DIVISION OF FRACTIONS.

Division of Fractions is naturally divided into the three following kinds; viz. the division of a fraction by a whole number; the division of a whole number by a fraction; and the division of a fraction by a fraction.

Division of fractions by a whole number was fully illustrated in Sec. 5th of the remarks introductory to this rule. It is, therefore, necessary here merely to repeat, that a fraction is divided by a whole number, either by dividing its numerator, or multiplying its denominator by that number.

It is obvious that the quotient arising from dividing a whole number by a fraction, must be as much larger than the number itself, as a unit or 1 is greater than the fraction; or, in other words, the given dividend must bear the same ratio to the required quotient, as the numerator of the fraction bears to its denominator. A unit or 1 is contained in 6, six times; is contained in the same number, twelve times, and, eighteen times; and, half as many times as §, viz. nine times. The operation to obtain this last quotient is as follows, 6x3=18, the number of thirds in six; and 18-2=9.

For dividing a whole number by a fraction, we have then the following rule:

Multiply the whole number by the denominator of the fraction, and divide the product by the numerator.

Ex. 1. Divide 9 by . Operation, 9×4=36, and 36÷3 =12, the quotient.

2. Divide 15 by . Operation, 15×6=90, and 90÷5= 18, quotient.

Operations of a similar character may be performed by canceling.

RULE FOR CANCELING.-Place the whole number above a horizontal line and invert the fractional divisor; that is, place the denominator above the line and the numerator below. Cancel, &c.

The following sums may be divided by either of the above

That the scholar may be enabled to commence understandingly the division of fractions by fractions, he may turn back and review the 7th section of the introductory remarks of this rule. It is there said, that a fraction is divided by another fraction by inverting the divisor and then multiplying them together as in multiplication.

1. Divide by . Ans. or 15. 2. Divide by . Ans. 16 or 17.

RULE FOR CANCELING.-Proceed in all respects as in multiplication of fractions, in arranging the terms of the dividend,

then invert the divisor; that is, place the numerators below and the denominators above the line. Proceed to cancel, &c.

Note. When the divisor is a compound fraction, each fraction in the divisor must be inverted.

4. Divide of by of 3. Statement,

3. 4. 4. 5
4. 5. 1. 3'

Canceled,

3. 4. 4. 5

4. 5. 1. 3

; therefore, or 4 is the answer required.

15. 11. 5. 7

5. Divide 15 of 1 by 3 of 9. Statement, 17. 12. 3. 6°

1925, or 17014
6. Divide of

Ans.

25. Bought 8 lb. of coffee for 24 of a dollar; what was the cost of one pound? Ans. or of a dollar.

26. Bought 9 pounds of sugar for 1⁄2 of a dollar; what was the price of a pound? Ans. 1 of a dollar.

27. In 8 weeks a family consumes 84 pounds of butter; how much is that per week? Ans. 10.

QUESTIONS.-What is a fraction? If a unit be divided into two equal parts, what is each of these parts called? How is it written? If it be divided into three equal parts, what is each part called, and how is it written? Similar questions should be asked respecting other fractions. When more parts than one are to be expressed, how is it done? What do fractions therefore express? How are they represented? What is the number below the line called? What does the denominator show? What is the number above the line called? What does the numerator show? What are the two numbers called, when spoken of collectively? How many kinds of fractions are there? What are they? What is a proper fraction? Give an example.

What is

an improper fraction? Give an example. What is a simple fraction? Give an example. What is a compound fraction? Give an example. What is a mixed number? Give an example. What is complex fraction? Give an example. What does the denominator show? If the denominator remain the same, how is the value of the fraction affected by increasing the numerator? How, by diminishing it? Give an illustration. What is therefore the value of a fraction? If the numerator of a fraction be less than the denominator, how is its value compared with a unit? If the numerator be equal to the denominator, how then does its value compare with a unit? And how, if the numerator be greater than the denominator? What is the only consideration which limits the value of a fraction? In what ratio is the value of a fraction increased? How then may fractions be multiplied? How is a fraction multiplied into a number equal to its denominator? In what form should the terms of the fraction always be preserved? What is necessary to increase or diminish the value of a fraction? How then may a fraction be multiplied by a whole number? In what two ways may fractions be multiplied by whole numbers? How may a fraction be divided by a whole number? How else may a fraction be divided by a whole number? In what two ways then may fractions be divided by whole numbers? In what case should the numerator always be divided? What operation is therefore performed on the value of a fraction, when the numerator is operated upon? And what, when the denominator is operated upon? How is a fraction multiplied by a fraction? How is a fraction divided by a fraction? If the numerator and denominator be both multiplied by the same number, how is the value of the fraction affected? What is the rule for reducing fractions to their lowest terms?

What is the rule for reducing a whole number or mixed quantity to an improper fraction? How is an improper fraction reduced to a whole or mixed number? In all cases of division, what disposition may be made of the remainder, when any occurs? How are compound fractions reduced to simple ones? What is the rule for canceling? What is note 1st? What is note 2d? What is note 3d? What is Case 5th ? How are fractions of low denominations reduced to those of higher denominations? What is the rule for canceling?

How are fractions of high denominations reduced to those of a lower value? What is Case 6th? What is the rule for it? What is Case 7th? What is the rule for it? What is the rule for canceling?

What is Case 8th? What is the rule for it? What is the note? What must be done before fractions can be added? What else requires to be done? What is the rule for the addition of fractions? What note follows? What is note 2d? What preparations are necessary before fractions can be subtracted? What is the rule for subtracting fractions? How is a fraction multiplied by a whole number? How is a fraction multiplied into a quantity equal to its denominator. What is the rule for multiplying fractions by fractions? What is the rule for canceling? Into what three kinds is division of fractions naturally divided? How is a fraction divided by a whole number? What is the rule for dividing a whole number by a fraction? How are fractions divided by fractions? What is the rule for canceling? What note follows the rule?

DECIMAL FRACTIONS.

In the preceding rule we have contemplated the unit as divided into any number of equal parts.

We are now to regard it as divided first into ten equal parts, then each of these into ten other equal parts, or the whole unit into one hundred equal parts; and these parts again, each into ten other parts; or the whole into a thousand equal parts, &c. The expressions obtained by these several divisions, therefore, decrease in value in the constant ratio of ten, from the left to the right, and are called decimals. Whole numbers, as was shown in Numeration, increase in the same ratio from the right to the left, and both commence their enumeration with the unit figure. The connection between them is therefore so intimate as to render them susceptible of being written together and subjected to the same operations. The only important consideration in writing them, in addition to what has already been explained, is to distinguish the one from the other. This is effected by the period, called in decimals, the point of separation, which is always placed between them. In the expression, 23.56, the 23 is the whole number, and the .56 the decimal.

It will be observed that decimals, although they express parts of units, do not, like vulgar fractions, require two terms to express them. The given decimal may however be regarded, as in truth it is, a numerator, with a denominator always under