347. We may observe that Art. 346 is merely a repetition of that which we proposed as an Example at the end of the Chapter on Miscellaneous Fractions. There is not indeed any necessity for us to weary the reader with examples on Ratio: for since we express a ratio by a fraction, nearly all that we might have had to say about Ratios has been anticipated in our remarks on Fractions.

348. The student may, however, work the following Theorems as Examples.

(1) If a b be a ratio of greater inequality, and x a positive quantity, the ratio a-x: b-x is greater than the ratio a: b.

(2) If a b be a ratio of less inequality, and x a positive quantity, the ratio a+x: b+x is greater than the ratio a : b.

(3) If a b be a ratio of less inequality, and x a positive quantity, the ratio a-x: b-x is less than the ratio a : b.

349. In some cases we may from a single equation involving two unknown symbols determine the ratio between the two symbols. In other words we may be able to determine the relative values of the two symbols, though we cannot determine their absolute values.

Thus from the equation 4x=3y,

Find the ratio of a to y from the following equations:

1. 9x=6y.

4. x2+2xy=5y3.

2. ax=by.

3. ax-by=cx+ dy.

5. x2-12xy=13y2. 6. x2+mxy = n2y2.

7. Find two numbers in the ratio of 3: 4, of which the sum is to the sum of their squares :: 7 : 50.

8. Two numbers are in the ratio of 6: 7, and when 12 is added to each the resulting numbers are in the ratio of 12 : 13. Find the numbers.

9. The sum of two numbers is 100, and the numbers are in the ratio of 7: 13. Find them.

10. The difference of the squares of two numbers is 48, and the sum of the numbers is to the difference of the numbers in the ratio 12: 1. Find the numbers.

11. If 5 gold coins and 4 silver ones are worth as much as 3 gold coins and 12 silver ones, find the ratio of the value of a gold coin to that of a silver one.

12. If 8 gold coins and 9 silver ones are worth as much as 6 gold coins and 19 silver ones, find the ratio of the value of a silver coin to that of a gold one.

350. Ratios are compounded by multiplying together the fractions by which they are denoted.

and

and

Thus the ratio compounded of a: b and c d is ac: bd,

EXAMPLES.-CXXX.

Write the ratios compounded of the ratios

1. 2:3 and 4: 5.

2. 37, 14: 9 and 4: 3.

3. x2-y2: x3+y3 and x2- xy + y2: x+y.

4. a2-b2+2bc-c2: a2-b2-2bc-c2 and a+b+c:a+b-c.

5. m3 + n3: m3-n3 and m-n: m+n.

6. x2+5x+6: y2-7y+12, and y3 − 3y : x3+3x.

351. The ratio a2: b2 is called the DUPLICATE RATIO of a: b.

Thus 100: 64 is the duplicate ratio of 10 : 8,

36x2: 25y2 is the duplicate ratio of 6x : 5y.

The ratio a3: b3 is called the TRIPLICATE RATIO of a: b.

Thus 64: 27 is the triplicate ratio of 4 : 3,

343x3: 1331y3 is the triplicate ratio of 7x : 11y.

352. The definition of Ratio given in Euclid is the same as in Algebra, and so also is the expression for the ratio that one quantity bears to another, that is, A: B. But Euclid cannot employ fractions, and hence he cannot represent the value of a ratio as we do in Algebra. In fact he does not treat of ratios except when one ratio is equal to another.

XXVIII. ON PROPORTION.

353. PROPORTION consists in the equality of two ratios. The Algebraic test of proportion is that the two fractions representing the ratios must be equal.

Thus the ratio a: b will be equal to the ratio cd,

and the four numbers a, b, c, d are in such a case said to be in proportion.

354. If the ratios a: b and c d form a proportion, we express the fact thus:

a: b=c : d.

This is the clearest manner of expressing the equality of the ratios ab and c : d, but there is another way of expressing the same fact, thus

which is read thus,

a:b::c:d,

a is to b as c is to d.

The two terms a and d are called the EXTREMES.

355.

When four numbers are in proportion,

product of extremes=product of means.

Let a, b, c, d be in proportion.

Multiplying both sides of the equation by bd, we get

ad=bc.

Conversely, if ad=bc we can shew that a: bc: d.

357. From this it follows that if any 4 numbers be so related that the product of two is equal to the product of the other two, we can express the 4 numbers in the form of a proportion.

The factors of one of the products must form the extremes. The factors of the other product must form the means.

358. Three quantities are said to be in CONTINUED PROPORTION When the ratio of the first to the second is equal to the ratio of the second to the third.

Thus a, b, c are in continued proportion if

a: b=b: c.

The quantity b is called a MEAN PROPORTIONAL between a and c.

Four quantities are said to be in Continued Proportion when the ratios of the first to the second, of the second to the third, and of the third to the fourth are all equal.

Thus a, b, c, d are in continued proportion when
ab=b:c=c; d.

359. We shewed in Art. 205 the process by which when two or more fractions are known to be equal, other relations between the numbers involved in them may be determined. That process is of course applicable to Examples in Ratio and Proportion, as we shall now shew by particular instances.

Ex. (1) If a b=c:d, prove that

a2+b2: a2-b2=c2+d2 : c2— d2.