CONTRACTIONS. I. If the divisor be an integer with any number of cyphers at the end; cut them off, and remove the decimal point in the dividend so many places farther to the left, as there were cyphers cut off, prefixing cyphers, if need be; then proceed as before. Here I first divide by 3, and then by 7, because 3 times 7 is 21. 2. Divide 41020 by 32000. Ans. 1'281875. Note. Hence, if the divisor be 1 with cyphers, the quotient will be the same figures with the dividend, having the decimal point so many places farther to the left, as there are cyphers in the divisor. EXAMPLES. 217 3-100=2173. 5'19 by 100000516. 416 by 10-41'9. 21 by 1000= 00021. II. When the number of figures in the divisor is great, the operation may be contracted, and the necessary number of decimal places obtained. RULE. 1. Having, by the 4th general rule, found what place of decimals or integers the first figure of the quotient will possess; consider how many figures of the quotient will serve the present purpose; then take the same number of figures on the left of the divisor, and as many of the dividend figures as will contain them less than ten times; by these find the first figure of the quotient. 2. And for each following figure, divide the last remainder by the divisor, wanting one figure to the right more than before, but observing what must be carried to the first product for such omitted figures, as in the contraction of Multiplication; and continue the operation till the divisor is exhausted. 3. When there are not so many figures in the divisor, as are required to be in the quotient, begin the division with all the figures as usual, and continue it till the number of figures in the divisor and those remaining to be found in the quotient be equal; after which use the contraction. EXAMPLES. i. Divide 2508 928065051 by 92*41035, so as to have four decimals in the quotient.—In this case, the quotient will contain six figures. Hence Common Way. 92*41035)2508*928065051(27 ̊1498 18482070 660721 06 64687 45 1384515 9241 035 4607 5800 3696 4140 911 16605 8369315 79 472901 73 928280 5 544621 2. Divide 721 17562 by 2'257432, so that the quotient may contain three decimals. Ans. 319 467. 3. Divide 12 169825 by 3*14159, so that the quotient may contain five decimals. Ans. 3*87377. 4. Divide 87'076326 by 9365407, and let the quotient contain seven decimals. Ans. 9*2976559. REDUCTION OF DECIMALS. CASE 1. To reduce a vulgar fraction to its equivalent decimal. RULE.* Divide the numerator by the denominator, annexing as many cyphers as are necessary; and the quotient will be the decimal required. * Let the given vulgar fraction, whose decimal expression is required, be. Now since every decimal fraction has 10, 100; EXAMPLES. 1. Reduce to a decimal. 4)5'000000 6)1'250000 *208333, &c. 2. Required the equivalent decimal expressions for 1 and. 3. What is the decimal of? Ans. 25, 5, and 75. Ans. 04. Ans. 015625. Ans. 071577, &c. To reduce numbers of different denominations to their equivalent decimal values. RULE.* 1. Write the given numbers perpendicularly under each other for dividends^ proceeding orderly from the least to the greatest. 2. Opposite to each dividend, on the left, place such a number for a divisor, as will bring it to the next superior name, and draw a line between them. 1000, &c. for its denominator; and, if two fractions be equal, it will be, as the denominator of one is to its numerator, so is the denominator of the other to its numerator; therefore 13:7:: 7X1000.&c. 70000, &c. 1000, &c.: =53846, the numerator of the decimal required; and is the same as by the rule. * The reason of the rule may be explained from the first ex■ ample; thus, three farthings are of a penny, which brought to 2 a decimal is '75; consequently 9|d. may be expressed 9*75d. but 9-75 is 975 of a penny =-^fe of a shilling, which brought to a decimal is '8125; and therefore 15s. 9|d. may be expressed 15-8125S. In like manner 15 ai25s. is V58125 of 3 shillihg= 188888 of a pound, by bringing it to a decimal, 7906251. a» by the rule. 8125 10000 3. Begin with the highest, and write the quotient of each division, as decimal parts, on the right of the dividend next below it; and so on till they are all used, and the last quotient will be the decimal sought. XXANFLSS. 1. Reduce 15s. 9 d. to the decimal of a pound. '790625 the decimal required. 2. Reduce 9s. to the decimal of a pound. 3. Reduce 19s. 5^d. to the decimal of a pound. Ans. 45 Ans. '972916. 4. Reduce 10oz. 18dwt. 16gr. to the decimal of a lb. Troy. Ans. '911111, &c. 5. Reduce 2qrs. 14lb. to the decimal of a cwt. Ans. '625, &c. 6. Reduce 17yd. 1ft. 6in. to the decimal of a mile. Ans. 00994318, &c. 7. Reduce 3qrs. 2nls. to the decimal of a yard. Ans. 875. 8. Reduce lgal. of wine to the decimal of a hhd. Ans. 015873. 9. Reduce 3bu. lpe. to the decimal of a quarter. Ans. 40625. 10. Reduce low. 2d. to the decimal of a year. CASE III. Ans. 1972602, &c. To find the decimal of any number of shillings, pence, and far' thhigs by inspection. RULE.* Write half the greatest even number of shillings for the first deqimal figure, and let the farthings in the given pence * The invention of the rule is as follows; as shillings are so many 20ths of a pound, half of them must be so many lOths, |