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much as the first, is = . If į be reduced to a decimal, it produces .3333 .. &c. It is plain, that whenever a single figure repeats, it is so many ninths. Charge .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 it, that is, t=; these must be added together. Å is , and makes zs =. 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. Change .3888 &c. to a common fraction. Change .3744 &c. to a common fraction. Change .46355 &c. to a common fraction. If y's be changed to a decimal, it produces .010101 &c. The decimal .030303 &c. is three times as much, therefore it must be in = 35. The decimal .363636 &c. is thirty-six times as much, therefore it must be =:

If yo be changed to a decimal, it produces .001001001 &c. The decimal .006006 &c. is 6 times as much, therefore it must be g = 5: The fraction .027027 &c. is twenty-seven times as much, and must be 27 = ti. The fraction .354354 &c. is 354 times as much, and must be

=}}. 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 mamy 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, fc. 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, &c. according to the place in which it commences.

Instead of writing the repeating figures over several times,

354 99

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.
Change .42 to a common fraction.
Change .539 to a common fraction.
Change .4745 to a common fraction.
Change .8374 to a common fraction.
Change .47647 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 Is.

If either the multiplicand or the multiplier be divisible by 9, it is evident the product must be so. Multiply 437 by 35. 437

81 times 437 = 35397 85

4 times 432 = 1728

4 times 5: 20 2185 3496

37145

Ans. 37145 85 = 81 + 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 81 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 two 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 be 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 1 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 ib. ? 2. If 2 bushels of corn cost 8s. 6d., what is that per bushel ?

3. If 2 lbs. of raisins cost ls. 10d., what is that per Ib. ?

4. If 3 bushels of potatoes cost 9s. 6d., what is that per bushel ?

5. If 4 gals. of gin cost 12s. 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£. 15s. 8d., what is that per barrel ?

9. If 4 gallons of gin cost 175. 8d., what is that per gallon ?

10. Ir 5 yards of cloth cost 6£. 10s. 5d., what is that per

yard ?

11. If 7 barrels of flour cost 17£. 8s. 7d., what is that per barrel ?

12. If 8 yards of cloth cost 20%. 188. 5., what is that per

yard ?

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13. A man had 4 cwt. 3 qrs. 14 lbs. of tobacco, which he put into 2 boxes, į of it in each; how much did he put in each box ?

14. Divide 13£. 8s. 5d. equally among 5 men.
15. Divide 8 cwt. 3 qrs. 17 lbs. into 3 equal parts.

16. Divide 16 cwt. I gr. 11 lbs. of flour equally among 7 men ; how much will each have ?

17. Divide 3 hhds. 42 gals. 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 g of 45£. 9s. 7d. ?

21. If 18 cwt. of sugar cost 56£. 13s. 8d. what will 53 cwt. cost ?

22. If of a ship is worth 943£. 7s. 8d., what is the whole ship worth?

23. If 84 cows cost 453£. 14ş. 8d., how much is that apiecé ?

24. If 3 cwt. of sugar cost 9£. 15s. 9d., what is that per cwt. ?

25. If 9barrels of flour cost 21£. 3s. 8d., what cost 175 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 43 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 o'clock in the

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afternoon of the same day, and travels at the rate of 62 miles per hour. The distance is 250 miles. Supposing them to travel constantly until they meei, at what time will they meet, and at what distance from each place ?

30. The distance from New-York to Baltimore is 197 miles. Two travellers set out at the same time in order to meet; A from New-York towards Baltimore, and B from Baltimore towards New-York. When they met, which was at the end of 6 days, A had travelled 3 miles a day more than B. How many miles did each travel per day ?

31. If when wheat is 7s. 6d. per bushel, the penny-loaf weighs 9 oz., what ought it to weigh when wheat is 6s. per bushel ?

32. Suppose 650 men are in a garrison, and have provisions sufficient to last them two months ; how many meno must leave the garrison in order to have the provisions last those who remain five months ?

33. If 8 boarders will drink a barrel of cider in 15 days, how long will it last if 4 more boarders come among them ?

34. A ship’s crew of 18 men is supposed to have provision sufficient to last the voyage, if each man is allowed 23 oz. per day, when they pick up a crew of 8 persons. What must then be the daily allowance of each person ?

35. How many yards of flannel that is 14 yard wide will line a cloak, containing 9 yards, that is yard wide ?

36. A garrison of 1800 men have provisions sufficient to last them 12 months ; but at the end of 3 months, the garrison was reinforced by 600 men, and 2 months after that, a second reinforcement of 400 men was sent to the garrison. How long did the provisions last in the whole ?

37. A regiment of soldiers, consisting of 1000, are to be new clothed; each coat to contain 24 yards of cloth 14 yard wide, and to be lined with flannel of yard wide. How many yards of flannel will line them?

38. I borrowed 185 quarters of corn, when the price was 19s. per quarter ; how much must I pay to indemnify the lender when the price is 17s. 4d. ? 39. If 7

en can reap 84 acres of wheat in 12 days, how many men can reap 100 acres in 5 days ?

40. If 7 men can build 36 rods of wall in 3 days, how many roʻ's can 20 men build in 14 days ?

41. Ii 20 bushels of wheat are sufficient for a family of 15

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