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SOLUTION BY ADDING TEN.—5 units from 7 units leave 2 units; we cannot take 6 tens from 4 tens, we will therefore add 10 tens to the 4 tens, making 14 tens; 6 tens from 14 tens, leave 8 tens: now since we have added 10 tens, or 1 hundred, to the minuend, our remainder will be 1 hundred too large, hence we must add 1 hundred to the subtrahend; 1 hundred and 3 hundreds are 4 hundreds; 4 hundreds from 6 hundreds leave 2 hundreds.

Rule.—I. Write the subtrahend under the minuend, placing terms of the same order in the same column, and draw a line beneath.

II. Begin at units, and subtract each term of the subtrahend from the corresponding term of the minuend, writing the remainder beneath.

III. If any term of the subtrahend is greater than the corresponding term of the minuend, add 10 to the latter and then subtract.

IV. Add 1 to the next term of the subtrahend (or subtract 1 from the next term of the minuend), and proceed as before.

Proof. Add the difference to the subtrahend, and if the work is correct the sum will equal the minuend.

SECOND METHOD.-Subtract the difference from the minuénd, and, if the work is correct, the result will equal the subtrabend. NOTE.—The method of proof by excess of 9’s will be explained hereafter. 2. From 987435 take 369476.

Ans. 617959. 3. From 70432065 take 64545406. Ans. 5886659. 4. From 80035007 take 7640094. Ans. 72394913. 5. From 95930—53428 take 5036. Ans. 37466. 6. From 55159 take 75138—61859. Ans. 41880. 7. From 604035—470647 take 90009—78087.

Ans. 121466. 8. From 888888—99999 take 810000—188888.

Ans. 167777. In the four following problems subtract the second number from the first, then from the remainder, etc., till the last remainder is less than the second number. 9. 610000 and 155555.

Ans. 143335. 10. 1000008 and 245679.

Ans. 17292.

11. 98765432 and 23456789.

Ans. 4938276. 12. 9834064 and 1277045.

Ans. 894749. PRACTICAL PROBLEMS. 1. If a farm was bought for $6770, and sold for $8025, what was the gain ?

OPERATION. SOLUTION.-If a farm was bought for $6770, and

$8025 sold for $8025, the gain was the difference between

6770 $6770 and $8025, which is $1255.

$1255 Ans. 2. Having $7570 in bank, I added enough to make my deposit $12000; how much did I add ? Ans. $4430.

3. A speculator lost $5675, then gained $4325, and then had $5000; how much had he at first ? Ans. $6350.

4. I received an invoice of goods amounting to $1876.25, and immediately forwarded a draft for $987.75 on account; what balance do I owe?

Ans. $888.50. 5. I invested $7680 in California wheat, giving my note for $2875, and paying the balance in cash ; how much cash did I pay ?

Ans. $4805. 6. A merchant owed a jobber $4653, and in payment gave him two checks for $1250 each ; what balance was due the jobber?

Ans. $2153. 7. A dealer in stocks gained $25,500 one year, and the following year lost $19,750, and had $50,000 remaining; what was the original capital ?

Ans. $44,250. 8. Plato was born 430 B. C., and Socrates 470 B. C.; the former died 347 B. C., the latter 400 B. C.; wbat was the difference in their ages?

Ans. 13 years. 9. What is the difference, in Roman characters, between M and LXXVDCCIII ? Ans. DCCCCXXIVCCXCVII.

10. What is the difference between 1 duodecillion and 1 quadrillion, and how many figures will express it ?

Ans. 39 figures. 11. What is the difference between 28 decillion 143 quadrillion 705 billion 96 million 6 thousand and 5, expressed in the English and French methods !

Ans. A number of 62 figures.

12. If a coal merchant's gains increase $5426 annually for six successive years, and the last year his gains amount to $39,765, what were the first year's gains? Ans. $7209.

13. An assignee found the assets of an estate supposed to be worth $20,000, $3575 more than the liabilities, which were $5860 less than the supposed value of the estate; what were the assets?

Ans. $17,715. 14. What is the difference between 5 septillion 10 sextillion 3 quintillion, expressed in the English and French methods, the answer to be written in the English method ?

Ans. 5 septillion 10 sextillion 2 quintillion 999994 quadrillion 989997 trillion.

