CASE IV.

235. To find the product of two mixed numbers whose frac tional parts are halves.

METHOD.-Take the product of the integers, increase this by of their sum, and by .

DEM. This method can be readily obtained by a process similar to the preceding.

236. To square a mixed number whose fractional part is 1. METHOD.-First, when the integer is EVEN, square the integer, add of itself and the square of 1.

Secondly, when the integer is ODD, square the integer, add of the next smaller number, and 1o·

237. To square a mixed number whose fractional part is §.

METHOD. First, when the integer is EVEN, square the integer, rease this by of the integer, and by the square of 4. Secondly, when the integer is ODD, square the integer, add of the next smaller integer, and also 21.

CASE VII.

238. To square a number whose unit figure is 5.

METHOD.-Multiply the part preceding the units by itself increased by a unit, and prefix the result to 25.

239. To square a number ending in 25.

METHOD. When the part preceding 25 is EVEN, square it, and add of it to the result, aud prefix the sum to 0625. When the part preceding 25 is ODD, square it, add of the next smaller number, and prefix the result to 5625.

240. To square a number ending in 75.

METHOD. We regard the 75 as 3, and then proceed as in the case of squaring a mixed number whose fractional part is 4.

NOTE. For other cases and methods, see the author's "METHODS OF TEACHING MENTAL ARITHMETIC, etc."

CONTRACTIONS IN DIVISION.

241. All the methods of multiplying under Case I. may be applied to division, by reversing the operations. by 2 we multiply by 4 and divide by 10, etc.

Divide

Thus, to divide

242. Ratio is the measure of the relation of two similar quantities, and is found by dividing the first quantity by the second. Thus, the ratio of 6 to 2 is 3, and of $10 to $5 is 2.

243. The Terms of a ratio are the two numbers compared. The first is called the Antecedent, the second the Consequent. The two together are called a couplet.

244. The ratio of two numbers is indicated in two ways: 1st. By a colon between the figures; thus, 6: 2 denotes the ratio of 6 to 2.

2d. By a fraction; thus, & denotes the ratio of 6 to 2.

245. Ratio exists only between similar numbers; thus, there is no ratio between $4 and 8 yards, or between 5 cows and 15 apples.

246. A Compound Ratio is the product of two or more simple ratios; as, 2 : 4 multiplied by 5: 6, or X 8.

NCTES.-I. The definition of ratio usually given is the relation of two numbers. This is indefinite, for the ratio is the numerical measure of the relation.

II. The correct method of determining a ratio is to divide the first term by the second. A few authors divide the second term by the first, calling it the French Method. The name is founded in error, since nearly all the French mathematicians divide the first term by the second. This is also the method used by the German and English mathematicians.

PRINCIPLES OF RATIO.

I. A ratio equals the quotient of the antecedent divided by the consequent.

II. Therefore, The antecedent equals the product of the consequent and ratio.

III. Hence also,-The consequent equals the quotient of the antecedent divided by the ratio.

4 488 to 61?

Ans. 5 7. : 3 ? % : 7 ?
5g 3: ?? /?

10

Ans. 8.18.5 15? 23:33? Ans. 14.

9 What is the value of the compound ratio

2:4

}

? 13:9

SOLUTION. This compound ratio equals (24) X (3: 9), which equals, Ans.

5 2

?

Ans. 31.

8

6 J

what is the

11. What is the value of the ratio {

12. The antecedent is 24, the consequent 8;

ratio?

Ans. 3.

13. The consequent is 8 and ratio 9; what is the antecedent?

Ans. 72. 14. The antecedent is 36 and ratio 4; what is the consequent? Ans. 9.

15. The consequent is 1 and ratio ; what is the antecedent? Ans. 7.

16. The antecedent is § and ratio ; what is the conseAns. 1.

quent?

PROPORTION.

247. Proportion is the comparison of numbers having equal ratios. Thus, the ratio 6: 3 and 8: 4 are equal, and if written thus, 6:38: 4, we have a proportion.

The equality of ratios is generally indicated by writing a double colon between the two ratios; thus, 6: 3 :: 8 : 4.

248. A proportion is read in several different ways. Thus, 6:38:4 may be read: 1st. The ratio of 6 to 3 equals the ratio of 8 to 4; 2d. 6 is to 3 as 8 is to 4. The last is the method generally used.

The first and

249. A proportion consists of four terms. fourth are called the extremes, the second and third the means. The first and second terms together are called the first couplet, the third and fourth the second couplet.

250. Proportion may be Simple or Compound. A Simple Proportion is the comparison of simple ratios. A Compound Proportion is the comparison of compound ratios.

PRINCIPLES OF PROPORTION.

PROP. I.-In every proportion the product of the means equals the product of the extremes.

DEM. In the proportion 2: 6:4: 12 we have; now if we multiply both these equals by 6 and then by 12, we have 2X12=4X6, which proves the proposition, since 2 and 12 are the extremes and 4 and 6 are the means.

DEM. 2D. Since the antecedent of a ratio equals the consequent multiplied by the ratio, every proportion may be expressed in the following form: 2d X ratio: 2d term:: 4th ratio: 4th term. Now in the means we have the factors 2d term, 4th term, and ratio, and in the extremes we have the same factors, hence the products are equal.

PROP. II.—Either extreme is equal to the product of the means divided by the other extreme.

PROP. III.—Either mean is equal to the product of the extremes divided by the other mean.

PROP. IV. The first term of a proportion equals the second term multiplied by the ratio of the third to the fourth.