7. A gentleman dying left his eldest son $7286, the next younger son $9111, and the youngest as much as both the others; what was the fortune? Ans. $32794.

8. Bought calico for a dress for $2, a neck-tie for $0.45, a parasol for $3.50, a pair of boots for $4.25, and a pair of gloves for $1.75; what was the whole cost? Ans. $11.95.

9. A speculator bought three city lots for $5750 each; for what must he sell them that he may gain $2367 by the investment? Ans. $19617. 10. Mr. Bowman paid $628 for flour, $382 for sugar, and $125 for potatoes; how much did he receive for all if he sold them at a gain of $324? Ans. $1459.

11. A boy bought an arithmetic for 95 cents, a grammar for 65 cents, a history for 75 cents, a geography for $1.88, a piece of India-rubber for 5 cents, and a pencil for 3 cents; what was the whole cost? Ans. $4.31.

12. A merchant pays his book-keeper $1500 a year, three salesmen $750 each, a porter $450, and a boy $200; what amount of salaries does he pay? Ans. $4400. 13. A young man paid $450 for a horse, $115 for a sleigh, $137.50 for a harness and bells, and $33.35 for a robe; what was the cost of the whole outfit? Ans. $735.85.

14. At an auction a woman bid off a marble-topped bureau at $19.50, a dozen cane-seat chairs at $15.00, a carpet at $31, a small rug at $1.50, and a dozen china dinner plates at $1.75; what was her bill? Ans. $68.75.

15. A schooner sailed from Milwaukee to Erie, having on board 12391 bushels new No. 3 Milwaukee wheat, 11420 bushels Mixed Spring Minnesota wheat, and 1239 bushels common Winter Red Western wheat; of how many bushels did her cargo consist? Ans. 25050 bushels.

16. A dry goods merchant, in purchasing his stock, paid $650 for silks, $875 for woolens, $345.75 for cotton dress goods. and $795.45 for other goods; what did his stock cost? Ans. $2666.20. 17. Mr. Anthony's balance in bank on Monday morning was $1546.75; that day he deposited $500; Tuesday he

deposited $350; Wednesday, $764.75; Thursday, $250; Friday, $640; Saturday, $54; how much did he have on deposit on the next Monday morning? Ans. $4105.50.

18. A builder bought a lot for $450, built upon it a house costing $3545, and a barn and carriage-house costing $847.50, fenced it at a cost of $127.50, and graded it at a cost of $77.25; for what must he sell the property in order to gain upon it $540? Ans. $5587.25.

19. A Normal school paid, in 1875, $18,622.17 for salaries of professors and teachers, $1350 for salaries of other officers, $34,721.03 for board, washing, etc., $11,806.95 for servants" hire, and $7159.15 for other expenses; what were the expenditures for the year? Ans. $73,659.30

20. In erecting an academy, the trustees paid $250 for digging the cellar, $580 for laying the foundation walls. $12,575 for the brickwork, and $10,650 for the woodwork, plastering, etc.; what was the entire cost? Ans. $24,055.

21. Mr. Johnson's house cost for brick $450; for lumber, $780; other materials, $350; digging the cellar, $87.50; masons' and bricklayers' work, $425; carpenters' work, $789.75; painting, glazing, etc., $350; what was the whole cost of the house? Ans $3232.25.

22. In the year ending June 1, 1875, Pennsylvania expended for school-houses, $2,059,464.83; for teachers' wages, $4,746,875.52; for fuel and contingent expenses, $2,448,315.78; for other expenses, $109,270.94; $77.324.32 extra appropriation to city of Pittsburg; $85,815.84 for Normal schools; and $423,693.76 for Soldiers' Orphan schools; what was the whole amount expended that year for education by the State? Ans. $9,950,760.99.

23. The cash value of farms in New York in 1870 was $1,272,857,766; in Pennsylvania, $1,043,481,582; in New Jersey, $257,523,376; in Ohio, $1,054,465,226; in Indiana, $634,804,189; in Illinois, $920,506,346; in Missouri, $392,908,047; in Wisconsin, $300,414,064; and in Michigan, $398,240,578; what is the total value of the farms in these States? Ans $6,275,201,174.

CONTRACTIONS IN ADDITION.

60. Contractions in Addition are abbreviated methods

of adding.

61. There are several methods of abbreviating the process of addition, a few of which will be stated:

1. In adding omit naming the numbers added; merely name the results.

2. When two or more terms of a column can be easily grouped together, use their sum instead of adding each separately; combining with especial reference to TENS.

