?
            Ans. 25 of rum, and 5 of brandy.

9. There are two numbers which are to each other as 3 to 2. If 6 be added to the greater and subtracted from the less, the sum and remainder will be to each other, as 3 to 1. What are the numbers?
              Ans. 24 and 16.

10. There are two numbers whose product is 320; and the difference of their cubes, is to the cube of their difference, as 61 to 1. What are the numbers?
              Ans. 20 and 16.

11. There are two numbers, which are to each other, in the duplicate ratio of 4 to 3; and 24 is a mean proportional between them. What are the numbers?
              Ans. 32 and 18.


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