16. A merchant having 336 pounds of rice, bought 248 pounds, and then sold 467 pounds; how many pounds remained? Ans. 117 pounds.

17. Lanterns were invented by King Alfred in 890; how many years from that time to 1876 ? Ans. 986 years.

18. Spectacles were invented by Spina in 1299; how many years from their invention to 1878? Ans. 579 years.

19. Watches were invented at Nuremberg in 1477; ho many years from that time to 1876? Ans. 399 years.

20. The circulation of the blood was discovered by Harvey in 1619; how long since that time?

21. The telescope was invented by Galileo in 1610; how many years since it was invented?

22. Printing is supposed to have been invented in 1441; how many years has the art been used?

23. The first steamship crossed the ocean in 1839; how long is it since that time?

24. The first line of telegraph was established in the United States in 1844; how long has the invention been in use?

25. The distance from Philadelphia to Pittsburgh is 353 miles, and to Harrisburg is 105 miles; what is the distance from Harrisburg to Pittsburgh? Ans. 248 miles.

26. A merchant had in bank $4500, and "checked out" $2345; how much will remain in bank? Ans. $2155. 27. An architect received $54,750 for building a markethouse, and he expended upon it $51,784; what was his profit? Ans. $2966. 28. Mr. Newton bought a house for $10,575, and paid for it in yearly installments of $1057.50; how many years did It require to pay for the house? Ans. 10 years.

29. At an election the successful candidate received 10.172 votes, and the unsuccessful candidate 34,789 votes; what was the successful candidate's majority?

Ans. 5383 votes.

CONTRACTIONS IN SUBTRACTION.

69. Contractions in Subtraction are abbreviated methods of subtracting.

70. When two or more numbers are to be successively subtracted from another, the operation can be abbreviated by the following rule:

Rule. Add the terms of the numbers to be subtracted, and write for a remainder a term which, added to this sum, will give a number having for its unit term the correspond ing term of the minuend.

1. Subtract 324 and 549 from 1024.

SOLUTION.-4 and 9 are 13, and since 1 more will make 14, we write 1 as the first term of the remainder; 4 and 1 are 5 and 2 are 7, and since 5 more make 12, we write 5 in the remainder; 5 and 1 are 6 and 3 are 9, and since 1 more make 10, we write 1 in the remainder.

WRITTEN EXERCISES.

OPERATION.

1024

324

549

151

6. A raised 785 bushels of grain, and having sold B 250 bushels, C 320 bushels, and D 169 bushels, he retained the remainder; how many bushels did he retain? Ans. 46 bu.

7. From a tract of land in a newly settled country containing 4850 acres, there were sold at one time 1560 acres; at another, 2490 acres; and at another 500 acres; how much of the tract then remained? Ans. 300 acres.

8. In five granaries there are 7874 bushels of wheat; in the first there are 1160 bushels; in the second 1729 bushels; in the third 1957 bushels; in the fourth 1433 bushels; how many bushels are contained in the fifth? Ans. 1595 bu.

9. An estate of $10,000 was divided as follows: the widow received $4397; the youngest son, $2498; the daughter, $3104; and the eldest son the remainder. What was the share of the latter?

WRITTEN EXERCISES.

IN ADDITION AND SUBTRACTION.

1. An estate worth $35,000 has two mortgages upon it. one for $15,000, and the other for $8000; what is it worth above the incumbrances? Ans. $12,000.

2. A merchant's gross profits last year were $9000; he pays for rent $1200, insurance $150, salaries $3500, and incidental expenses, $225; what is his net profit? Ans. $3925.

3. A clerk receives $800 a year; he pays $150 a year for house rent, his butcher's bill is $177.50 and grocer's bill $145.45, and he has expended for clothes and other expenses $250; what does he save in the year?

Ans. $77.05.

4. Mr. Wainwright bought a farm which cost him $25,000; but afterwards getting into difficulties, he burdened the property with a mortgage for $7500; for what must he sell the farm to gain $3500? Ans. $21,000.

5. Mr. Hatton receives for goods sold a check for $15,495.25 on the Girard National Bank; he wishes to deposit $13,500 in the same bank; what will the bank return him when he presents the check? Ans. $1995.25. 6. I had in bank on Monday morning, Dec. 4, $11,275, I checked out Monday afternoon $745 Tuesday, I deposited $1500; Wednesday, I deposited $475 Thursday, I checked out $873.50; Friday, I again checked out $563.75; Saturday, I deposited $560; what is the amount of my deposit on Monday morning, Dec. 11? Ans. $11,627.75.