3. When a term is repeated several times in a column, multiply it by the number of times it is repeated, and use the result.

1. What is the sum of 8752+3687+2573+8576+2857 +6872?

1ST METHOD.-We add thus: 2, 9, 15, 18, 25, 27; write the 7 and carry the 2, 2, 9, 14, 21, 28, 36, 41; write the 1 and carry the 4; etc.

2D METHOD. Thus: 9, 15, 25, 27, in which the 3 and 7 are grouped and used as 10. We may also group 2 and 6, 7 and 3, and then 7 and 2; thus, 8, 18, 27, etc. 3D METHOD. In the second column, we may take the three 7's, then the two 5's, and then the 8; thus, 21, 31, 39, etc.

WRITTEN EXERCISES.

OPERATION.

8752

3687

2573

8576

2857

6872

33317

442

INTRODUCTION TO SUBTRACTION.

MENTAL EXERCISES.

1. If I have 12 books and sell 5 of them, how many books shail I have remaining?

SOLUTION.-If I have 12 books and sell 5 of them, I have remaining 12 books minus 5 books, which are 7 books.

2. James had 14 cents and spent 9 of them; how many cents had he remairing?

3. Susan has 25 plums and Jane 13; how many more has Susan than Jane?

4. In a school numbering 45 pupils, 15 are absent; how many pupils are present?

5. A watch was bought for 28 dollars and sold for 21 dollars; wha! was the loss?

6. A cow was bought for 18 dollars and sold for 28 dollars; what was the gain?

7. Begin at 2 and count by 2's to 40; begin at 40 and count by 2'a backward to 2.

8. Begin at 45 and count by 3's backward to 3; begin at 44 and count by 3's backward to 2.

9. Count by 4's from 48 back to 4; from 55 back to 3; from 54 back to 2; from 53 back to 1.

10. Count by 5's from 60 back to 5; from 64 back to 4; from 63 back to 3; from 62 back to 2; from 61 back to 1.

11. In a similar manner begin at different numbers, and count backward by 6's, 7's, 8's and 9's.

12. Take the number 3, add 5, subtract 6, add 7, subtract 5, add 8, subtract 7, add 9, subtract 4, and name the result.

13. Take the number 11, add 4, subtract 5, add 5, subtract 4, add 6, subtract 5, add 7, subtract 6, add 8, subtract 7, and name the result

14. How many are 3 plus 5 minus 7? 4 plus 7 minus 8? 5 plus 6 minus 4? 8 plus 5 minus 9? 9 plus 10 minus 12? 12 plus 13 minus 15? 15. How many are 8 plus 12 minus 13? 9 plus 16 minus 14? 10 plas 15 minus 16? 16 plus 17 minus 18? 18 plus 19 minus 20? The process of finding the difference between two numbers is called subtraction. The sign of subtraction is, and is read minus. 16. Required the value of the following:

5+3-2

13+14-15

2+5-4 9+8-6 | 10+ 7-12

.

6+4-3

3+6-5

8+6-7

11+ 8-13

14+15-17

SUBTRACTION.

62. Subtraction is the process of finding the difference between two numbers.

63. The Difference between two numbers is a number which added to the less, equals the greater.

64. The Minuend is the number from which we subtract. he Subtrahend is the number to be subtracted.

65. The Sign of Subtraction is and is read minus. It denotes that the number immediately following it is to be subtracted from the number preceding it.

NOTES 1. The Sign of Subtraction is a short line in the line of writing. 2. The symbol was introduced by Stifelius, a German mathematician, in a work published in 1544.

PRINCIPLES,

1. The minuend and subtrahend must be similar numbers. 2. The difference is a number similar to the minuend and subtrahend.

CASE I.

OPERATION.

486

66. To subtract when no term of the subtrahend is greater than the corresponding term of the minuend. 1. What is the difference between 486 and 243? SOLUTION.-We write the subtrahend under the minuend, placing terms of the same order in the same column, draw a line beneath, and begin at the right to subtract. 3 units from 6 units leave 3 units, which we write under the units; 4 tens from 8 tens leave 4 tens, which we write under the tens; 2 hundreds from 4 hundreds leave 2 hundreds, which we write under the hundreds Therefore, the difference between 486 and 243 is 243.

243

243

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

II. Begin at the right and subtract each term of the subtrahend from the corresponding term of the minuend, writ ing the remainder beneath.