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OPERATION.

1. Divide 791 by 24, using the factors 2, 3, and 4. SOLUTION.—Dividing by 2 we find that 791 equals 395 twos and 1 remaining ; di

2)791 viding 395 twos by 3, we find 395 twos equals 131 sixes and 2 twoa, or 4, remaining; di

3)395

1 viding by 4, we find Chat 131 sizes consists 4)131, 2 two8 = 4 of 32 twenty-fours and 3 sixes, or 18, remain

32, 3 sixes =18 ing. Hence the true remainder is 18+4+ 1, which is 23. Hence, to find the correct

True remainder, 23. remainder we have che following

Rule.- Multiply each, remainder by all the divisors pre ceding the one which obtained it, and take the sum of the products and the remainder arising from the first division Divide the following and find the true remainder: 2. 582 by 15.

Rem. 12. 3. 503 by 12.

Rem. 11. 4. 2497 by 20.

Rem. 17. 5. 4507 by 14.

Rem. 13. 6. 3717 by 30. 2, 3, 5.

Rem. 27. 7. 13853 by 105. 3, 5, 7.

Rem. 98. 8. 41837 by 180. 4, 5, 9.

Rem. 77. 9. 47117 by 308.

4, 7, 11.

Rem. 301. 10. 96711 by 310.

2, 5, 31.

Rem. 301. 11. 87831 by 720. 2, 3, 4, 5, 6.

Rem. 711.

OPERATION.

CASE II. 98. When there are ciphers at the right of the divisor. 1. Divide 8254 by 600.

SOLUTION.-6 hundreds are contained in 82 hundreds 13 times, and 400 remaining; 600 is not contained in

6100)82 54 54, hence the entire remainder is 400+54, or 454. From this solution we may derive the following

13 454 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.

NOTES.—1. 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 igures at the left.

2. When the part of the divisor at the left of the naughts is greater than 12, divide by long division. 2. Divide 876 by 50.

Ans. 17; Rem. 26. 3. Divide 953 by 400.

Ans. 2; Rem. 153. 4. Divide 1733 by 500.

Ans. 3; Rem. 233. 5. Divide 2765 by 700.

Ans. 3; Rem. 665. 6. Divide 7859 by 800.

Ans. 9; Rem. 659. 7. Divide 9763 by 900.

Ans. 10; Rem. 763. 8. Divide 14873 by 1900. Ans. 7; Rem. 1573. 9. Divide 25075 by 2300. Ans. 10; Rem. 2075. 10. Divide 187654 by 14700.

Rem. 11254. 11. Divide 269856 by 237000.

Rem. 32856. 12. Divide 5767220 by 4730000. Rem. 1037220.

EXERCISE UPON THE PARENTHESIS. 99. The Parenthesis ( ), denotes that the quantities included are to be subjected to the same operation; thus, (8+6—4) 3 denotes that the value of 8+6–4, which is 10, is to be multiplied by 3.

The vinculum, thus 8+6–4X3, is often used in place of the parenthesis.

1. What is the value of (12+9—7)X5?

SOLUTION.—12+9 equals 21, and 21 minus 7 equals 14, and 14 mul. tiplied by 5 equals 70. Therefore, etc. Required the value 2. Of (25+23–18) x 7.

Ans. 210. 3. Of (46+97-82) x 9.

Ans. 549. 4. Of (98-75+87) x 14.

Ans. 1540. 5. Of (89+96-47)-6.

Ans. 23. 6. Of (145-110+117):-8.

Ans. 19. 7. Of (396-128 +483) x 32.

Ans. 24032. 8. Of (860+980—1120)=45.

Ans. 16. 9. Of (320–98) X (860—145).

Ans, 158730. 10. Of (689-327+986—397) x 428. Ans. 407028. 11. Of (729+487-244)-(247-210+71). Ans. 9. 12. Of (3014-2601)*(2477–1325)-59. Ans. 8064. Note.-In a series of numbers connected with symbols, the sign X denotes the closest connection, the sign • next, thus, 12+8+2-5X2 =6; also, 16--4X2=2, rather than 8.

WRITTEN EXERCISES.

ON THE FOUR FUNDAMENTAL RULES. 1. The minuend is 4160, and the subtrahend is 3425; what is the remainder ?

Ans. 735. 2. The minuend is 9164 and the remainder is 3426 ; what is the subtrahend ?

Ans. 5738. 8. The subtrahend is 3872 and the remainder 4648; what is the minuend?

Ans. 8520. 4. The multiplicand is 745 and the multiplier 456; what is the product?

Ans. 339720. 5. The multiplicand is 2463 and the product 854661; what is the multiplier ?

Ans. 347. 6. The product is 881919 and the multiplier 981; what is the multiplicand ?

