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30. What will 369 gallons of molasses cost, at $0.375 a gallon ?
Ans. $138.37,5. 31. What will 97.48cwt. of hay cost, at $1.125 per cwt. ?
Ans. $109.66,5. 32. What will 63.5 bushels of corn be worth, at $ 0.78 per bushel ?
Ans. $ 49.53.
DIVISION OF DECIMALS.
EXAMPLES. OPERATIONS. VULG. FRAC. DECIMALS. 1. Divide & by to 8XL0 = = 18 = 2. Divide 3 by fo.
1x1 = 3. Divide 36 by to 3 X = 168 4. Divide 4 by 4. 18x1 = 48 5. Divide by looo x 100 = = 6. Divide Teo by 1000. Tio X100e=90.00 = 1 = 7. Divide 1.728 by 1.2. 8. Divide 1728 by 14. OPERATION BY DECIMALS. OPERATION BY VULGAR FRACTIONS. 1.2)1.728(1.44 Ans. 1768x12=12888=1228=199=14 12
Rule, — Divide as in whole numbers, and point off as many decimals in the quotient as the number of decimals in the dividend exceed those of the divisor; but if the number of those in the divisor exceed that of the dividend, supply the defect by annexing ciphers to the dividend. And if the number of decimals in the quotient and divisor together are not equal to the number in the dividend, supply the defect by prefixing ciphers to the quotient. 9. Divide 780.516 by 2. 43.
Ans. 321.2. 10. Divide 7.25406 by 9.57.
Ans. .758. 11. Divide .21318 by .38.
Ans. .561. 12. Divide 7.2091365 by .5201.
Ans. 13.861+. 13. Divide 56.8554756 by .0759.
14. Divide 30614.4 by .9567.
Ans. 32000. 15. Divide .306144 by 9567.
Ans. .000032. 16. Divide four thousand three hundred twenty-two and four thousand five hundred seventy-three ten thousandths by eight thousand and nine thousandths.
Ans. .5403+.. 17. Divide thirty-six and six thousand nine hundred fortyseven ten thousandths by five hundred and eighty-nine.
Ans. .0623. 18. Divide three hundred twenty-three thousand seven hundred sixty-five by five millionths. Ans. 64753000000.
19. Divide 119109094.835 by 38123.45. Ans. 3124.3. 20. Divide 1191090.94835 by 3812345. 21. Divide 11910909483.5 by 38.12345. Ans. 22. Divide 11.9109094835 by 381234.5. Ans. 23. Divide 1191.09094835 by 3.812345. 24. Divide 11910909483,5 by .3812345. Ans. 25. Divide 1.19109094835 by 3.812345. 26. Divide .119109094835 by .3812345. Ans.
REDUCTION OF DECIMALS.
CASE I. To reduce a vulgar fraction to its decimal. 1. Reduce to its decimal. OPERATION. That the decimal .375 is equal to may be 8)3.000 shown by writing it in a vulgar fraction and re
3975 Ans. ducing it; thus, 100% = 20 = = Ans.
RULE. — Divide the numerator by the denominator, annexing one or more ciphers to the numerator, and the quotient will be the decimal required.
NOTE. — It is not usually necessary that decimals should be carried to more than six places. 2. Reduce to a decimal.
Ans. .625. 3. Reduce je to a decimal.
Ans. .5. 4. Reduce , , , 11, 16, 25, ana š to decimals.
Ans. .666+, .75, .833+, .91666+, .1875, .04, .125.
CASE II. To reduce denominate numbers to decimals. 1. Reduce 15. 9fd. to the decimal of a £. Ans. .790625. OPERATION.
The 3 farthings are of a penny, and 4 3.00
these reduced to a decimal are .75 of a 12 9.75000 penny, which we annex to the pence, and 2015.81250 proceed in the same manner with the other .790625 Ans.
terms. RULE. — Write the given numbers perpendicularly under each other for dividends, proceeding orderly from the least to the greatest ; opposite to each dividend, on the left hand, place such a number for a divisor, as will bring it to the next superior denomination, and draw a line between them. Begin at 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 divided; and the last quotient will be the decimal required.
CASE III. To find the decimal of any number of shillings, pence, and farthings, by inspection.
RULE. — Write half the greatest even number of shillings for the first decimal figure, and if the number of shillings be odd, annex to the decimal the figure 5. Then write underneath the number of farthings contained in the given pence and farthings, setting the left-hand figure in the second place, if there be more than one figure, and the single figure in the third place, if there be but one, and increasing the number by 1 when it exceeds 12, and by 2 when it exceeds 36. The sum of the whole will be the decimal required.
.7 = 1 of 14s.
2 = for excess of 36.
.791 2. Find the value of 13s. 6 d. by inspection. Ans. .678. 3. Find the value of 19s. 8 d. by inspection. Ans. .984.
4. Value the following sums by inspection, and find their total : 19s. 11 d., 16s. 9 d., ls. 11d., 3s. 0 d., 17s. 5 d., 13s. 41d., 18s. 84d., 19s. 1178., 13s. 34d., 16s. Ojd., 17. 7 d.
Ans. 7.91£. Note. — As shillings are so many twentieths of a pound, it is evident, that by taking one half of their number, we obtain their value in tenths or decimals of a pound. Thus, 16s.=£.
In like manner, any number of farthings are so many nine hundred sixtieths of a pound. So that, in order to obtain their value in the denomination of pounds, we write the number of farthings for the numerator and 960 for a denominator, as 17 farthings = 7£. But, in order to treat this fraction decimally, we must raise the denominator to 1000, which in the fraction o£. is done by adding 40 to the denominator and 1 to the numerator, and in the fraction 990 by adding 2 to the numerator and 40 to the denominator. Q. E. D.
CASE IV. To find the value of a decimal in integral or whole numbers.
1. What is the value of .790625£. ? OPERATION. .790625 Now it is evident, that .790625£. expressed
20 in terms of a shilling must be the product of 15.812500
.790625£. multiplied by 20, and that to con
tinue the reduction to the lowest terms we must 9.750000
multiply by the same number as in common reduction.
RULE. — Multiply the given decimal by the number which will bring it to the next lower denomination, and cut off for a remainder as many places on the right as there are places in the given decimal.
Multiply this remainder by the number which will bring it to the next lower denomination, cutting off for a remainder as before, and thus proceed till the reduction is carried to the denomination required. The several integral numbers, standing at the left hand, will be the answer sought in the different lower denominations.
2. What is the value of .625 of a shilling? Ans. 77d. 3. What is the value of .6725 of a cwt. ?
Ans. 2qr. 191b. 5 oz. 4. What is the value of .9375 of a yard? Ans. 3qr. 3na. 5. What is the value of .7895 of a mile ?
Ans. 6fur. 12rd. 10ft. 64in. 6. What is the value of .9378 of an acre ?
Ans. 3R. 30p. 13ft. 9 in. 7. Reduce .5615 of a hogshead of wine to its value in gallons, &c.
Ans. 35gal. Iqt. Opt. 313 gi. 8. Reduce .367 of a year to its value in days, &c.
Ans. 134da. 1h. 7m. 19 sec. 9. What is the value of .6923828125 of a cwt. ?
Ans. 2qr. 21lb. 8oz. 12dr. 10. What is the value of .015625 of a bushel ?
Ans. 1 pint. 11. What is the value of .55 of an ell English ?
Ans. 2qr. 3na: 12. What is the value of .6 of an acre ? Ans. 2Ř. 16p.