his daughter, whose fortune was 15 thousand, 15 hundred and 15 pounds; what was the son's portion, and what did the whole estate amount to?

(12.) What number, deducted from the 32d part of 3072, will leave the 96th part of the same? Ans. 64. (13.) What is the difference between thrice five, and thirty, and thrice thirty-five? Ans. 60. (14.) Suppose the quotient arising from the division of two numbers to be 5379, the divisor, 37625; what is the dividend, if the remainder come out 9357-?

Ans. 202394232. (15.) A privateer of 175 men took a prize which amounted to 59 pounds a man, besides the owner's half; what was the value of the prize? Ans. £20650.

(16.) There are two numbers; the greater of them is 25 times 78, and their difference is 9 times 15; their sum and product is required.

Ans. 1950 the greater; 1815 the less; 3765 the sum; and 3539250 the product.

(17.) A merchant began trade with 25327 dollars; for 6 years together, he cleared 1253 dollars per annum; the next 5 years, he cleared 1729 dollars per annum; but, the last 4 years, he had the misfortune to lose 3019 dollars per annum; what was he worth at the 15 years' end? Ans. 29414 dollars.

(18.) If 9000 men march in a column of 750 deep, how many march abreast? Ans. 12.

(19.) The least of two numbers is 19418, and the difference between them is 2384; what is the greater, and sum of both?

Ans. 21802 the greater, and 41220 the sum. (20.) What number added to the 27th part of 6615, will make 570 ? Ans. 325. (21.) What is the difference betwen six dozen dozen, and half a dozen dozen; and what is their product, and the quotient of the greater by the less?

Ans. 792 difference, 62208 product, and 12 quotient. (22.) Divide 151200 lbs. of beef equally among an army, consisting of 27 regiments, each regiment 7 companies, and each company 100 men. Ans. 8 lbs.

PART II.

FRACTIONS.

§ 45. 1. IN DIVISION, at the close of an operation, we frequently had á remainder, which, being smaller than the divisor, we were unable to divide, with the means then possessed (§ 37. 3, (3). The quotient, however, being incomplete without that division, we indicated the actual result, and so completed the quotient, by writing the divisor under the remainder, with a-horizontal line between them, at the right of the quotient figures previously obtained (§ 37. 3).

2. Such expressions have given rise to a distinct class, or modification of numbers, called Vulgar Fractions.

3. The origin of them, thus given, constantly borne in mind, will materially aid in their consideration.

4. A VULGAR FRACTION represents a part, or parts of some thing, or number regarded as a whole.

5. It is expressed simply as an unexecuted division, with the number corresponding to the divisor below the other, and a horizontal line between them (§ 37. 3).

6. The number above the line, corresponding to the dividend, is called the Numerator; that below the line, corresponding to the divisor, is called the Denominator; and the quotient, whatever it may be, represents the value of the fraction.

7. The numerator and denominator together, are called terms of the fraction.

8. The numerator shows how many parts are denoted by the fraction.

9. The denominator shows how many of those parts make unity.

10. The denominator, therefore, gives name to the fraction. Thus, when the denominator is 2, resulting from

the division of unity into two parts, the fraction is read and called one half; when the denominator is 3, thirds; 4, fourths; and so on; as, , one half; , two thirds; 2, three fourths.

§ 46. 1. The numerator is said to be both a dividend or number to be divided (§ 45. 6), and to denote, as the result of a division, how many parts, such as the denominator shows make unity, the fraction expresses (§ 45. 8).

2. These two objects, which it is said to serve, appear to be contradictory; and, therefore, need explanation.

3. The origin of vulgar fractions has been shown to be in a division which could not be executed-in a remainder, at the close of a division, too small to be divided by the divisor (§ 45. 1, 2). This remainder was clearly a part of the original dividend, and so was itself, a dividend, but too small to contain the divisor (§ 37. 3, (3). We then wrote it at the right of the quotient figures, previously obtained, with the divisor underneath; thus indicating a division which we were not able to execute.

