it is necessary to explain the method of finding it.
1. Divide 2243 by 84, using the factors 3, 4, and 7.
SOLUTION.-Dividing by 3 we find that
2243 quals 747 threes, and 2 remaining;
dividing by 4 we find 747 threes equals
186 twelves and 3 threes, or 9 remaining;
dividing by 7 we find that 186 twelves

equals 26 eighty-fours and 4 twelves or 48.

10

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.

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 12-258

Rule.-I. Cut off the ciphers at the right of the divizor, 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.

OPERATION.

37)86795(2345

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 the 6; 2 times 3 are 6 and 1 to carry are 7; 7 and 1 are 8; we write the one under th 8, and bringing down 7, the next figure of the dividend, we have 127 fo 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

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.

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 7957838, 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 dif ference 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.

(9+5) means 18 minus the sum of 9 and 5.

Thus, 18

is used for the same

134. The Vinculum, or bar, 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

8. Of (9324 +2461-7275) ÷ 3471-2432+1216 >> (6789-2507+3364)?

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

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?

4. What is the nearest number to 25000 that can be divided by 575 without a remainder?

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

6. Find the value of 29+348÷6+217 x 25+43873 added to 19224+ (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?