Then XXX1 =

13X1X 7X6

18-11 the Answer.

3X5X32X1

Ans. 20

Ans. 7.

Ans.

2. Multiply by.

3. Multiply 5 by J.

4. Multiply of 5 by 3 of 4.

5. Multiply of by of of 113.

6. Multiply 93, of 3, and 124 continually together.

7. What is the continual product of 2 of 4, 5, 7 and

3. What is the continual product of 7,,of, and 31?

Ans. 11

Another method for the Multiplication of mixed Quantities.

CASE I.

To multiply a whole number by a fraction, or a fraction by a whole number.

RULE.

Multiply the whole number by the numerator of the fraction and divide the product by the denominator: But if the numerator be 1, divide by the denominator only.

6.

7.

To multiply a whole number by a mixed one.

RULE.

Multiply by the fraction as in Case 1st; then multiply by the whole number, and add the two products, as in the examples-or, to multiply a mixed number by a whole one, change the place of the factors, and proceed as the rule directs.-See example 6.

6.

CASE III.

To multiply a mixed number by a mixed number.

RULE.

Multiply the integral part of the multiplicand by the denominator of its fractional part, and add thereto its numerator: Then multiply by the mixed multiplier, by Case 2d, and divide the product by the denominator of the fractional part of the multiplicand, as in the following example.

Mult. 423 S

By 8

1st. 423=213) which mult. by 8

3)426

After this manner may feet and inches be multiplied, calling 1 inch of a foot, 2 inches 1, 3 inches 1, 1424 inches, 5 inches, 6 inches 1, 1704 7 inches 7, 8 inches, 9 inches 3, 10 inches, 11 inches of a foot.

5)1846

Product = 3691 J

DIVISION OF VULGAR FRACTIONS.

RULE.*

Prepare the fractions as before then, invert the divisor and proceed exactly as in Multiplication: The products will be the quotient required.

* The reason of the rule may be shewn thus. Suppose it were required to

DECIMAL FRACTIONS.

A DECIMAL FRACTION is a fraction whose denominator is a unit with so many cyphers annexed as the numerator has places of figures.

As the denominator of a decimal fraction is always 10, or 100. or 1000, &c. the denominators need not be expressed: For the numerator only may be made to express the true value: For this purpose it is only required to write the numerator with a point before it at the left hand, to distinguish it from a whole number, when it consists of so many figures as the denominator has cyphers annexed to unity, or 1: So is written 5; 10% 33; 100% .735, &c.

The point prefixed is called the Separatrix.

33

735

But i the numerator has not so many places as the denominator has cyphers, put so many cyphers before it, viz. at the left hand, as will make up the defect; so write thus, 05; and thus, 006, &c. Thus do these fractions receive the form of whole numbers.

We may consider unity as a fixed point, from whence whole numbers proceed infinitely increasing toward the left hand, and decimals infinitely decreasing toward the right to 0, as in the following

TABLE.*

hand

10

It will be very apparent to the learner from the nature of decimals, and what has been said of Federal Money, that this money is purely decimal; and, the dollar being the money unit, the lower denominations are plainly so many decimal parts of a dollar; thus 9 dollars and 8 dimes are expressed 9.8 9.8 doll 12 dollars, 4 dimes, and 7 cents thus, 12:47 12 47 doll.-20 dollars, 3 dimes, 4 cents and 5 mills, thus 20 345 20 345 doll.-100 dollars and 9 mills, thus 100.009 100- Q doll. and 50 dollars, 5 cents, thus 500550, doll. wherefore, it is, in all respects, added, subtracted, multiplied and divided, the same as decimals; and, of all coins, it is the most simple.

1000

It may also be observed that the sum exhibits the particular number of each different piece of money contained in it, viz. 455997 mills 45599,7 cents === E. D. d. c. m.

45597 dimes

10

45599 dollars 45,5997 eagles 4 5 5 9 9 7.