This expression we have regarded as a vulgar fraction, and named the numbers composing it, terms; one the numerator, the other the denominator (§ 45. 6, 7). The origin of the expression, therefore, shows the numerator to be a dividend.

4. Now we will show that the same expression also numbers the parts of unity denoted by the fraction (§ 45. 8).

Any fraction may be regarded as the product of a whole number into unity, divided by another number. Thus,

7X; where 7 is the numerator, counting the parts, and equals 7×1, and 18 is the denominator, showing those parts to be eighteenths of the unit.

Now if we suppose our remainder after division to be unity, or 1, and the divisor to be 8, we shall have the expression (§ 37. 3). This 1 above the 8 is to be divided.

If we suppose this to be done, the 1 will be divided into 8 equal parts, and these, divided by the 8, would give us 1 of the 8 parts as the result; or, one eighth of 1. This, then, would be a fraction, of 1; or, as the 1 is once written in the numerator, it need not be repeated; we then abridge the expression, and write simply. In this case the numerator expresses how many parts, such as the de

nominator shows make unity, the fraction denotes; in other words, counts, or numbers, the parts (§ 45. 8).

5. The division denoted, and the division executed, then, furnish us precisely the same expression. Therefore, both assertions relative to the numerator, are consistent and true.

6. If our remainder were 7 instead of 1, and the divisor 8, 1 of the 7 divided, would, as above, give us ; but having 7 instead of 1, the result would be 7 times what it would be in the case of the 1, or 1X7=3. And so universally.

§ 47. 1. The value of the parts, as shown by the denominator, may evidently be as much varied as the numbers themselves; therefore, vulgar fractions have not, like whole numbers but a single mode of expression for the same quantity.

2. By a whole number the same quantity can be expressed but in one form; while, by vulgar fractions, the same quantity may be expressed in an infinite variety of forms. Thus, 1, 1, 3, 1, 15, 11, 7, and so on, indefinitely, denote precisely the same quantity.

4

3. The fraction is called Proper, when the numerator is less than the denominator, and, therefore, cannot be divided by it; the name denoting that the dividend (the numerator) not containing the divisor (the denominator), the expression is properly a fraction, or a proper fraction; as,, 1, 1, 4.

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4. It is called Improper, when the numerator equals or exceeds the denominator, and, therefore, can be divided by it; the name denoting that the dividend (the numerator) containing the divisor (the denominator), the expression is improperly a fraction, or an improper fraction; as, 告,号,号.

5. From the principle and nature of fractions, thus developed, we deduce six fundamental propositions.

§ 48. 1. PROPOSITION I. As many times as the numerator of a fraction is made larger, the denominator remaining unchanged, so many times the value of the fraction is made larger.

2. Illustration. ×6=48. Here we multiply the numerator 3 by 6, preserving the denominator unchanged, and we obtain the result, .

3. Explanation. When the numerator is multiplied by any number, as many more parts are denoted by it, as there are units in that number; or, as the numerator is the number to be divided-the dividend-when it is increased, so much greater is the number to be divided, and, ́ 'consequently, the divisor-the denominator-remaining unchanged, so much greater is the quotient which represents the value of the fraction; hence, the value of the fraction is so many times increased.

4. One simple principle of division holds through this and the following propositions; that the larger the dividend, the divisor remaining the same, the larger the quotient; the smaller the dividend, the smaller the quotient.

2. Illustration. ÷8. Here we divide the numerator 8 by 8, preserving the denominator unchanged, and we obtain the result, 17.

3. Explanation. When the numerator is divided by any number, as many less parts are denoted by it as there are units in that number; or, as the numerator is the number to be divided-the dividend-when it is decreased, so much less is the number to be divided, and, consequently, the divisor-the denominator-remaining unchanged, so much less is the quotient, which represents the value of