8. Bought a ship for 42650 dollars and sold it for 49000 dollars; what did I gain? Ans. 6350 dollars. 9. A gentleman gave 12462 dollars for a house and some land; the house alone was worth 9375 dollars; what was the the value of the land? Ans. 3087 dollars.

10. A lumberman, having 650000 feet of boards, sold 162372 feet of them; how many feet then remained? Ans. 487628 feet. 11. The battle of Gettysburg, in 1863, was 48 years after the battle of New Orleans; in what year was the latter?

12. A man having 100000 dollars, gave away 365 dollars; how much had he left? Ans. 99635 dollars. 13. A merchant owns property to the amount of 45563 dollars, and owes 21209 dollars; how much is he worth more than he owes?

14. If two candidates for office received in the aggregate 73462 votes, and the successful one had 45309 votes, how many did the other have? Ans. 28153 votes.

15. Illinois contains 55405 square miles and Iowa 50914 square miles; how many more square miles does the one contain than the other?

16. Mount Sorata, in South America, is 25380 feet high, and 19146 feet higher than Mount Washington in New Hampshire; how high is Mount Washington? Ans. 6234 feet.

17. Girard College, in Philadelphia, is said to have cost 1422800 dollars, and Trinity Church, in New York City, 338000 dollars; how much more did the one cost than the other?

18. If the value of the annual products of the industry of. Massachusetts is 266000000 dollars, and that of Pennsylvania is 285500000 dollars, how much do the products of the one State exceed those of the other? Ans. 19500000 dollars.

19. If the population of Ohio was 45365 in 1800, and 2339502 in 1860, how much was the increase?

REVIEW QUESTIONS. What is Addition? (37) to be observed in Addition? (39)

Ans. 2294137.

What principles are

REVIEW EXERCISES.

1. 5736 thousand+6 million +507963 +1245= how many? Ans. 6515781.

2. If the minuend is eight million six hundred seventy-three thousand four hundred and one, and the subtrahend six million seven hundred twenty thousand seven hundred and thirty, what is the difference? Ans. 1952671.

3. If the larger of two numbers is 100101 and their difference 9902, what is the smaller number? Ans. 90199.

4. A man owning 4605 acres of land, gave to one of his sons 1420 acres and to another 1280 acres; how many acres had he remaining?

SOLUTION. If he gave to one son 1420 acres and to another 1280 acres, he must have given to both the sum of 1420 acres and 1280 acres, or 2700 acres.

If he had 4605 acres and gave away 2700 acres, he must have had remaining the difference between 4605 acres and 2700 acres, or 1905 Therefore, he had remaining 1905 acres.

acres.

5. A grain dealer bought 6000 bushels of wheat; he afterwards sold to one man 1575 bushels, and to another 3600 bushels; how many bushels remain unsold?

6. A man died leaving 24000 dollars, of which he gave his wife 8000 dollars, one daughter 3500 dollars, another 4500 dollars, and the residue to his son; what was the son's portion?

7. Mr. Jones had in a bank 16830 dollars, drew out 9460 dollars, and afterward put in 2000 dollars; how much had he then in the bank? Ans. 9370 dollars.

8. A farmer had a horse worth 275 dollars, and exchanged it for a yoke of oxen and two cows; the oxen he sold for 125 dollars, one of the cows for 75 dollars, and the other for 58 dollars. How much did he lose by the trade? Ans. 17 dollars.

REVIEW QUESTIONS. What is Subtraction? (42) What principles ara to be observed in Subtraction? (44)

MULTIPLICATION,

46. 1. How many dollars will 6 tons of coal cost, at 7 dol ars a ton?

SOLUTION. Since 1 ton of coal costs 7 dollars, 6 tons must cost 6 times 7 dollars, which are 42 dollars. Therefore, 6 tons of coal, at 7 dollars a ton, will cost 42 dollars.

2. How many cents will buy 5 pencils, at 8 cents each?

3. When berries are 8 cents a quart, how much must be paid for 4 quarts?

4. If a boy can walk 3 miles in an hour, how many miles can he walk in 5 hours?

5. A farmer had 10 cows in each of 3 pastures; how many had he in all of them?

6. If 1 horse will eat 4 tons of hay in a given time, how many tons will 7 horses eat in the same time?

The preceding operations are called MULTIPLICATION. Hence,

47. Multiplication is the process of finding the result of taking one of two given numbers as many times as there are units in the other.

The MULTIPLICAND is the number to be multiplied.

THE MULTIPLIER is the number by which the multiplicand s to be multiplied.

The PRODUCT is the result of the multiplication.

The FACTORS OF THE PRODUCT are the multiplicand and multiplier.

48. The SIGN OF MULTIPLICATION is an inclined cross, X, read multiplied by. Thus, 5 × 4 is read, 5 multiplied by 4.

What is Multiplication? The Multiplicand? The Multiplier? The Product? The Factors of the Product? The Sign of Multiplication?

Any number of times 0 is 0, and 0 times any number is O Thus,

0X1=0,0×2=0, etc.; 1X0=0, 2 × 0=0, etc.

༡

49. The process of Multiplication is based upon the following

Repeat the column once 1 is 1. What is any number of times 0 ?

2 times 1 are 2. 3 times 1 are 3, etc. O times any number?

PRINCIPLES.

1. The product and multiplicand must be like numbers.

Thus,

4 times 6 men are 24 men.

times 7 cents are 21 cents.

2. The multiplier must always be regarded as an abstract number. Thus,

In finding the cost of 6 tons of coal at 7 dollars a ton, the 7 dollars are taken 6 times, and not multiplied by 6 tons.

3. The product of two or more factors is the same in whatever order they are taken.

Thus,

The product of 6× 3, or 3 × 6, is 18, and the product of 5 × 3 × 2, or 2 X3 X5, or 3 × 5 × 2, is 30.

50. From the definition of Multiplication, it follows, that

Multiplication, when the size or value of a single thing, or unit, is given, enables us to find the size or value of any number of things of the same kind.

51. To multiply one number by another.

1. Let it be required to multiply 564 by 7.

OPERATION.

For convenience, we write the multiplier Multiplicand, 564 under the units in the multiplicand, and begin with units to multiply.

Multiplier

Product,

7

7 times 4 units are 28 units, which equal 2 3948 tens and 8 units; we write the 8 units, and reserve the 2 tens to add to the next product.

7 times 6 tens are 42 tens, which with the two tens added are 44 tens, or 4 hundreds and 4 tens, we write the 4 tens, and reserve the 4 hundreds to add to the next product.

7 times 5 hundreds are 35 hundreds, which with the 4 hundreds added are 39 hundreds, or 3 thousands and 9 hundreds; which we write.

Therefore, the product is 3 thousand 9 hundred and 48, or 3948.

What is the first Principle? Multiplication enable us to find?

The second? The third? What does