nearly the same, since the excesses and the defects will then probably balance each other. As the last figure, however, cannot be depended on, it is proper to work for one figure more than it is necessary to have true, and to reject it at the conclusion: and in lengthened computations, such as many of those in compound interest and annuities, it may be right to work for two or three additional figures. Exam. 4. Multiply 681472 by 01286, so that the decimal in the product may contain five figures. In this example, since the multiplier contains no integer, a cipher is placed below the fifth figure of the multiplicand. and then, the multiplier being written in reversed order, the work proceeds as in the last example. Exam. 5. Multiply 7.94 by 3.69, so that there may be four places of decimals in the product. Here the multiplicand is carried out to four places; and by a process similar to those which precede, the answer is found to be 29.315, which is quite correct. ⚫681472 682100 681 + 136 + 55 4+ 00876, product. 7.9444 Answers. 1.262643 267.706650 5.782154 2.45975 ⚫002521 Exer. 22, 23, 24. Required the product, true to six places of decimals, of the numbers given in Exercises 4, 6, and 14, in the article on multiplication of fractions; the several fractions being previously reduced to decimals. Answ. 113445, 467480, and 2.393162. 25. The sun's diameter is 111.454 times the equatorial diameter of the earth, which is 7925 648 miles. Required the sun's diameter in miles. Answ. 883,345 miles. 26. The moon's mean distance from the earth is 29.982175 times the equatorial diameter of the earth. Required that distance in miles. (See the last exercise.) Answ. 237628-165 miles. 27. In consequence of the vast mass of matter contained in the sun, a body at his surface weighs 27.9 times as much as it would at the surface of the earth. How much, then, would a person who weighs here 161 lbs. (14375 cwt.) weigh at the sun's surface ? Answ. 2 tons 0 cwt. qr. 11.9 lbs. DIVISION IN DECIMAL FRACTIONS. RULE I. (1.) If the divisor and dividend do not contain the same number of places of decimals, supply the deficiency by annexing ciphers, or, in a periodical decimal, the next figures of the period. (2.) Then, rejecting the separating points, divide as in whole numbers, and the quotient will be a whole number. (3.) If there be a remainder, after all the figures of the dividend have been used, ciphers or periodical figures may be annexed, till nothing remains, or till as many figures are found as may be judged necessary. The part of the quotient thus obtained, will be a decimal. If, after the rejection of the separating points, the divisor be greater than the dividend, the quotient will contain no whole number, and the work will proceed according to Rule I. in reduction of decimals. Exam. 1. Divide 1346 5 by 43.68. Here, by annexing a cipher to the dividend, and rejecting the points, we have for divisor 4368, and for dividend 134650. Hence, dividing in the common way, we find 30 for the integral part, and annexing ciphers to the remainders, and continuing the operation, we get 826465, &c. The answer, therefore, is 30 826465, &c. The work is left for the learner to perform. With respect to the reason of the operation, the value of 1346·5 is not changed by the annexing of a cipher; and the removal of the points merely multiplies each of the given numbers by 100. (See page 181.) It is evident, therefore, that the value of the quotient will not be affected; since, while the dividend is multiplied by 100, the divisor is increased in the same ratio. We might also consider the dividend as the numerator, and the divisor as the denominator of a fraction; and then the reason of the process would depend on Proposition 1, established in page 50. The reason of removing the points is to make the dividend and divisor whole numbers, and thus to render the operation, as much as possible, the same as in simple division. Exam. 2. Divide 1342 by 67·1. Here, by annexing three ciphers to the divisor, and rejecting the points, we get for divisor 671000, and for dividend 1342. Then, the divisor being the greater, the quotient will contain no integral part; and the annexing of a cipher to the dividend gives one cipher for the quotient: the annexing of a second gives another cipher; but the addition of a third gives 2. Hence, the quotient is .002. When the number of places of decimals in the divisor is not greater than in the dividend, the number of figures of decimals in the quotient will be equal to the difference between the number of places in the divisor and dividend, as is evident from multiplication of decimals; and in this way the number of decimal figures in the quotient is often easily determined. RULE II. When the divisor consists of many figures, the work will be shortened, if, instead of annexing a cipher, or a periodical figure, to each remainder, a figure be cut off from the divisor. In this case, each product is to be increased by carrying from the product of the figure last cut off, and of the figure last placed in the quotient. It may facilitate the use of this important contraction, if, after the rejection of the separating points, so many figures be annexed to the divisor and dividend, or taken from them, that the divisor may contain one or two figures more than are required to be in the quotient. Other directions might be given, but the following examples and illustrations will perhaps be preferable. Exam. 3. Divide 2.3748 by 1.4736, so that the quotient may contain three places of decimals. In this example, the numbers being prepared according to Rule I., and the first figure of the quotient being found, instead of adding a cipher to the remainder, that, 3 is cut off in like manner, and then 7. The quotient is found to be 1611, or more nearly 1.612, because the remainder 8 is rather more than the half of 14. The annexed operation at full length will explain the reason of the contracted process, the vertical line cutting off the rejected part. Exam. 4. Divide 73.64 by 432, so that the quotient may have four places of decimals. Here, it is easy to see, that the quotient will contain three places of whole numbers. (This would be seen by dividing 736 by 4.) Hence the quotient must contain seven figures. Extending, therefore, the divisor to eight places, and the dividend to the same number of places of decimals, the process will stand as in the margin. In the work, instead of bringing down the last two ciphers of the dividend, two figures may be cut in succession from the divisor, and the rest of the operation will proceed as before. 43232323)7364000000(170-3355 43232323 30407677 30262626 145051 129697 15354 12970 2384 2162 222 216 The pupil should work all the following exercises by Rule II., and several of them by Rule I. In those in which the divisor consists of few figures, the work must be commenced by Rule I., and it may be finished by Rule II. Exer. 20. The sun contains 354,936 times as much matter as the earth, and 1048-69 times as much as Jupiter. From these data, find how many times as much matter Jupiter contains as the earth. Answ. 338 456 times. 21. How many times as much matter does the earth contain as the moon, the matter in the moon being represented by 0.0125172, when that in the earth is denoted by 1? Answ. 79.89 times. After the full illustration of the multiplication and division of decimals, which has been given in the preceding pages, it appears unnecessary to give their application in the rule of proportion; as, in thus applying them, the pupil can feel no difficulty, the terms being arranged in the manner already explained, and the product of the second and third terms, in like manner, divided by the first. PRACTICE. An ALIQUOT PART of a quantity is such a part às, when taken a certain number of times, will exactly make that quantity. Thus, 5 is an aliquot part of 20, 3 of 12, &c. What is generally called PRACTICE in mercantile arithmetic, is only an abridged method of performing operations in the rule of proportion by means of aliquot parts; and it is chiefly employed in computing the prices of commodities. |