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When the numerator is greater than the denominator, the value is greater than a unit, and the expression is called an improper fraction.

Thus, each of the expressions, 3, 5, 18, 13, &c., is equal to a unit.

Each of the expressions, 1, 4, 3, 5, 4, t, &c., is a proper fraction.

Each of the expressions, f,, §, tt, 4, 17, &c., is an improper fraction.

When a whole number and fraction are connected, the expression is called a mixed number. Thus, 4, 37, 57, 229, &c., are mixed numbers. The whole number is called the integral part of the expression, and the fraction is called the fractional part.

When several fractions are connected by the word of, the expression is called a compound fraction. The expressions,of of 5, of of of 1, of 2 of 3 of 5, &c., are compound fractions.

Any number may be made to assume the form of an improper fraction, by writing under it a unit for the denominator. Thus, 2, 3, 4, 5, 7, &c., are the same as f, i, 4, 4, 7, &c.

Fractions sometimes occur, in which the numerator, or denominator, or both, are themselves fractional; such expressions are called complex fractions.

Thus,

3층 4 2층 10ᄒ

&c., are complex fractions. 4, 7, 37, 9,

A fraction is said to be inverted when the numerator and denominator exchange places. Thus: the fractions, 4, 4, 7, 1, 4, 7, when inverted, become 4, 4, 4, V, †, 7.

What is a vulgar fraction? Which is the numerator of a vulgar fraction? Which the denominator? What does the denominator show? What does the numerator show? In the vulgar fraction five eighths, which is the numerator, and which the denominator? How is it read? What may a vulgar fraction be considered a concise

way of expressing? In a vulgar fraction, which part corresponds to the dividend, and which to the divisor? What is the value of the fraction, when the numerator is equal to the denominator? When is the value less than a unit? What is the fraction then called ? When is the value greater than a unit? What is the fraction then called? Give examples of proper fractions. Give examples of improper fractions. When a whole number and fraction are connected, what is the expression called? Give ex amples. When several fractions are connected by the word of, what kind of a fraction is it then called? Give examples. When the numerator, or denominator, or both, are already fractional, what are they called? Give examples. When is a fraction said to be inverted? Give examples.

REDUCTION OF FRACTIONS.

34. In division, the divisor, dividend and quotient are so related, that the product of the divisor and quotient is always equal to the dividend. Hence, the divisor and quotient may be interchanged; that is, if the dividend be divided by the quotient, the result will be the divisor. It is also obvious, that, with the same divisor, twice as great a dividend will give twice as great a quotient; thrice as great a dividend will give thrice as great a quotient; and in general, the effect of multiplying the dividend by any number is to multiply the quotient by the same number. On the other hand, if the dividend remain the same, multiplying the divisor by any number produces the same effect as dividing the quotient by the same number. Consequently, if we multiply both dividend and divisor by the same number, it will produce no change in the quotient.

Again, it is obvious, that with the same divisor, half as great a dividend will give but half as great a quotient; one-third as great a dividend will give one-third as great

a quotient; and in general, the effect of dividing the dividend by any number, is to divide the quotient by the same number. On the other hand, if the dividend remain the same, dividing the divisor by any number produces the same effect as multiplying the quotient by the same number. Consequently, if we divide both dividend and divisor by the same number, it will produce no change in the quotient.

If, now, we call to mind that the value of a fraction is the quotient arising from dividing the numerator by the denominator, we readily infer the following

PROPOSITIONS.

I. That, multiplying the numerator by any number is the same as multiplying the value of the fraction by the same number.

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II. That, multiplying the denominator by any number is the same as dividing the value of the fraction by the same number.

III. That, multiplying both numerator and denominator by any number does not alter the value of the fraction.

IV. That, dividing the numerator by any number is the same as dividing the value of the fraction by the same number.

V. That, dividing the denominator by any number is the same as multiplying the value of the fraction by the same number.

VI. That, dividing both numerator and denominator by the same number does not alter the value of the fraction.

GREATEST COMMON DIVISOR.

35. The greatest common divisor of two or more numbers, is the greatest number which will divide them without any remainder.

Before proceeding to find the greatest common divisor of two numbers, we will show that any number which will divide two numbers exactly, will also divide their difference.

Suppose we have a common divisor of 636 and 276; this will also exactly divide 360, their difference. For, 636 is made up of the two parts 276 and 360, so that any number which will exactly divide 636, will also divide 276+360; if a divisor of 636 will at the same time divide one of its parts, 276, it will of necessity divide the other part, 360. Hence a common divisor of 636 and 276 is also a divisor of their difference, 360.

As the divisor which is common to 636 and 276, is also a divisor of 360, it must be a common divisor of 360 and 276, and consequently of 84, the difference between 360 and 276; and in general, when any two numbers have a common divisor, and we subtract any number of times the smaller number from the larger, the remainder will be exactly divisible by this common divisor.

What, now, is the greatest common divisor of 360 and 276.

The greatest divisor cannot exceed the less number, 276. But 276 will not divide the other number, 360, without a remainder, 84. Hence, the greatest divisor of 276 and 84 must be the greatest common divisor of 360 and 276. Again, dividing 276 by 84, we find 3, quotient, and 24, remainder. So the greatest common divisor of 84 and 24 is also the greatest common divisor of 276 and 84, and consequently of 360 and 276. Now, dividing 84 by 24, we find the quotient 3, and remainder 12. Finally, dividing 24 by 12, we find it is contained exactly twice; so that the greatest common divisor of 24 and 12 is 12: consequently, 12 is the greatest common divisor of 360 and 276.

We will exhibit in one point of view the above.

OPERATION.

276)360(1
276

84)276(3

252

24)84(3

72

12)24(2

24

0

Hence, to find the greatest common divisor of two num bers, we deduce this

RULE.

Divide the greater number by the less, then the less number by the remainder; thus continue to divide the last divisor by the last remainder, until there is no remainder. The last divisor will be the greatest common divisor.

NOTE.-When there are more than two numbers whose greatest common divisor is required, we must find the greatest common divisor of any two, and then find the greatest common divisor of this divisor thus found, and one of the remaining numbers; and thus continue until all the different numbers have been used.

What is the greatest common divisor of two or more numbers? Repeat the rule for finding the greatest common divisor of two numbers. How do you proceed when there are more than two numbers?

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