Rule.-Multiply each remainder by all the divisors preceding the one which obtained it, and take the sum of the products and the remainder arising from the first division. Divide and find the true remainder. 2. 13225 by 105 (3, 5, 7).

Ans. 125. Rem. 100.

130. When there are ciphers at the right of the divisor.

1. Divide 9856 by 800.

SOLUTION.-8 hundreds are contained in 98 hundreds 12 times with a remainder of 200; 800 is not contained in 56, hence the entire remainder is 200+56, or 256.

OPERATION.

8100)98156 12-256

Rule.-I. Cut off the ciphers at the right of the divisor, and as many terms at the right of the dividend.

II. Divide the remaining part of the dividend by the remaining part of the divisor.

III. Prefix the remainder to the part of the dividend cut off, and the result will be the true remainder.

NOTE.-When the divisor is a unit of any order with ciphers, the remainder will be the figures cut off at the right, and the quotient the figures at the left.

131. When the remainders are obtained without writing the products and subtracting.

1. Divide 86795 by 37.

OPERATION.

37)86795(2345

127

169

215

30 Rem.

SOLUTION.-We divide 86 by 37 and find a quotient of 2; we then multiply 37 by 2, but instead of writing the product and subtracting it from the partial dividend, we observe what numbers must be added to the product to give the terms of the partial dividend, and write them for the remainder, thus: 37 is contained in 86, 2 times; 2 times 7 are 14 and 2 are 16; we write the 2 under the6; 2 times 3 are 6 and 1 to carry are 7; 7 and 1 are 8; we write the one under the 8, and bringing down 7, the next figure of the dividend, we have 127 for the next dividend; 37 is contained in 127, 3 times; 3 times 7 are 21, and 6 are 27; hence we write the 6 under the 7; 3 times 3 are 9, and 2 to carry are 11, which increased by 1 make 12; we write the 1 under the 2, and bringing down, we have 169 for the next dividend, etc.

Rule.-I. Obtain the quotient figures in the usual man

ner.

II. Obtain the remainders by observing what number must be added to each partial product to obtain the terms of the partial dividend.

III. Bring down the terms of the dividend in the usual manner, and thus proceed until the division is complete.

132. When the divisor is a little less than 100, 1000, etc.

1. Divide 7639521 by 96.

OPERATION.

76395 21
3056 01
122 25
5 13

33

SOLUTION.-Dividing 7639521 by 100 (or 96+4) by cutting off two figures at the right of the dividend, we obtain for the first partial quotient 76395, and a remainder 21. Since the divisor used is 4 more than the real divisor, the remainder is too small by 4 times 76395. Adding 4 times 76395, or 305580, to the remainder, we find it to be 305580+21 305601, which contains the divisor. Dividing again, we have a quotient 3056 and a remainder 1. Adding to this remainder 4 times 3056, or 12224, we have a remainder 12225, which still contains the divisor. Dividing again, we have a quotient 122 and remainder 25, to which remainder adding 4 times 122 or 488, we obtain a

=

79578

fourth remainder 513, which being again divided and increased by 4 times 5, gives the true remainder 33. Adding the several partial quotients, and annexing the remainder, we have 7957833, the quotient required. Hence the following

Rule.-I. Cut off from the right of the dividend by a vertical line as many terms as there are in the divisor, multiply the part on the left of the line by the difference between the divisor and 100, 1000, etc., and add the product to the number on the right for a true remainder, of which we make a new dividend.

II. Divide as before, multiply the new quotient by the difference between the divisor and 100, 1000, etc., add the product to the remainder for a true remainder, and thus proceed until the remainder is less than the given divisor ; the sum of the several quotients with the last remainder, if any, will be the quotient required.

THE PARENTHESIS AND VINCULUM.

133. The Parenthesis, (), denotes that the quantities included are subjected to the same operation. Thus, 18(9+5) means 18 minus the sum of 9 and 5.

134. The Vinculum, or bar, —, is used for the same purpose as the parenthesis, the numbers under it being considered as one quantity. Thus 12-9-3 means that the difference of 9 and 3 is to be subtracted from 12.