7. The day before Christmas I went into a fancy store and bought for Christmas presents, a writing-desk at $4.25, a work-box at $2.75, a glove-box at $1.50, a handkerchiefbox at $1.75, a case of perfumery at $2.25, and a gentle man's dressing-case at $5.75; I handed the clerk a twenty. dollar bill; what change should I receive? Ans. $1.75.

8. A farmer took to the store 2 dozen eggs, $0.70; 10 pounds of butter, $5; 2 barrels of apples, $11.50; he bought 25 pounds crushed sugar, $3.50; 2 gallons molasses, $0.60; 10 yards red flannel, $3.75, and one piece unbleached sheeting, $10; how much cash must he pay

y?

Ans. $0.65.

INTRODUCTION TO MULTIPLICATION.

MENTAL EXERCISES.

1. Repeat the table of 2 times; of 3 times; of 4 times; of 5 times of 6 times; of 7 times; of 8 times; etc.

2. If one barrel of flour costs 6 dollan, what will 5 barrels cost at the same rate?

SOLUTION.-If one barrel of flour costs 6 dollars, 5 barrels will cost 5 times 6 dollars, which are 30 dollars.

3. What will 7 gold pens cost at 3 dollars apiece?

4. At 20 cents apiece what will 6 spelling-books cost?

5. What is the cost of 11 yards of calico at 12 cents a yard?

6. George has 15 dollars and James has 6 times as much; how many dollars has James?

7. A farmer bought 6 cows at 25 dollars apiece; how much did he pay for them?

8. If 6 men can do a piece of work in 20 days, how long will it take 1 man to do it?

9. If 8 men mow a field of grass in 15 days, how long will it take 1 man to mow it?

10. In an orchard there are 16 rows of trees and 10 trees in each row; how many trees are there in the orchard?

11. John borrowed 12 cents from a friend, and then earned 6 times as much as he borrowed; how many cents had he then?

12. Arnold earned 18 dollars a week, and paid 9 dollars a week for his board; how much could he save in 8 weeks?

13. How many are 3 times 5, plus 6? 5 times 6, plus 7? 6 times 7, plus 8? 7 times 8, plus 9? 9 times 10, plus 11? 12 times 10, plus 13? 14. How many are 4 times 6, minus 8? 6 times 7, minus 5? 8 times 9, minus 10? 7 times 11, minus 9? 9 times 12, minus 13?

15. How many are 6 and 6 times 7? 7 and 7 times 9? 8 and 8 times 10? 9 and 9 times 8? 10 and 10 times 11? 11 and 11 times 12?

16. The process of taking one number as many times as there are units in another is called multiplication. The number multiplied is the multiplicand; the number by which we multiply is the multiplier; the result is the product. The sign of multiplication is X, and is read "multiplied by."

17. Read the result of the following:

MULTIPLICATION.

71. Multiplication is the process of finding the product of two numbers.

72. The Product of two numbers is the result obtained by taking one number as many times as there are unite in another.

73. The Multiplicand is the number to be multiplied. 74. The Multiplier is the number by which we multiply. 75. The Sign of Multiplication is ×, and is read multiplied by, times, or into. When placed between two numbers it denotes that one is to be multiplied by the other.

NOTES.-1. The Sign of Multiplication consists of two short lines of equal length, bisecting each other at an angle of 45 degrees with the line of writing.

2. The symbol X was introduced by Wm. Oughtred an English mathe matician, born in 1574.

PRINCIPLES.

1. The multiplier is always an abstract number.

2. The multiplicand may be abstract or concrete.

3. The product is always similar to the multiplicand.

CASE I.

76. When the multiplier is not greater than twelve.

1. Multiply 353 by 8.

SOLUTION.-We write the multiplier under the multiplicand, draw a line beneath, and begin at the right to multiply. 8 times 4

OPERATION.

Multiplicand, 854
Multiplier,

2832

Product,

8

units are 32 units, or 2 units and 3 tens; we write the units in units place, and reserve the tens to add to the next product: 8 times 5 tens are 40 tens, plus the 3 tens equal 43 tens, or 3 tens and 4 hundreds; we write the 3 tens in tens place, and reserve the 4 hundreds o add to the next product: 8 times 3 hundreds are 24 hundreds, plus he 4 hundreds equal 28 hundreds, or 8 hundreds and 2 thousands, which we write in their proper places, and we have 2832. Hence the following

Rule.-I. Write the multiplier under the multiplicand, and draw a line beneath.

II. Begin at the right, and multiply each term of the multiplicand by the multiplier, carrying as in addition.