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OPERATION.

112345678 Multiplicand.
288144486 Multiplier.

674074068 = Product by 6 units.

5392592544 = product X 8 tens for product by 48 tern.

l £177777 632 ^ Pro<*uct * * tbouiaiuU for product by 144

32355555^64 = M Product x2 »ilHoai for product by 288

3 237 1 787 64 1 63 1 508 = Product by 288144486.

3. Multiply 61370913 by 96488. Ans. 5921556653544.

4. Multiply 8649347864 by 1325769612.

Ans. 11467042561708308768.

94. When the multiplier is any number of nines.

Ex. 1. Multiply 87654 by 999. Ans. 87566346.

Operation. By annexing three ci

87654000 = 87654X1000 phers to the multiplicand

8765 4 = 87654X 1 we teke i4 1000 times, or

„ _ „ , 1 time more than is re

875 6 6 346 = 87654X 999 quired by the given mul

tiplier. We therefore from this result subtract the multiplicand taken once, and thus obtain the product by 999. In like manner we may multiply by any number of nines. Hence the

Rule. Annex as many ciphers to the multiplicand as there are nines in the multiplier, and from the number thus produced subtract the given multiplicand.

Note. — To multiply by any number of threes, find the product for the same number of nines, by the rule, and take one third of it by dividing by 3; and to multiply by any number of sixes, take twice the product of the same number of tlirees, by multiplying it by 2.

2. Multiply 7777777 by 9999. Ans. 77769992223.

3. Multiply 416231 by 99999.

4. Multiply 987654 by 333333. Ans. 329217670782.

5. Multiply 876543 by 66666. Ans. 58435615638.

6. Multiply 999999 by 9999. Ans. 9998990001.

7. Multiply 32567895 by 3333.

8. Multiply 66666 by 66666. Ans. 4444355556.

9. Multiply 912345678 by 99. Ans. 90322222122.

10. Multiply 1234567 by 9999. Ans. 12344435433.

11. Multiply 98123452 by 999999. Ans. 98123353876548.

CONTRACTIONS IN DIVISION.

95i When the divisor is a convenient part of a number of tens, hundreds, or thousands.

Ex. 1. Divide 19641075 by 25. Ans. 785643.

Operation. By multiplying both divisor and

19641075 dividend by 4, which does not change

4 the relation of the one to the other

. (Art. 83), the divisor becomes 100, 785o43i00 Quotient. enables us to perform the di

vision by simply cutting off two figures at the right of the dividend (Art. 79). In like manner, if the divisor had been 3J, by taking both it and the dividend 3 times as large, we could have performed the division by simply cutting off one figure at the right of the dividend; or, if the divisor had been 125, by taking both it and the dividend 8 times as large, we could have performed the division by simply cutting off three figures at the right of the dividend. Hence the

Rule. Multiply both divisor and dividend by that number which will change the divisor to a number of tens, hundreds, or thousands, and then divide.

Examples.

2. Divide 89630 by 3$. Ans. 26889.

3. Divide 123450 by 16§.

4. Divide 18621 by 12J. Ans. 1489-&V

5. Divide 317121 by 2£. Ans. 126848TV

6. Divide 876735 by 33f Ans. 26302T^.

7. Divide 123456 by 125.

8. Divide 61678500 by 250. Ans. 246714.

9. J. Gushing bought a number of horses, at 166§ dollars each, for $ 9500 ; required the number bought.

Ans. 57 horses.

10. A company has received 12000 dollars from the sale of piano-fortes, at an average price of 333£ dollars each; required the number sold.

11. How many shares of the Illinois Central Railroad, at 125 dollars each, can be bought for 150000 dollars?

Ans. 1200 shares.

12. How many cows, at 33£ dollars each, may be purchased for 383£ dollars.

13. How many books, at 2£ dollars each, may be purchased for 120 dollars? " Ans. 48 books.

14. A certain magazine contains 616350 pounds of powder, in kegs of 25 pounds each; required the number of kegs.

Ans. 24654 kegs.

96.° When the divisor is any number of nines.

Ex. 1. Divide 316234 by 99. Ans. 3194f f.

