shall not notice them. Sufficient accuracy may always be attained without them. I shall show, however, how the true value of them may always be found in common fractions. The fraction reduced to a decima!, is .1111 ... &c. Therefore, if we wish to change this fraction to a common fraction, instead of calling it t, u, or uit, which will be a value too small, whatever number of figures we take, we must call it. This is exact, because it is the fraction which produces the decimal. If we have the fraction .2222 .. &c. it is plain that this is twice as much as the other, and must be called . If be reduced to a decimal, it produces .2222 .. &c. if we have .3333 .. &c. this being three times as much as the first, is = . If be reduced to a decimal, it produces .3333.. &c. It is plain, that whenever a single figure repeats, it is so many ninths. Change .4444 &c. to a common fraction. Ans. . Change .5555 &c. to a common fraction. Change :6666 &c. to a common fraction. Change .7777 &c. to a common fraction, Change .9999 &c. to a common fraction. Change .5333 &c. to a common fraction. This begins to repeat at the second figure or hundredths. The first figure 5 is ; and the remaining part of the fraction is of ho, that is, jó = jo; these must be added together. g is jy, and makes jy = 2. The answer is . If this be changed to a decimal, it will be found to be .5333 &c. If a decimal begins to repeat at the third place, the two first figures will be so many hundredths, and the repeating figure will be so many ninths of another hundredth. Change .4666 &c. to a common fraction. If zo be changed to a decimal, it produces .010101 &c. The decimal. 030303 &c. is three times as much, therefore it must be = : The decimal .363636 &c. is thirty six times as much, therefore it must be If it be changed to a decimal, it produces.001001001 &c. The decimal .006006 &c. is 6 times as much, therefore it must be ŠT. The fraction .027027 &c. is twenty seven times as much, and must be : The fraction .354354 &c. is 354 times as much, and must be 13 = If This principle is true for any number of places. Hence we derive the following rule for changing a circulating decimal to a common fraction: Make the repeating figures the numerator, and the denominator will be as many 9s as there are repeating figures. If they do not begin to repeat at the first place, the preceding figures must be called so many tenths, hundredths, &c. according to their number, then the repeating part must be changed in the above manner, but instead of being the fraction of an unit, it will be the fraction of a tenth, hundredth, fc. according to the place in which it commences. Instead of writing the repeating figures over several times, they are sometimes written with a point over the first and last to show which figures repeat. Thus .333 &c. is written .3. 2525 &c. is written 25. .387387 &c. is written .387. .57346346 &c. is written .57346. Change .24 to a common fraction. Note. To know whether you have found the right answer, change the common fraction, which you have found, to a decimal again. If it produces the same, it is right. Proof of Multiplication and Division by casting out 9s. If either the multiplicand or the multiplier be divisible by 9, it is evident the product must be so. Multiply 437 by 85. 437 81 times 437 = 35397 85 4 times 432 = 1728 4 times 20 2185 3496 37145 5 = Ans. 37145 85 81 to 4 and 437 = 432 + 5. 81 is divisible by 9, and 85 being divided by 9 leaves a remainder 4. 432 is divisible by 9, and 437 leaves a remainder 5. 81 times 437, and 4 times 432, and 4 times 5, added together, are equal to 85 times 437. 81 times 437 is divisible by 9, because 1 is so, and 4 times 432 is divisible by 9, because 432 is so. The only part of the product which is not divisible by 9, is the product of the iwo remainders 4 and 5. This product, 20, divided by 9, leaves a remainder 2. It is plain therefore that if the whole product, 37145, be divided by 9, the remainder must be 2, the same as that of the product of the remainder. Therefore to prove multiplication, divide the divisor and the dividend by 9, and multiply the remainders together, and divide the product by 9, and note the remainder ; then divide the whole product by 9, and if the remainder is the same as the last, the work is right. Instead of dividing by 9, the figures of each number may be added, and their sum divided by 9, as in Art. XXI., (and for the same reason) and the remainders will be the same as if the numbers themselves were divided. In the above example, say 7 and 3 and 4 are 14, which, divided by 9, leaves a remainder 5; then 5 and 8 are 13, which, divided by 9, leaves a remainder 4. Then 4 times 5 are 20, which, divided by 9, leaves a remainder 2. Then adding the figures of the product, 5 and 4 and 1 and 7 and 3 are 20, which being divided by 9 leaves 2, as the other. Instead of dividing 14 and 13 by 9, these figures may be added together, thus 4 and i are 5; 3 and 1 are 4. Since in division the quotient multiplied by the divisor produces the dividend; if the divisor and quotient be divided by 9 and the remainders multiplied together, and this product divided by 9, and the remainder noted; and then the dividend be divided by 9; this last remainder must agree with the other. N. B. If there is a remainder after division, it must be subtracted from the dividend before proving it, Miscellaneous Examples. 1. If 2 lbs. of figs cost 2s. 8d., what is that per lb.? 2. If 2 bushels of corn cost Ss. 6d., what is that per bushel? 3. If 2 lbs. of raisins cost 1s. 10d., what is that per Ib.? 4. If 3 bushels of potatoes cost Is. 6d., what is that per bushel ? 5. If 4 gals. of gin cost 125. 8d., what is that per gal.? 6. If 2 barrels of flour cost 3£. 4s., what is that per barrel ? 7. If 2 gallons of wine cost 1£. 10s. 4d., what is that per gallon ? 8. If 2 barrels of beer cost 1£. 155. 8d. what is that per barrel? gallon? 9. If 4 gallons of gin cost 17$. 8d., what is that per 10. If 5 yards of cloth cost 6£. 10s. 5d., what is that per yard? ii. If 7 barrels of flour cost 17£. 8s. 7d., what is that per barrel! 12. If 8 yards of cloth cost 20£. 18s. 5., what is that per yard? 13. A man had 4 cwt. 3 qrs. 14 lbs. of tobacco, which be put into 2 boxes, of it in each ; how much did he put in each box ! 14. Divide 13£. Gs. 5d. equally among 5 men. 16. Divide 16 cwt. 1 qr. 11 lbs. of flour equally among 7 men; how much will each have ? 17. Divide 3 hhds. 42 galls. 2 qts. into 5 equal parts? 18. If 12 yards, 3 qrs. 2 nls. of cloth will make 7 coats, how much will make 1 coat? How much will make 13 coats ? 19. If 5 yards of cloth cost 19£. 3s. 4d., what cost 17 yards? 20. What is of 45£. 9s. 7d. ? 21. If 18 cwt. of sugar cost 56£. 13s. Sd. what will 537 cwt. cost? 22. If 4 of a ship is worth 943£. 7s. 8d., what is the whole ship worth? 23. If 84 cows cost 453£. 14s. 8d, how much is that apiece? 24. If 31 cwt. of sugar cost 9£. 15s. 9d. what is that per cwt. ? 25. If 97 barrels of flour cost 21£. 3s. 8d., what cost Le 179 barrels ? 26. If a staff 4 feet long cast a shade on level ground 6 ft. 8 in., what is the height of a steeple which casts a shade 173 feet at the same time? 27. If 57 gallons of water in one hour run into a cistern containing 258 gallons, and by another cock 42 gallons run out in an hour, in what time will it be filled ? 28. A and B depart from the same place, and travel the same road; but A starts 6 days before B, and travels at the rate of 28 miles a day; B follows at the rate of 13 miles a day. In how many days will B overtake A ? 29. A sets out from Boston to New-York, at 20 min. past 8 in the morning, and travels at the rate of 5 miles an hour; and B sets out from New-York to Boston at 3 |