degrees in 1 day; find how many degrees it will move in 2.52035 days, carrying the answer to two decimal places. Ans. 360.00 degrees. 11. A gallon of distilled water weighs 8.33888 pounds; how many pounds in 35.8756 gallons? 12. One French metre is equal to how many yards in 478.7862 metres. Ans. 299 16 pounds. 1.09356959 English yards; Ans. 523.58 yards. 13. The polar radius of the earth is 6356078.96 metres, and the equatorial radius, 6377397.6 metres; find the two radii, and their difference, to the nearest hundredth of a mile, 1 metre being equal to 0.000621346 of a mile. DIVISION. 223. In division of decimals the location of the decimal point in the quotient depends upon the following principles: I. If one decimal number in the fractional form be divided by another also in the fractional form, the denominator of the quotient must contain as many ciphers as the number of ciphers in the denominator of the dividend exceeds the number in the denominator of the divisor. Therefore, II. The quotient of one number divided by another in the decimal form must contain as many decimal places as the number of decimal places in the dividend exceed the number in the divisor. 1. Divide 34.368 by 5.37. OPERATION. 5.37) 34.368 ( 6.4 2 148 2 148 PROOF. 1000 539 = 11 ANALYSIS. We first divide as in whole numbers; then, since the dividend has 3 decimal places and the divisor 2, we point off 3 - 2 = 1 decimal place in the quotient, (II). The correctness of the work is shown in the proof, where the dividend and divisor are written as common fractions. For, when we have canceled the denominator of the divisor from the denominator of the dividend, the denominator of the quotient must contain as many ciphers as the number in the dividend exceeds those in the divisor. 224. Hence the following RULE. Divide as in whole numbers, and from the right hand of the quotient point off as many places for decimals as the decimal places in the dividend exceed those in the divisor. NOTES.-1. If the number of figures in the quotient be less than the excess of the decimal places in the dividend over those in the divisor, the deficiency must be supplied by prefixing ciphers. 2. If there be a remainder after dividing the dividend, annex ciphers, and continue the division: the ciphers annexed are decimals of the dividend. 3. The dividend should always contain at least as many decimal places as the divisor, before commencing the division; the quotient figures will then be integers till all the decimals of the dividend have been used in the partial dividends. 4. To divide a decimal by 10, 100, 1000, etc., remove the point as many places to the left as there are ciphers on the right of the divisor. 14. If 25 men build 154.125 rods of fence in a day, how much does each man build? 15. How many coats can be made from 16.2 yards of cloth, allowing 2.7 yards for each coat? 16. If a man travel 36.34 miles a day, how long will it take him to travel 674 miles? Ans. 18.547+days. 17. How many revolutions will a wheel 14.25 feet in circumference make in going a distance of 1 mile or 5280 feet? CONTRACTED DIVISION. 225. To obtain a given number of decimal places in the quotient. In division, the products of the divisor by the several quotient figures may be contracted, as in multiplication, by rejecting at each step the unnecessary figures of the divisor, (220). 1. Divide 790.755197 by 32.4687, extending the quotient to two decimal places. FIRST CONTRACTED METHOD. 32.4687) 790.755197 (24.35 649 4 141 3 129 9 11 4 97 17 16 1 SECOND CONTRACTED METHOD. 32.4687) 790.755197 141 3 17 COMMON METHOD. 32.4687) 790.7 55198 (24.35 649 3 74 141 3 811 129 8.748 11 5 0639 9 7 4061 1 765787 1 6 23435 142352 ANALYSIS. In the first method of contraction, we first compare the 3 tens of the divisor with the 79 tens of the dividend, and ascertain that there will be 2 integral places in the quotient; and as 2 decimal places are required, the quotient must contain 4 places in all. Then assuming the four left hand figures of the divisor, we say 3246 is contained in 7907, 2 times; multiplying the assumed part of the divisor by 2. and carrying 2 units from the rejected part, as in Contracted Multiplication of Decimals, we have 6494 for the product, which subtracted from the dividend, leaves 1413 for a new dividend. Now, since the next quotient figure will be of an order next below the former, we reject one more place in the divisor, and divide by 324, obtaining 4 for a quotient, 1299 for a product, and 114 for a new dividend. Continuing this process till all the figures of the divisor are rejected, we have, after pointing off 2 decimals as required, 24.35 for a quotient. Comparing the contracted with the common method, we see the extent of the abbreviation, and the agreement of the corresponding intermediate results. In the second method of contraction, the quotient is written with its first figure under the lowest order of the assumed divisor, and the other figures at the left in the reverse order. By this arrangement, the several products are conveniently formed, by multiplying each quotient figure by the figures above and to the left of it in the divisor, by the rule for contracted multiplication, (222), and the remainders only are written as in (112). 226. From these illustrations we derive the following RULE. I. Compare the highest or left hand figure of the divisor with the units of like order in the dividend, and determine how many figures will be required in the quotient. II. For the first contracted divisor, take as many significant figures from the left of the given divisor as there are places required in the quotient; and at each subsequent division reject one place from the right of the last preceding divisor. III. In multiplying by the several quotient figures, carry from the rejected figures of the divisor as in contracted multiplication. NOTES. 1. Supply ciphers, at the right of either divisor or dividend, when necessary, before commencing the work. 2. If the first figure of the quotient is written under the lowest assumed figure of the divisor, and the other figures at the left in the inverted order, the several products will be formed with the greatest convenience, by simply multiplying each quotient figure by the figures above and to the left of it in the divisor. EXAMPLES FOR PRACTICE. 1. Divide 27.3782 by 4.3267, extending the quotient to 3 decimal places. Ans. 6.328 ±. 2. Divide 487.24 by 1.003675, extending the quotient to 2 decimal places. 3. Divide 8.47326 by 75.43, extending the quotient to 5 decimal places. 4. Divide .8487564 by .075637, extending the quotient to 3 decimal places. Ans. 11.221 ±. 136 5. Divide 478.325 by 1.433, extending the quotient to mal places. Ans. 332.94 6. Divide 8972.436 by 756.3452, extending the quotie decimal places. 7. Divide 1 by 1.007633, extending the quotient to 6 places. Ans. .99242 8. Find the quotient of .95372843 divided by 44.73654 to 8 decimal places. 9. Reduce to a decimal of 4 places. Ans. .744 CIRCULATING DECIMALS. 227. Common fractions can not always be exactly expre the decimal form; for in some instances the division will exact if continued indefinitely. 228. A Finite Decimal is a decimal which extends a number of places from the decimal point. 229. An Infinite Decimal is a decimal which exte unlimited number of places from the decimal point. 230. A Circulating Decimal is an infinite decimal in a figure or set of figures is continually repeated in the same as .3333+, or .437437437+. 231. A Repetend is the figure or set of figures cont repeated. When a repetend consists of a single figure, it dicated by a point placed over it; when it consists of mor one figure, a point is placed over the first, and one over t figure. Thus, the circulating decimals .55555+ and .324324 are written, 5 and 324. 232. A repetend is said to be expanded when its figu continued in their proper order any number of places towa right; thus, .24, expanded is .2424+, or .242424242+. 233. Similar Repetends are those which begin at th decimal place or order; as .37 and .5, .24 and .375, 1.56 and 234. Conterminous Repetends are those which end same decimal place or order; as .75 and 1.53, .567, and 3. NOTE.-Two or more repetends are Similar and Conterminous when th |