Operation. "We first divide by 100, or 99 -f- 1, by cut

3 1 6 2 3 4 ting off two figures at the right of the dividend,

3 19 6 obtaining for the first partial quotient 3162

12 7 and a remainder 34. Since the divisor used

2 g was 1 larger than the given divisor, 99, the

quotient obtained denotes that an excess of

3 1 9 4 f| Ans. 3162, or a number equal to the quotient itself, must be added to the 34 for the true remainder, 34 -\- 3162 = 3196, which exceeding the divisor 99, we write it for a second dividend; and dividing by 100, or 99 -J- 1, as before, we obtain for the second partial quotient 31, and a remainder 96. To the remainder 96 we add 31, the excess denoted by the last quotient, and obtain 127 for a third dividend, which being divided by 100, or

99 -\- 1 gives the third partial quotient 1, and a remainder 27. To the remainder 27 adding 1, the excess denoted by the last quotient, we have for the true remainder 28, which is JJ. The sum of the partial quotients with the final remainder annexed gives 3194f|, the quotient required.

The above process is based'upon the principle that 10 = 9 -f- 1;

100 = 99 -{- 1; 1000 = 999 1, etc.; consequently 20 = (2 X 9) -f- 2, and 37 = (3 X 9) + 3 -f- 7; 200 = (2 X 99) -f- 2; 3859 = (38 X 99) + 38 + 59; 15987 = (15 X 999) + 15 + 987, etc. Hence, 316234 = 316200 -f- 34 = (31C2 X 99) + 3162 -f- 34; 3162 -f- 34 = 3196 = 3100 -f- 96 = (31 X 99) -f- 81 + 96; 31 -f 96 = 127 = 100 + 27 = (1 X 99) -+- 1 + 27; 1 + 27 = 28 = ff; and 3162 -f- 31 -f- 1 -f- |f = 3194||, the answer, as before obtained. By like process, and upon the same principle, may the quotient be (bund for any number of nines.

Rule. Add 1 to the given divisor, and, by it thus increased, divide by cutting off figures at the right of the dividend. To the figures cut. off on the right add those on the left for a true remainder, of which, if it equal or exceed the given divisor, make a second dividend, and divide as before. Proceed thus till there shall be no remainder as large as the given divisor; and the sum of the several quotients, with the last remainder, if any, will be the answer required.

Note. — When the last remainder is the same as the given divisor, it must be cancelled, and 1 written as a partial quotient.

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Ans. 12345.

4. Divide 12332655 by 999.

5. Divide 987551235 by 9999.

6. Divide 9123456779876543211 by 999999999.

Ans. 9123456789.

97.° Abbreviated metbod of long division. Ex. 1. Divide 34634 by 134.

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Examples.

2. Divide 39006 by 44. Ans. 886ff.

3. A certain orchard contains 1088 trees in 34 rows; how many trees in each row? Ans. 32 trees.

4. A speculator sold 191 mules at a gain of 5157 dollars; what was the gain on each?

5. The population of Massachusetts, in 1855, was 1,133,123; how many would that be to each of its 7750 square miles of surface? Ans. 146fffg.

98i When the divisor or dividend has a fraction annexed. Ex. 1. Divide 3692 by 8§. Ans. 426.

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OPERATION.

13 6 9 2 We bring the divisor and divi

3 dend to the same fractional parts

o c \ i i A n a / A o a A 88 those denoted by the given 2 b ) 1 1 0 7 b ( 4 I b Ans. fraction, which are thirds, by mul

10 4 tiplying them both by 3, the num

jTZ ber of thirds in 1. We thus change

„ the divisor to 26 thirds, and the

0 dividend to 3692 thirds, without

1 5 6 changing the quotient (Art. 83); 1 5 6 and dividing obtain 426, the an- swer required.

Rule. Reduce the divisor and dividend to the same fractional parts as are denoted by the given fraction, and then divide as in whole numbers.

Note. — In case there should be a remainder after the division, its true value may be found as in Art. 76.

Examples.

2. Divide 13120 by 9f Ans. 1418£f

3. Divide 76672? by 36. Ans. 2129££g.

4. Divide 20901- by 205. Ans. lO^ff

5. If 16J feet make a rod, how many rods are there in 10626 feet?

6. If 69^-miles make a degree, how many degrees in 12450 miles? Ans. 180 degrees.

7. How many barrels of pork, at 17f dollars a barrel, may be bought for 5591£ dollars?

8. A merchant has sold 2667 yards of silk in dress patterns of 10£ yards each; required the number of patterns.

Ans. 254 patterns.

9. How many plats, each of 272£ square feet, are equal in extent to 136125 square feet? Ans. 500 plats.

10. How many days will 119 pounds of bread last a man, if "he consume 2? pounds per day? Ans. 49 days.

11. How many casks of 31^ gallons each will be required to hold 12968£ gallons of cider? Ans. 415 casks.

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