3. Divide 3 bushels by 3 quarts.-First, 3 bush.=96 qts., then 96-3-32 bushels, Ans.

4. Divide 4 acres by 5 rods.-First, 4 acres=640 rods; then 640-5=128 acres, Ans.

NOTE 1. The preceding examples exist only in theory. 2. The two foregoing problems and their examples, involve the principles of cross multiplication and the reverse.

3. See the operation of the thirty-fourth question in Proportionals.

TO ABBREVIATE THE OPERATIONS OF

ARITHMETIC.

PROB. I. To abbreviate operations in Multiplication and Division.

RULE. 1.-Draw a perpendicular line, and place the numbers to be multiplied on the right, and the numbers by which you are to divide, on the left hand.

2. If there be two equal numbers on each side of the line, cross them out, and omit them in the operation.

3. If a number on one side of the line will divide a number on the other side, without a remainder, erase both numbers, and substitute for the larger the number of times it contains the smaller. Multiply the remainders together, on the right, for a dividend, and the remainders on the left, for a divisor.

EXAMPLE.-Multiply 8 by 9, and divide by 8; multiply the quotient by 6, and divide the product by 3.

N.B. For want of proper type, a dot is | placed atop the figures to be erased.

Operation.

8

9

3

6 2

18, Ans.

PROB. II. To abbreviate the operation of the Multiplication and Division of Fractions by whole numbers-whole numbers by Fractions, or Fractions by Fractions.

RULE. Draw a perpendicular line and place all those figures, which are to be multiplied together for a numerator, or dividend, on the right of the line, and those figures which are to be multiplied together for a denominator, or divisor, on the left hand of the line; also, the numerators of fractions, by which a division is to be made, on the left.

The question thus stated, equals on each side of the line may be crossed out.

When no two numbers remain, one on each side of the line, capable of being divided by any one figure, (see preceding operation,) multiply the figures on the right of the line for a numerator, or dividend, and those on the left for a denominator, or divisor, and the result will be the answer in the lowest terms of the fraction.

PROB. III. To abbreviate the operation of all proportional questions.

RULE. 1.-Draw a perpendicular line, and place the sign of the answer on the left, at the top of the line, and that number which is of the same kind with the answer on the right, at the bottom; and as this number is greater or less than the answer sought, place the greater or less of the two remaining numbers on the right, and the other on the left, and proceed in all respects as before instructed, if Direct or Inverse; but, if Double Proportion, place any two of the same kind, of the remaining numbers, one on the right and the other on the left, according to directions for Direct or Inverse Proportion.

2. Then cross out, multiply, and divide, as before directed. NOTE 1.-When the answer is required in a different denomination from that given in the supposition, follow the tables from the denomination given, to the denomination required.

NOTE 2.-Mixed numbers must be reduced to improper fractions, and the numerators placed on that side of the line where the whole numbers, standing in the place of the fraction, would be placed.

EXAMPLE.

1. If 1 pint cost 10d. what will 3 hhds. cost in pounds?

Explanation.

Here, the answer is required in a different denomination than that given in the supposition. (See Note 1.)

Therefore, follow the tables, until you

find the name of the answer required, observing to commence each successive step on the left with the denomination last placed on the right.

C

This method of operation renders the process of Reduction Descending and Ascending, entirely unnecessary.

2. A merchant bartered 58 cwt. of sugar, at 62d. per lb., for tea, at 8gs. per pound. How much tea did he receive?

To discover to the student the utility of the abbreviating method, in operations, and the superiority it has over all other modes of performing such solutions as may be performed by the abbreviating process, let the preceding question be solved by the two different operations.

FRACTIONAL OPERATION OF BOTH METHODS.

cwt., 3 of 112-5936 | 58-cwt.For this part of this 67d., 27X5936-160372 63=27d. Method of operas., of d. 288=s. tion, see Note 2d.

58

8

4. If 3 men can build 360 rods of wall in 24 days, how many rods can 8 men build in 27 days?

NOTE 2. The ingenious pupil will easily discover to which proportion each of the preceding questions belong.

REMARK.-Persons intending to teach, should become thoroughly acquainted with the three preceding problems, if they wish to enjoy the desirable advantage of but one mode of solution, to the questions of several different rules.

Considerable should be said by committees and teachers, in favor of pupils being taught this advantageous method of performing solutions.

Abbreviating operations is an amusement; and the necessary combination of numbers, to close a question by this process, is the simplest of the different methods, and when once practised, it can never be supplanted from the memory.

The scholar will find the most difficult questions to yield readily to this mode of solution, and has the satisfaction of proceeding upon a principle which is evidently unerring.

It is applicable to all Proportional Questions, embracing the Rules of Three, Single and Double, Direct and Inverse; Interest, Discount, Barter, Loss and Gain, Exchange, Reduction, Multiplication and Division of Fractions, &c. &c.

X. DECIMAL FRACTIONS.*

The reason of what is most difficult to understand in Decimal Fractions, is that of Multiplication, Division, and Reduction.

*Decimal is derived from the Latin word decem, which signifies ten. Fraction is derived from the Latin word frango, which signifies to break.

MULTIPLICATION AND DIVISION.

DEMONSTRATION.

1. It is obvious, that multiplying whole numbers by any fraction, is taking a certain part of the multiplicand for the product; consequently, multiplying one fraction by another, must produce a fraction smaller than either of the factors. And it is evident, that ,25 is or, and,5 is or ; and it is obvious, that of of, is of 12, which is,125, because decimals read the same as whole numbers, and are the same as their equivalent vulgar fractions; therefore, ,25X,5= ,125; consequently, the number of decimal places in any product, must be equal to the number of decimal places in both the factors of that product: Hence the RULE.

2. The preceding shews, that the product must have as many decimal places as both its factors.

The Multiplicand and Multiplier, in proving Multiplication, becomes the divisor and quotient in Division. Therefore, the number of decimal places in the quotient, must be equal to the difference between the number of decimal places in the dividend, and the number of decimal places in the divisor: Hence the Rule.

REDUCTION.

DEMONSTRATIONS,

1. To Reduce a Vulgar Fraction to its equivalent Decimal. Let the given fraction, whose decimal expression is required, be 15.

Now, since every decimal fraction has 10, 100, 1000, &c. for its denominator: and, if two fractions be equal, it will be, as the denominator of one is to its numerator, so is the denominator of the other to its numerator.

Thus, as 15:9:10:6,=, the numerator required, and is the same as by the rule.

2. To Reduce Numbers of different denominations to their equivalent decimal values, and the contrary.

This is only expanding or contracting the ratios in question, as they are larger or less than the ratio required; and what is taken from one ratio, to make it equal with the tenfold one, is only giving to another place what would have been left in its preceding place, had the common ratio been equal to the one in question.