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CHAPTER XXIII.

RATIO. PROPORTION. VARIATION.

235. Definitions. The relative magnitude of two quantities, measured by the number of times which the one contains the other, is called their ratio.

Concrete quantities of different kinds can have no ratio to one another: we cannot, for example, compare with respect to magnitude miles and tons, or shillings and weeks.

The ratio of a to b is expressed by the notation a: b ; and a is called the first term, and b the second term of the ratio.

Sometimes the first and second terms of a ratio are called respectively the antecedent and the consequent.

It is clear that a ratio is greater than, equal to, or less than unity, according as its first term is greater than, equal to, or less than the second.

A ratio which is greater than unity is sometimes called a ratio of greater inequality, and a ratio which is less than unity is similarly called a ratio of less inequality.

236. Magnitudes must always be expressed by means of numbers; and the number of times which one number is contained in another is found by dividing the one by the other. Hence

a

a: b is equal to 2

b

Thus ratios can be expressed as fractions.

237. A fraction is unaltered in value by multiplying its numerator and denominator by the same number. [Art. 158.]

Hence also a ratio is unaltered in value by multiplying each of its terms by the same number.

Thus the ratios

2:3, 6:9 and 2 m : 3 m

are all equal to one another.

Again, the ratios 4: 5, 7:9 and 11: 15 are equal respectively to 36:45, 35: 45 and 33: 45.

Hence the ratios 4:5, 7:9 and 11: 15 are in descending order of magnitude.

238. A ratio is altered in value when the same quantity is added to each of its terms.

For example, by adding 1, 10 and 100 to each of the terms of the ratio 4: 5, we obtain respectively the ratios

5:6, 14: 15 and 104: 105;

and these new ratios are different from the given ratio and from each other.

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we see that by adding the same quantity to each of the terms of the ratios 4:5, a new ratio is obtained which becomes more nearly equal to unity as the quantity added becomes greater.

This is a particular case of the following general proposition.

239. Any ratio is made more nearly equal to unity by adding the same positive quantity to each of its terms.

By adding to each term of the ratio a:b, the ratio (a + x): (b + x) is obtained. We have to show that (a + x)/(b+x) is more nearly equal to 1 than is a/b.

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and it is clear that the absolute value of (a - b) / (b+x) is less than that of (a - b)/b, for the numerators are the same and the denominator of the first is greater than that of the second: this proves the proposition.

Now when a is greater than b, a/b is greater than 1, and so also is (a + x)/(b+x); hence, as (a + x)/(b+x) is more nearly equal to unity than a/b is, it follows that (a + x)/(b+x) is less than a/b.

Thus a ratio which is greater than unity is diminished by adding the same positive quantity to each of its terms. If, however, a is less than b, a/b is less than 1, and so also is (a + x)/(b + x); but (a + x)/(b + x) is more nearly equal to unity than a/b is, and therefore (a + x)/(b+x) is greater than a/b.

Thus a ratio which is less than unity is increased by adding the same positive quantity to each of its terms.

Also, when x is very great, the fraction (a−b)/(b+x) is very small; and we can make (a - b)/(b+x), which is the difference between (a + x)/(b+x) and 1, as small as we please by taking a sufficiently great. This is expressed by saying that the limit of (a+x)/(b+x) when x is very great, is unity.

240. The following definitions are sometimes required: The ratio of the product of the first terms of any number of ratios to the product of their second terms is called the ratio compounded of the given ratios.

Thus ac bd is called the ratio compounded of the

:

ratios ab and c: d.

The ratio a2: b2 is called the duplicate ratio of a : b.

The ratio a3: b3 is called the triplicate ratio of a : b. The ratio va√b is called the sub-duplicate ratio of a: b.

241. Incommensurable Numbers. The ratio of two quantities cannot always be expressed by the ratio of two whole numbers; for example, the ratio of a diagonal to a side of a square cannot be so expressed, for this ratio is √2, and we cannot find any fraction which is exactly equal to √2.

When the ratio of two quantities cannot be exactly expressed by the ratio of two whole numbers, they are said to be incommensurable.

Although the ratio of two incommensurable numbers cannot be found exactly, the ratio can be found to any degree of approximation which may be desired; and the different theorems which are proved with respect to ratios of commensurable numbers can be proved to be true also for the ratios of incommensurable numbers.

EXAMPLES LXX.

1. Arrange the ratios 5:6, 7: 8, 41: 48, and 31: 36 in descending order of magnitude.

2. For what value of x will the ratio 3+x: 4 + x be equal to 5:6?

3. For what value of x will the ratio 15 + x: 17 + x be equal to ?

4. What must be added to each of the terms of 3: 4 to make the ratio equal to 25: 32 ?

5. Find two numbers whose ratio to one another is 5: 6, and whose sum is 121.

6. Two numbers are in the ratio 3 to 8, and the sum of their squares is 3577: find them.

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8. If 4x2 + y2 = 4 xy, find the ratio of x to y.

9. Find xy, having given x2 + 6 y2 = 5 xy.

10. A certain ratio will be equal to 2:3 if 2 be added to each of its terms, and it will be equal to 1:2 if 1 be subtracted from each of its terms: find the ratio.

11. Find two numbers such that their sum, their difference, and the sum of their squares are as 7: 1: 75.

12. What is the least integer which must be added to the terms of the ratio 9: 23 in order to make it greater than the ratio 7: 11 ?

13. Write down the ratio compounded of the ratios 2:3 and 15: 16; also the ratio compounded of the ratios 5: 6 and 18: 25.

14. Write down two quantities which are in the duplicate ratio of 2x:3y.

15. Find x in order that x + 1: x + 4 may be the duplicate ratio of 3: 5.

16. The ages of two persons are as 3:4 and thirty years ago they were as 1:3, what are their present ages?

17. Show that, if from each term of a ratio the inverse of the other be taken, the ratio of the differences will be equal to the original ratio.

18. Show that, if a and x be positive and a>x, then a2 — x2: a2 + x2 will be greater than a -x: α + x.

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