1. What is the value of (572-14)-376-35?

SOLUTION.—572—14 equals 558; 376-35 equals 341, and 558 — 341 equals 217. Therefore, etc.

2. Of (84793-45832)-(76345-46247)?

3. Of (534-46)-7640-6989+472-12?

Ans. 8863.

Ans. 297.

4. Of (7000700-2999299)-40040-37737+572?

Ans. 3999670.

5. Of 8796-2437+210x (8761-5672+6912)?

Ans. 65696569.

6. Of (656+397)÷(247.—166)+25×670 ?

Ans. 16763.

7. Of 7945-5340 × (549+751)÷(5789-5529)?

Ans. 13025.

8. Of (9324+2461-7275)÷3471-2432+1216 × (67892507+3364)? Ans. 15292.

PRACTICAL EXAMPLES.

1. The product of two numbers is 415638, and one of them is 7697; what is the other?

Ans. 54.

2. The product of three numbers is 2237984, and two of them are 103 and 97; what is the third?

3. The dividend is 274500, the quotient remainder 243; what is the divisor?

Ans. 224. 983, and the

Ans. 279.

4. What is the nearest number to 25000 that can be

divided by 575 without a remainder?

Ans. 24725.

5. What is the nearest number to 37401 that can be divided by 784 without a remainder?

Ans. 37632.

6. Find the value of 29+348÷6+217 × 25—438 ÷ 73 added to 192÷24+(225-102) × 26. Ans. 8724.

7. A man paid a debt of $105.45 with an equal number of dollars, dimes and cents; how many were there of each kind?

Ans. 95.

8. Find the value of (9097+6956-2364)-(8765-2721 +2917)÷6432+5832× 99—(3278-118503÷297—2790) } Ans. 24.

(12965-5273+7391-8771+3349).

9. The product of three numbers is 196790480, the smallest is 365, and the product of this and the largest is 396755; required the other two factors. Ans. 496; 1087.

10. A New Jersey farmer, wishing to go West, sold his farm of 150 acres at $84 an acre, and bought prairie land in Illinois for $45 an acre; how many acres did his new farm contain? Ans. 280 acres.

. 11. A farmer sold an equal number of ducks and turkeys; for the ducks he received $2 each, and for the turkeys $3.50

each; and the whole amount received was $44; how many of each did he sell? Ans. 8.

12. The first edition, 2800 copies, of a book of 480 pages cost me $1543; what did I pay a page for stereotyping, if the press work cost me $125, the paper about 12 cents a copy, and the binding 15 cents a copy? Ans. $1.35.

13. A cistern containing 13500 gal. is filled by two pipes, one discharging 250 gal. an hour and the other 300 gal., but, by a leak in one of the pipes, 100 gal. are lost in an hour; how long will it take to fill the cistern? Ans. 30 h.

14. Prove and illustrate that the sum or difference of two numbers, divided by any number, will equal the sum or difference of the quotients found by dividing those two numbers by the same number.

15. The day before Christmas, a butcher sold an equal number of ducks and turkeys, and three times as many chickens; he received for the chickens $1.75, for the ducks $2.25, and for the turkeys $4 each, and the whole amount was $92; what was the number of each? Ans. 24; 8; 8.

16. The keeper of a restaurant, counting the currency received from one day's sales, found it to amount to $31.50, one-ninth being in fifty-cent notes, and the rest made up of an equal number of twenty-five-cent and ten-cent notes; how many were there of each? Ans. 7; 80; 80.

17. A farmer's wife took to the store 6 lb. of butter at 45 cents a pound, 4 doz. eggs at 25 cents a dozen, and 2 pair of spring chickens at $1.25 a pair; she received in exchange groceries amounting to $1.75, a pair of scissors at 50 cents, needles and thread at 20 cents, and delaine at 25 cents a yard; how many yards of delaine did she receive? Ans. 15.

18. James Green purchased Norristown Railroad stock to the amount of $5814, and sold part of it for $2756 at $53 a share, losing $4 on a share; but some years after, the road being leased by the Reading Railroad, he sold out at a gain on the whole transaction of $1492; for what did he sell a share?

Ans. $91.