15. A gentleman having an estate of $253,760, bequeathed $20,200 to each of his two sons, $24,000 to his daughter, to his widow as much as to all the children, and the remainder of his estate to a college; what did the widow and college receive ? Ans. Widow, $64,400; College, $124,960.

16. A man bought a horse for $200, but seeing another he liked better traded him off, giving $25 to boot; receiving a good offer for this one, be traded again, receiving $75; he traded again, paying $60, and finally sold the last one for $175 ; what did he lose by his bargains ? Ans. $35.

17. Find the balance of the following ledger account: DR. JONES & MARSTON.

CR.

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CONTRACTIONS IN SUBTRACTION. 99. Contractions in Subtraction are abbreviated methods in subtracting.

CASE I. 100. To subtract a number by subtracting from 10 and then adding. 1. Subtract 25368 from 63524.

OPERATION. SOLUTION.—8 from 10 leaves 2, and 4 are 6; 6 and

63524 1 are 7, 7 from 10 leaves 3, and 2 are 5; 3 and 1 are

25368 4, 4 from 5 leaves 1; 5 from 10 leaves 5, and 3 are 8; 2 and 1 are 3, 3 from 6 leaves 3.

38156 Rule.- When a term of the subtrahend is greater than the corresponding term of the minuend, subtract the former from 10, and add the remainder to the latter; when not greater, subtract by the ordinary rule.

NOTES.–1. We may apply this rule to all the terms by omitting to carry when the sum is greater than 9.

2. The reason for this method will be readily seen. The pupil may be required to explain it.

2. Subtract 54062 from 82547; 68975 from 97460; 79240 from 107725 ; 34651 from 63136.

Ans. 28485. 3. Subtract 346978 from 543217; 556324 from 752563 ; 789313 from 985552; 153454 from 349693. Ans. 196239.

4. Subtract 54321 from 93213; 679543 from 718435; 439765 from 478657; 974512 from 1013404. Ans. 38892.

CASE II. 101. To subtract two or more numbers at a single operation.

1. Subtract 546 and 365 from 1364.

OPERATION.

SOLUTION.—5 and 6 are 11, and since 3 more will

1364 make 14, we write 3 for the first term of the remainder; 6 and 1 are 7 and 4 are 11, and since 5 more

546 makes 16, we write 5 in the remainder; 3 and 1 are 4

365 and 5 are 9, and since 4 more makes 13, we write 4 in the remainder.

Rule.- Write the several subtrahends under the minuend, add each column of the subtrahends, and write for the re

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(3)

Take

123786

mainder a term which, added to this sum, will give a number having for its unit term the corresponding term of the minuend. NOTE.—Observe the same rule in carrying as in addition. (2)

(4) From 7845 From 12978 From 856346 From 1000000 (1472 ( 7934 432157

5 13655 Take 3456 Take 2561

154337 Take 314133 (864 1278

142212

( 79584 6. A speculator bought a lot of land for $413, a second lot for $420, a third for $519, and a fourth for $607; be sold the whole for $2200; what was his gain? Ans. $241.

7. A man having an estate of $60,523, left $19,500 to his son, and $19,500 to his daughter; several small legacies and the expenses amounted to $1519; and what remained was to be the share of the widow; how much did the widow receive?

Ans. $20,004. 8. The area of the land surface of the globe is said to be 48,998,388 square miles; the area of Asia is 15,086,000 square miles, of Africa 10,936,000, of North America 8,160,000, of South America 6,552,000, of Europe 3,764,388; what is the area of Oceanica ?

Ans. 4,500,000. PRACTICAL PROBLEMS. 1. I received a bill of goods amounting to $1575.75, and remitted two drafts of $578.25 and $692.85 respectively; what is the balance ?

Ans. $304.65. 2. A man had in bank $15,000, deposited $3875, drew out $8725, and then put in enough to make his deposits $20,000; what was his last deposit ?

Ans. $9850. 3. A speculator lost $18,000 in oil, then gained $15,750 on grain, and then lost $19,250 in "Erie,” which exhausted his capital; what was his capital?

Ans. $21,500. 4. A speculator gained $1050 one year, the next year he lost as much again as he had gained, and the next year

gained as much again as he had lost, and then had $50,000 : · with what capital did he begin ?

Ans. $46,850.

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