RULE.

Q. How do you multiply Vulgar Fractions? A. All the numerators must be multiplied together for a new numerator, and all the denominators must be multiplied together for a new denominator, which will give the product required. If there be mixed numbers, they must first be reduced to equivalent fractions.

EXAMPLES

For Theoretical Exercise on a Slate.

1. Multiply by 4. Ans. 8

EXPLANATIONS.

13 15

In this example, you first multiply the 4 4 and 2, the numerators, for a new numera- 2 tor, making 8; and then multiply the 5 and 3, the denominators, for a new denominator, 8 15 8 making 15, which make, the required

& Ans.

product. Multiplying the denominator of a fraction by any number, is the same as dividing the numerator by the same number. You will readily perceive, that the value of the fraction is increased as many times as the numerator of a fraction is increased; thus, when you multiply the numerator of the fraction, by 4, the fraction is increased four times; but you do not want to increase the value of the fraction four times, but as much less than four as the denominator, 5, indicates; and when you multiply the denominator of the fraction by 5, it makes the value of the fraction five times less; for it takes five times the number of parts to make a unit.

Q. What is Division of Vulgar Fractions? A. Division of Vulgar Fractions teaches to find how often a part, or the parts, of an integer is contained in a given sum.

RULE.

Q. How do you divide Vulgar Fractions?

A. The fractions must be prepared as in Multiplication; then the divisor must be inverted, and you must proceed as in Multiplication, and the products will be the quotient required.

In this example, you must first invert the divisor, which will then be 3, and you must then multiply the 4 and the 3 together, and the 2 and the 7 together, which will give the quotient 12=

2. Divide by 43. Ans. 20.

Q. What is a DECIMAL FRACTION?

A. A Decimal Fraction is a part of a whole number, or integer, whose denominator is a unit, with a cipher, or ciphers, annexed to it. A Decimal Fraction, however, is usually expressed by writing the numerator only, with a comma, or point, prefixed at the left hand of the fraction; thus, ,5 tenths is the same as; and ,25 hundredths is the same as 25 &c.

10 09

EXPLANATIONS.

The integer, or whole number, is always divided either into 10, 100, or 1000, &c., equal parts; and, consequently, the denominator of the fraction will always be either 10, 100, 1000, &c., which, being understood, need not be expressed; for the true value of the fraction may be expressed by writing the numerator only with a point before it on the left hand; thus,

is written,5; 65,65, &c. Whole numbers and decimals may be written in the same line, with a point between

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them, called the separatrix; thus, 864 is written 86,4; and 9.27 is written 9,27. You must always remember, that the denominator is repeated in the expression when it is not written; thus, you say, 4 tenths, and 27 hundredths, &c., although you have no denominator expressed in the fraction. Decimals decrease in a tenfold proportion from the left hand to the right, or as they are removed, or recede from the place of units; thus, ,5 is only one tenth of the value which it would express in the place of units, if you should take away the decimal point; and ,05 is only one tenth as much as .5, and

so on.

1000

10

100

When ciphers are placed at the right hand of Decimal Fractions, they do not increase, or diminish their value, as every significant figure continues to possess the same value; thus, 5, 50, 500, being 5 five tenth parts, 50 fifty hundredth parts, 500 five hundred thousandth are all equal in value; for when you annex a cipher to the decimal, the denominator assumes one, consequently, it is multiplying the numerator and denominator by the same number; and, therefore, the proportion between them must ever remain the same. But when ciphers are placed at the left hand of a Decimal Fraction, they diminish the value of the decimal in a tenfold proportion; thus,,5,05,005, are the same as 5,150, 1000, in value, because, in the first example, ,5 shows that a unit is divided into ten parts, and that the fraction contains five of those parts, that is, five tenths; and the second example, ,05, shows that a unit is divided into one hundred parts, and the fraction contains only five of those parts, that is, five hundredths, &c. Hence, it is very evident, that the magnitude of a Decimal Fraction, compared with another, does not depend upon the number of its figures, but upon the value of its first left hand figure.

I presume, from your knowledge of federal money, you will be able to understand this perfectly, for federal money is purely decimal money, of which the dollar is the unit; and the inferiour, or lower denominations, the decimal parts. Thus, 5 dollars and 36 cents are expressed, $5,36, or $5,36. You must remember, that it takes ten tenths, or one hundred cents to make a dollar; and, therefore, when a dollar is divided into one hundred parts, the parts are cents consequently, the,36 hundredths are cents, of which it takes one hundred to make

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a unit, or dollar. In federal money, therefore, tenths represent dimes; hundredths represent cents, and thousandths represent mills; but the decimals are commonly expressed, where the unit is a dollar, in cents and mills; or taken together they represent thousandths of a dollar,

By paying particular attention to the preceding EXPLANATIONS, you will be able perfectly to understand the nature of Decimal Fractions, and clearly to perceive wherein they differ from Vulgar Fractions, and also from WHOLE NUMBERS OF

INTEGERS.

You will remember, that you learned, in the notation and numeration of figures, or WHOLE NUMBERS, to count from the right hand to the left; and also that they increase in a tenfold proportion from the right to the left; but in the notation and numeration of Decimal Fractions, you must learn to count them from the left to the right, also that they decrease in a tenfold proportion from the left to the right.