Ans. 899. 7. The dividend is 518077 and the divisor 763 ; what is the quotient?

Ans. 679. 8. The dividend is 801222 and the quotient 3257 ; what is the divisor?

Ans. 246. 9. The divisor is 587 and the quotient 8723; what is the dividend ?

Ans. 5120401. 10. The dividend is 72987 and divisor 45; required the quotient and remainder.

Ans. 1621 ; 42. 11. The dividend is 7972, the quotient is 274, and remainder 26; what is the divisor ?

Ans. 29. 12. The divisor is 26, the quotient 372, and remainder 23; what is the dividend ?

Ans. 9695. 13. Thomas read 789 pages of history in a week, which lacks 324 of being as many as Walton read; how many did both read ?

Ans. 1902 pages. 14. A freight car ran 365 miles one week, and 3 times as far, lacking 246 miles, the next week; how far did it run the second week ?

Ans. 849 miles. 15. A sold 8318 bushels of wheat, then bought 2514 bushell, and then had 3146 bushels; how many bushels had be at first?

Ans. 8950 bushels. 16. My barn cost $3156; my house cost 3 times as much as my barn, and my farm cost as much as both ; what was the cost of all ?

Ans. $25,248.

17. The value of 5 horses and 7 mules is $2436; now if the value of each mule is $208, what is the value of each horse ?

Ans. $196. 18. A man left $2535 each to his four children, but one of them dying, the three remaining children divided the money; how much did each receive ?

Ans. $3380. 19. Mr. Smith left $6264 to each of three sons and $7240 to each of two daughters, but one daughter dying, her share was equally divided among the remaining children; what did each receive ? Ans. Son, $8074; daughter, $9050.

20. The income of a man who “struck oil” was $480 a day; how many teachers would this employ at a salary of $438 a year?

Ans. 400. 21. A stock dealer bought 325 cows at $28 each, and sold 124 of them at cost; how much must he receive a head for the remainder to gain $804 ?

Ans. $32. 22. Mr. Galton buys a farm of 110 acres at $75 an acre, $2200 to be paid down and the remainder in five yearly installments; what must be pay each year? Ans. $1210.

23. A farmer raised 765 bushels of oats, of which he kept 65 bushels for seed, and after retaining enough for the use of his borses till next harvest, allowing 60 bushels to each horse, sold the balance at 85 cents a bushel, and received $442; how many horses had he?

Ans. 3 horses. 24. Mr. Milman bequeathed $6500 to each of two sons, to a third son $1000, $5000 to each of 3 daughters, and the balance of his estate, amounting to $25,000, to several benevolent institutions; the will, however, being set aside, the property was divided equally among his children; what was the share of each ?

Ans. $9000. 25. If a soldier enlisting in the late war for 3 years, received a bounty of $930; then served one year as a private, at $13 & month, 6 months as a corporal, at $14 a month, and 18 months as a sergeant at $17 a month ; what was the whole amount of his pay and his average pay per month?

Ans. $41 a month.

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1. The sum of all the parts equals the whole.

2. The whole diminished by one or more parts equals the sum of the other parts.

PRINCIPLES OF SUBTRACTION. 1. The Remainder equals the Minuend minus the Subtrahend. 2. The Minuend equals the Subtrahend plus the Remainder. 3. The Subtrahend equals the Minuend minus the Remainder.

PRINCIPLES OF MULTIPLICATION. 1. The Product equals the Multiplicand into the Multiplier. 2. The Multiplicand equals the Product divided by the Multiplier. 3. The Multiplier equals the Product divided by the Multiplicand.

PRINCIPLES OF DIVISION. 1. The Quotient equals the Dividend divided by the Divisor. 2. The Dividend equals the Divisor multiplied by the Quotient. 3. The Divisor equals the Dividend divided by the Quotient.

4. The Dividend equals the Divisor multiplied by the Quotient plus the Remainder.

5. The Divisor equals the Dividend minus the Remainder, divided by the Quotient.

OTHER PRINCIPLES OF DIVISION. 1. Multiplying the Dividend or dividing the Divisor by any num. ber, multiplies the Quotient by that number.

2. Dividing the Dividend or multiplying the Divisor by any number, divides the Quotient by that number.

3. Multiplying or dividing both Dividend and Divisor by the same number, does not change the Quotient.

4. A Generaļ Law. A change in the Dividend by multiplication or division produces a similar change in the Quotient; but such a change in the Divisor produces an opposite change in the Quotient.

NOTE TO TEACHER.-Let the pupils be required to show the reason for the above principles, and give illustrations of them. No demonstratious are given, since it is better for the pupil to learn to depend somewhat upon himself, that he may becomo, not a mere imitator, but an original thinker.

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