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which is in the place to its right in the multiplicand; do not set down the figure of this product, but carry its nearest ten to the next, and proceed.

3. Place the first figures of all the products under one another; add as usual, and mark off the required number of decimal places.

EXAMPLE.

Multiply .03281674 by 234.781 reserving six places of decimals.

.03281674

187432

6563348

984502

131267

22971

2625

33

7.704746 Ans.

VII. DIVISION OF DECIMALS.

Rule-1. To divide a decimal by 10, 100, 1000, &c. Remove the deci mal point as many places to the left as there are ciphers in the divisor, supplying, if necessary, ciphers on the left.

2. To divide one decimal by another. Make the number of decimal places in the dividend at least equal to that of those in the divisor, by annexing ciphers on the right, if necessary, preceded by a point, if the dividend be a whole number. Divide as in whole numbers; and when all the figures of the dividend have been used, annex a cipher to each succeeding remainder, as far as the division may be carried. Mark off in the quotient as many figures for decimals as the number of decimal places in the dividend (including the ciphers annexed to the remainders) exceeds that of those in the divisor.

EXAMPLES.

1. Divide 71634.3069 by 10, 1000, 1000000, &c.

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VIII. ABBREVIATED MEthod of divISION OF DECIMALS, WHEN ONLY A CERTAIN NUMBER OF DECIMAL PLACES ARE REQUIRED IN THE QUOTIENT.

Rule-1. Proceed to find the figures of the quotient by the ordinary rule, till the number that remain to be found is less than the number of figures in the divisor. If this be so at first, find one figure of the quotient by the ordinary rule.

2. Instead of bringing down the next figure to the remainder, cut off the last figure of the divisor, and divide the remainder by the remaining figures to obtain the next figure of the quotient.

3. In forming the subtrahend, take care to carry the nearest ten from the product of the quotient-figure, and the figure cut off.

4. Cut off another figure of the divisor, and repeat the process, and so proceed till no figures of the divisor are left.

5. If the divisor at first contain more figures than the quotient, the excess may be rejected from the beginning, attention however being paid to (3).

EXAMPLE.

Divide 2.71828180 by 3.1415927 to 8 places of decimals.

3.1,4,1,5,9,2,7)2.71828180(.86525596

2 51327416

20500764

18849556

1651208

1570796

80412

62832

17580

15708

1872

1571

301

283

18

19

Ans. .86525596.

IX. TO REDUCE A VULGAR FRACTION TO A DECIMAL.

Rule. Write the numerator as a decimal by annexing ciphers preceded by a decimal point, and divide by the denominator, by the preceding Rule. If the denominator be composed by the product of 2's and 5's only, the division will terminate, and the decimal is then a terminating, or nonrecurring decimal. If the denominator have any other prime factor than

or 5, the division will not terminate, and must be carried on till the emainder, and therefore the quotients begin to recur.

Obs. A non-terminating decimal, if the figures recur, is called a reurring, or circulating decimal. A dot is placed over the first and last f the recurring figures.

EXAMPLES.

1. Express as decimal fractions 3, 7, 51, 2}.

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X.

TO CONVERT A RECURRING DECIMAL INTO A VULGAR FRACTION.

Rule. For the numerator take the whole decimal diminished by the on-recurring part, and for denominator as many nines as there are re-urring figures, followed by as many ciphers as there are non-recurring figures.

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Def. 1. Ratio is the relation which exists between two magnitudes of the same kind, it being considered what multiple, part or parts, the 1st is of the 2nd.

A ratio is written thus :- 3: 4 (= the ratio of 3 to 4.)

Def. 2. The former of the two magnitudes composing a ratio is called the antecedent, the latter the consequent.

Def. 3. A ratio of greater inequality is one in which the antecedent is greater than the consequent. A ratio of less inequality is one in which the antecedent is less than the consequent.

Def. 4. If, by multiplying together the antecedents of two or more ratios for a new antecedent, and the consequents for a new consequent, a new ratio be formed, this ratio is said to be compounded of the former ratios.

Def. 5. If two ratios are equal, the four magnitudes are proportional. A proportion is written thus 3: 46: 8 or 3: 4 :: 6 : 8.

Def. 6. Magnitudes are said to be continually proportional, when the 1st 2nd 2nd: 3rd :: 3rd 4th : : &c.

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Def. 7. The 1st and 4th terms of a proportion are called the extremes, the 2nd and 3rd the means.

Def. 8. The antecedents of the ratios are called homologous terms of a proportion, as also the consequents.

Def. 9. If by multiplying together the corresponding terms of two or more proportions, a new proportion be formed, this proportion is said to be compounded of the others.

Def. 10. One quantity is said to vary directly as another, when, if the former be changed, the latter is changed in the same ratio. Thus if A vary directly, or vary, as B, and A1, B1, be other corresponding values of A and B, A: A1B: B1. This is expressed thus:-A ∞ B.

Def. 11. One quantity is said to vary inversely as another, when if the former be changed, the reciprocal of the other is changed in the same ratio.

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