Explanation. In the operation of this example, after obtaining two places of decimals in the quotient, we find that the same remainder occurs each time, and the same figure is found for a quotient in each succeeding division. This would be the case were we to continue the operation ever so far, for multiplying the same divisor by the same figure each time, we have the same subtrahend which will leave a remainder equal to the preceding one. After the same remainder is repeated once or twice, the operation may be discontinued and as many figures like the last found in dividing, may be set in the quotient as is thought necessary. It sometimes happens that different remainders repeat alternately. In examples of this kind, an exact answer in decimals cannot be obtained. When an exact answer in decimals cannot be found, they are called Circulating decimals. If there be a remainder after the operation is discontinued, the sign of addition is annexed, which implies that the answer is not exact. Explanation. being less than 1 tenth, there will be no decimal in place of tenths. We also observe that after adding two ciphers, the denominator cannot be taken from the numerator, which shows that Tis less than 1 hundredth. It may be observed generally, that as many ciphers wanting one must be set on the right of the decimal point as we annex ciphers to the numeratar. before we can divide, which removes the first decimal to its proper place. 13. Change to a decimal. Ans. ,0079+: ཙ ? Ans. 133+. . 16. 3 farthings equal of a penny. mal of a penny is equal 3 farthings? 17. 10 pence equal of a shilling. What deciAns. .75. What deci Ans. ,8333+. mal of a shilling is equal to 10 pence ? of a day. 18. 1 hour is of a day; change it to a decimal Ans. ,0416+. 19. What decimal part of a day is 11 h. 4 m. ? Explanation. The 11 h. and 4 m. must first be reduced to a vulgar fraction of a day by Explanation to question 58th, page 153. Ans. ,461+. 20. If I sell 3 oz. 15 grs. of silver, what decimal part of a pound do I sell? Ans. 2526+. 21. What decimal part of a mile are 11 inches ? Ans. ,000173+. 22. Reduce 2 A. 3 R. 28 r. 27 ft. to the form of a decimal. Explanation. No operation is to be performed upon the 2 acres, the highest denomination. Reduce the less denominations to the decimal of an acre. Ans. 2 A. and 2+, or 2,925+A. 23. Change 6 C. 100 ft. 1000 in. to the form of a decimal. Ans. 6,78+C. 24. Reduce 5 yd. 3 qr. 2 na. to the form of a decimal. Ans. 5,875 yd. 25. Reduce 7 E. E. 4 qr. 1 na. to the form of a decimal. Ans. 7,85 E. E. 26. Change 12 to the form of a decimal. Ans. 12,461+. to the form of a decimal. Ans. 100,99+. to the form of a decimal. Ans. 676,0026+ ADDITION OF DECIMAL FRACTIONS. RULE. Set the numbers under each other so that every figure may stand in a place of its own name, whether decimals or whole numbers. Add as in simple numbers. The decimal point in the sum must stand directly under those in the given numbers. 1. What is the sum of 16, 25, 130 275, 4000, ? 13 Illustration. A figure in any decimal place is so many tenths of 1 in the next left hand place; therefore, when we obtain 10 in adding any column of decimals, 1 added to the next left hand place equals the 10 so obtained, the same as in simple numbers. In the adjoining operation, 6, the sum of the thousandths, not being equal to 1 hundredth, is written in the place of thousandths. 12 hundredths are equal to 1 tenth and 2 hundredths. 1 tenth is added to the column of tenths for 10 of the hundredths, and the two remaining hundredths set in the place of hundredths. 10 tenths equal an unit; in the 22 tenths, we have 2 units and 2 tenths over. The two units added to the column of units, and the 226 thousandths equal all the given decimals. 2. What is the sum of 61,4+113,12+2579,7+ 190,0793 ? Ans. 2944,2993. N. B. When a whole number and a decimal in connexion are expressed in words, the words denoting decimal parts, will generally be found printed in italics, to distinguish them. This observation applies only to the remaining part of this Section. 3. What is the sum of two hundred and nine, and sixty-three hundredths, thirty-five and five hundredths, and fifteen hundredths? Ans. 244,83. 4. Add together two thousand five hundred and seventeen, and two hundred and forty-one thousandths, two hundred and-nineteen thousandths, thirty-six thousand, and one thousand and one hundred and one thousandths? Sum, 39517,561. 5. A man bought three tons and nine hundredths of hay at one time; four tons and sixty-seven hundredths, at another; fifteen tons and six hundred and seven thousandths, at another. How much hay did he buy at the three purchases? Ans. 23,367 T. 6. The elevation of a certain mountain above a river near its base, was taken at three observations ; one at the base, which was found to be 417 feet and nine hundred eighteen thousandths, higher than the river; one upon the side, which was 2000 and eighteen thousandths above the base; and one at the summit, which was 3002 and seven tenths higher than the place of the second observation. How much higher than the river was the summit of the mountain? Ans. 5420,636 feet. Note 1.--If no whole numbers are given, the point must be kept at the left hand of decimals, to show that they are decimals; and as many 1's set in the place of units as there are 10's in the column of tenths. 7. What is the sum of ,9+,17+,018+,0037+ ,90708. Ans. 1,99878. 8. What is the sum of seven hundred and six thousandths, two thousand and seventy-three hundred-thousandths and eight hundred and sixty-seventh millionths? Ans. 727597. 9. Add together one million, two thousand, nine hundred and fifty-five ten-millionths, seventeen thousand and ten ten-millionths, thirty-five ten-millionths, and eight hundred and ninety-eight thousandths. Ans. 1. 10. What is the weight of 3 bales of cotton; the 1st weighing 3194 lb., the 2d, 3714 lb; and the 3d, 400 lb. ? Explanation.-Change the given fractions to de cimals. Ans. 1091,5 lb. 11. What is the weight of 4 boxes of raisins weighing as follows: 473 lb., 542 lb., 513 lb., and 553 lb.? Ans. 208,925 lb. 12. How much wood in 4 loads which measure as follows: 17, 1, 8, and 1 cords? Ans. 5,35912+C. FEDERAL MONEY. Until 1786, pounds, shillings, pence and farthings, were the currency of the United States. In 1786 it was enacted by Congress, that the Money of the United States should consist of eagles, dollars, dimes and cents as whole coins,* and that a tenth part of a cent, which is imaginary, should be called a mili. It was also established, that the value of these coins should vary in the same proportion as simple numbers and decimal fractions. The general name of eagles, dollars, dimes, cents, and mills, is FEDERAL MONEY. * Coin is money stamped with a legal impression. |