13X1X 7X6

Then XXX1=

11 the Answer.

3x5x32x1.

2. Multiply

by

3. Multiply 5 by f.

4. Multiply of 5 by of.

5. Multiply of by of of 114.

6. Multiply 9, of 3, and 124 continually together.

Ans. 64

747

18

Ans. 2413.

7. What is the continual product of 3 of 3, 51, 7 and of?

Ans. 45%

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

Ans. 1.

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.

1.

2.

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.

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

1st. 423=213] which mult, by 84

3)426

After this manner may feet and inches be multiplied, calling 1 inch

of a foot, 2 inches 4, 3 inches, 1424 inches, 5 inches, 6 inches, 1704 7 inches a 8 inches 3, 9 inches 2, 10 inches, 11 inches of a foot.

5)1846

Product 369

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

4

4

1 4

4

2 i

divide by Now 2 is manifestly of ofor; but of 2;

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 denominater 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; 3333; 735% .735, &c.

The point prefixed is called the Separatrix.

But if the numerater 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 10% 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 deci mals infinitely decreasing toward the right to 0, as in the following

TABLE.*

hand

It will be very apparent to the learner from the nature of decimals, and

4;

what has been said of Federal Money, that this money is purely decimal; and, the dollar being the moucy unit, the lower denominations are plainly so many decimal parts of a dollar; thus 9 dollars and 8 dimes are expressed 9·89,8 doll.—12 dollars, 4 dimea, and 7 cents thus, 12:47 12,47 doll.-20 dollars, 3 dimes, 4 cents and 5 mills, thus 20 345 20 345 doll.-ico dollars and 9 mills, thus 100 009 100 doll. and 50 dollars, 5 cents, thus 5005-50,0 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.

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

455927 dimes 455,297 dollars:

45697 eagles 4 55 99 7.

18050

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.

%; 3 =

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 = TOUT, &c. And if a decimal be expressed by several gures, the denominator is 1, with so many cyphers as the lowest figure is distant from unit's place. So 357 signifies, and 0053 53 &c.

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 =

50

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. 5,1; 05, 150; 005, 13%‰·

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. 76% 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.

100

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, 060, 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 denominator with 459, it will be 500.

or 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 so 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 denominator, that of the decimal part. So 45.309 is equal to 45303, as is evident from the method given to reduce a mixed number to an improper fraction:

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+0197581-3478-112.358.

19.073
2.3597

223.

0197581

3478 1

12.358

3734.9104581 the sum.

2. Required the sum of 4294-21-37+355-003+107+1-7?

Ans. 808-148.

3. Required the sum of 5 3+11-973+43+9+1-7314+34-39

Ans. 103-2044.

4. Required the sum of 973+19+1-75+93-7164-4-9501 ? Ans. 1088.4165.

SUBTRACTION OF DECIMALS.

RULE.

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

EXAMPLES.

1. Find the difference of 1793-13 and 817-05693 ?

From 1793-13

Take 817-05693

Remainder 976-07307

2. From 171-195 take 125.9176.

3. From 219-1384 take 195-01.

1. From 400 take 945-0075.

Ans. 45-2774.

Ans. 23-2284. Ans. 234-9925.