10000

Also, the names of the coins, less than a dollar, are significant of their values. For the mill, which stands in the 3d place at the right hand of the separatrix

From this table it is evident, that in decimals, as well as in whole numbers, each figure takes its value by its distance from unit's place: If it be in the first place after units (or the separating point) it signifies tenths; if in the second, hundredths, &c. decreasing in each place in a tenfold proportion.

Every single figure expressing a decimal, has for its denominator a unit or 1, with so many cyphers as its place is distant from unit's place: Thus 2 in the decimal part of the table

180; 4 = 100, &c. To, &c. And if a decimal be expressed by several figures, the denominator is 1, with so many cyphers as the lowest figure is distant from unit's place. So 357 signifies 35, and 0053

10001

&c.

10009

Cyphers, placed at the right hand of a decimal fraction, do not alter its value, since every significant figure continues to possess the same place: So 5, 50, and 500, are all of the same value, and each

1 == 500

100 1000

But cyphers, placed at the left hand of a decimal, do alter its value, every cypher depressing it to of the value it had before, by removing every significant figure one place further from the place of units. So 5, 05, 005, all express different decimals, viz. 5,105, 1; 005, 15%

Hence may be observed the contrary effects of cyphers being annexed to whole numbers, and decimals.

It is likewise evident from the table, that since the places of decimals decrease in a tenfold proportion from units downwards, so they consequently increase in a tenfold proportion from the right hand toward the left, as the places of whole numbers do: For, ten hundredth parts make one tenth, ten tenths make 1; ten units, ten; ten tens, one hundred, &c. viz. 18=%, 18=1, and 1×10 10, which proves that decimals are subject to the same law of Notation, and consequently of operation, as whole numbers are.

10

Decimal fractions of unequal denominators are reduced to one common denominator, when there are annexed to the right hand of those, which have fewer places, so many cyphers, as make them equal in places with that which has the most. So these decimals, ∙5, 06, ·455, may be reduced to the decimals, ·500, 0, and ·455, which have, all, 1000 for their denominator.

Of Decimals, that is the greatest, whose highest figure is greatest, whether they consist of an equal or unequal number of places : Thus, 5 is greater than, 459, for if it be reduced to the same de nominator with 459, it will be ·599.

ar place of thousandths, is contracted from mille the Latin for thousand: Cent, which occupies the second place, or place of hundredths, is an abbreviation of centum, the Latin for bundred: And dime, which is in the first place or place of tenths, is derived from disme, the French for tenth.

Such being the nature of Federal Money, its operations can in no other way be AO well understood as in obtaining a good knowledge of decimals, and applying their several rules to the various cases of money matters.

In sums of Federal Money, it is customary to read it in dollars, cents and mills. Thus the above example is read 455 dolls. 99 cents and 7 mills.

A mixed number, viz. a whole number, with a decimal annexed, is equal to an improper fraction, whose numerator is all the figures of the mixed number, taken as one whole number, and the denom inator, that of the decimal part. So 45-309 is equal to 4530, is evident from the method given to reduce a mixed number to an improper fraction:

as

Thus, 45×1000+309-45309 as above.

ADDITION OF DECIMALS.

RULE.

1. Place the numbers, whether mixed, or pure decimals, under each other, according to the value of their places.

2. Find their sum as in whole numbers, and point off so many places for decimals, as are equal to the greatest number of decimal places in any of the given numbers.

EXAMPLES.

1. Find the sum of 19 073+2 3597+223+01975813479 1 12358.

19.073
2.3597

223.

⚫0197581

3478 1

12.358

3734.9104581 the sum.

2. Required the sum of 429+21-37+355-003-107+1-7?

Aus. 808-148.88/4

3. Required the sum of 5 3+11·973+43+9+1·7314+343 ?

Ans. 103:2044.

4. Required the sum of 973+19+1-75+93-7164+9501?

Ans. 1088.4165.

SUBTRACTION OF DECIMALS.

RULE.

Place the numbers according to their value; then subtract as in whole numbers, and point off the decimals as in Addition.