CONTRACTED MODES OF MULTIPLICATION.

Long Multiplications may sometimes be contracted.

1 st. By contracting the partial products, when the whole of the figures are not required in the result.

2 nd. By the use of reciprocal divisors.

3 rd. By proportioning the result of an approximate multiplier.

METHOD 1.

Rule. Mark that figure of the multiplicand which corresponds with the place or value of the lowest figure to be retained in the product.

Place the unit figure of the multiplier under the marked figure of the multiplicand, and keeping this figure in this place, invert the figures in the multiplier.

Then multiply by each of the given figures, beginning each operation with the figure immediately above it in the multiplicand, and placing the right hand figure of each product in the same column, and find the amount of the whole.

Obs. 1. For the sake of greater exactness, it may be proper to mark the multiplicand one place lower than the lowest figure to be retained.

Obs. 2. If the multiplicand does not contain a sufficient number of decimal places, annex as many ciphers as may be necessary. Obs. 3. If there is not any unit figure in the multiplier, put a cipher in its place.

Obs. 4. In taking each partial product, such a number must be added to the lowest figure, as is equal to the number of the tens in the product of the highest figure rejected.

Obs. 5. Each multiplying figure must be separately used.

Ex. 1. To multiply 47.3485 by 5.164 retaining 4 places of decimals in the product.

Four places of decimals being required to be retained, the 5 is marked, and the unit figure, 5, of the multiplier is placed under it.

To the third product we add 5 for 6 times the 8 rejected, and to the 4 th product we add 2 for 4 times the 4 rejected.

Example 2. To multiply 62.807 by 13.486, and by 0.13486, retaining 4 places of decimals in the product.

In the second operation 5 places are used, (that is, 1 more than the 4 places required in the product,) according to the first of the preceding observations.

METHOD 2.

By the use of a reciprocal divisor.

A reciprocal divisor is the decimal value of a fractional part, the denominator of which is the same number as the multiplier; thus 0.5, which is equal to, is the reciprocal divisor for 2 as a multiplier.

Instead of using these decimal values, we may consider the reciprocal as a fraction of which the denominator is equal to the division of the numerator by the given number; as

10 for 2, 190 for 25, 1990 for 125, 1000

10000 for 625,

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which are the chief numbers for which these substitutions can conveniently be made.

But for 125 we may also take 100 times and add 1-4 th, for 625 we may take 6 times 100 times, and for 3125 we may take 3 times 100 times.

Examples of Approximate products.

To multiply 249.68 by 612 and by 587.

To multiply 43.708 by 99.9, and by 99.7.

We here multiply the 1000 times by 5, placing each of the figures in the result

3 places to the right of the figure multiplied.

CONTRACTED MODES OF DIVISION.

The contractions that may be practised for shortening the operations of Long Divisions, are,

1 st. The cancelling of the dividing figures, or the contraction of the partial products and remainders.

2 nd. The use of reciprocal multipliers. 3 rd. The use of approximate divisors.

METHOD 1.

Rule. Correct the dividend if necessary, for the decimal in the divisor; then find how many places of whole numbers there will be in the quotient, and add to the number as many places of decimals as the quotient may be required to contain.

Then, if the divisor contains as many figures, instead of bringing down ciphers in the lower dividend figures, drop or cancel one figure of the divisor at each step in the operation.

If the divisor contains a greater number of figures, drop, at the commencement, all the superfluous figures.

If the divisor contains a less number of figures, commence and continue the division by the ordinary method, as many steps as there are places deficient.

Obs. As each of the figures is dropped it may be cancelled or marked. To the first figure of each product, is to be added the number of tens produced from the cancelled figure, and for the sake of greater precision, one more place may be reckoned than the number of places required to be retained.

Ex. 1. To divide 62.873654 by 3.17646 to retain 4 places of decimals.

317646) 6287365.4 ( 19.7937

311090

25209

2974

116

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When this dividend has been corrected for the 5 places of decimals in the given divisor, it appears that there will be 2 places of whole numbers in the quotient; and as there are to be 4 places of decimals, there will be 6 figures in all, the same as the number of figures in the given divisor.

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Therefore after having found the first quotient figure, we drop the 6 in the divisor in the next step of the division, instead of bringing down the 5 from the dividend; but in taking 9 times the 31764, we add 5, for the tens in 9 times 6; saying, 9 times 6 are 54, carry 5; 9 times 4 are 36 and 5 are 41 from 50, there remains 9, and carry 5; 9 times 6 are 54 and 5 are 59, &c.

The 4 is dropped in the 3 rd step, and 3 is carried for 4 times 7; the 6 is dropped in the 4 th step, and 5 is carried for the tens ; the 7 is dropped in the 5 th step, and 2 is carried for the tens ; and in the 6 th step, the quotient figure is taken at 7, as giving the nearest product.

Ex. 2. Divide 67854.34 by 2784 to 4 places of decimals.

2784) 67854.34 (24.3729

12174

10383

2031

82

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As there are here to be 6 figures in the quotient, and there are only 4 in the divisor, the contraction does not begin until 2 figures have been brought from the dividend.

Ex. 3. Divide 63.84715 by 436.27827 to 5 places of decimals.

43627827) 6384715.0 (0.14634

20219

2768

151

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As there are not any places of whole numbers in the quotient, and only 5 places of decimals are required, we begin the operation with only the five highest figures in the divisor, and drop 1 figure at each step.