PROOF.

Add the remainder to the subtrahend. If their sum is equal to the minuend the work may be regarded as right.

Q. How do you set down the numbers for subtraction? Where do you begin to subtract? How do you subtract? How do you prove subtraction?

Ans.

Ans. 2935621.

Ans.

Remainders, 9996

9. From 2637804 take 2376982.

10. From 3762162 take 826541.

11. From 78213609 take 27821890.

12. From thirty thousand and ninety-seven, take one thousand six hundred and fifty-four. Ans. 28443.

13. From one hundred million two hundred and fortyseven thousand, take one million four hundred and nine.

14. Subtract one from one million.

APPLICATIONS.

Ans.

Ans.

1. Suppose John were born in eighteen hundred and fifteen, and James in eighteen hundred and twenty-five: what is the difference of their ages? Ans. 10 years.

2. A man was born in 1785: what was his age in 1830? Ans. years. 3. Suppose I lend a man 1565 dollars, and he dies, owing me 450 dollars: how much had he paid me? Ans. 1115 dollars,

4. In five bags are different sums of money to the amount in all of 1000 dollars. In the first there are 100 dollars; in the second, 314 dollars; in the third, 143 dollars; and in the fourth, 209 dollars: how many dollars does the fifth contain? Ans. dollars.

5. America was discovered by Christopher Columbus in the year 1492. What number of years has since elapsed? 6. George Washington was born in the year 1732, and died in 1799: how old was he at the time of his death? Ans. years. 7. The declaration of independence was published July 4th, 1776: how many years to July 4th, 1838?

Ans. years.

8. By the census of 1840, it appeared, that the white population of the United States was 14,189,108, and the number of blacks 2,873,458 : how much did the white population exceed the black? Ans. 11,315,650.

9. In 1840 there were in the State of New York 2,428,921 inhabitants, and in the State of Pennsylvania 1,724,033 inhabitants: how many more inhabitants were there in New York than in Pennsylvania?

Ans.

10. The revolutionary war began in 1775; the late war in 1812 what time elapsed between their commencements? Ans. years.

11. In 1840 there were in New York, (which is the largest city in the United States,) 312,710 inhabitants, and in Philadelphia, (the next largest city,) 258,037: how many more inhabitants were there in New York, than in Philadelphia ? Ans. 54,673.

12. A man dies worth 1200 dollars; he leaves 504 to his daughter, and the remainder to his son: what was the son's portion? Ans. dollars.

13. Suppose a gentleman has an income of 3090 dollars a year, and pays for taxes 150 dollars, and expends besides 253 dollars: how much does he lay up?

Ans. dollars. 14. A merchant bought 500 barrels of flour for 3500 dollars; he sold 250 barrels for 2000 dollars: how many

barrels remained on hand, and how much must he sell them for, that he may lose nothing?

Ans. 250 barrels remained, and he

must sell for 1500 dollars.

APPLICATIONS IN ADDITION AND SUBTRACTION.

1. A merchant buys 19576 yards of cloth of one person, 27580 yards of another, and 375 yards of a third: he sells 1050 yards to one customer, 6974 yards to another, and 10462 yards to a third: how many yards has he remaining ?

Ans.

2. A person borrowed of his neighbour at one time 355 dollars, at another time 637 dollars, and 403 dollars at another time: he then paid him 977 dollars. How much did he owe him? Ans. 418.

3. I have a fortune of 2543 dollars to divide among my four sons, James, John, Henry, and Charles. I give James 504 dollars, John 600 dollars, and Henry 725: how much remains for Charles?

Ans.

dollars.

4. I have a yearly income of ten thousand dollars. I pay 275 for rent, 220 dollars for fuel, 35 dollars to the doctor, and 3675 dollars for all my other expenses: how much have I left at the end of the year? Ans. 5795.

5. A man pays 300 dollars for 100 sheep, 95 dollars for a pair of oxen, 60 dollars for a horse, and 125 dollars for a chaise. He gives in return 100 bushels of wheat worth 125 dollars, a cow worth 25 dollars, a colt worth 40 dollars, and pays the rest in cash: what amount of money does he pay? Ans. dollars.

MULTIPLICATION OF SIMPLE NUMBERS.

$21. If Charles gives 2 cents apiece for two oranges: how much do they cost him?

Ans. 4 cents. If Charles gives 2 cents apiece for 3 oranges: how much do they cost him?

Ans. cents.

If he gives 2 cents apiece for 4 oranges: how much do they cost him? Ans. 8 cents. If he give 2 cents apiece for 5 oranges: how much do they cost him? Ans. 10 cents. The cost in each case, may be obtained by adding the price of the separate oranges; thus,

2+2=4 cents, the cost of 2 oranges, 2+2+2=6 cents, the cost of 3 oranges, 2+2+2+2=8 cents, the cost of 4 oranges, 2+2+2+2+2=10 cents, the cost of 5 oranges.

In the first case 2 is repeated two times, in the second case it is repeated three times, in the third, four times, and in the fourth it is repeated five times; and in a similar manner any number may be repeated as often as we please by adding it continually to itself.

MULTIPLICATION is a short method of repeating one number as many times as there are units in another.

The number to be repeated is called the multiplicand. The number denoting how many times the multiplicand is to be repeated, is called the multiplier.

The number arising from repeating the multiplicand as many times as there are units in the multiplier, is called the product.

The multiplicand and multiplier are called factors, or producers of the product.

The sign X, placed between two numbers, denotes that they are to be multiplied together. It is called, the sign of multiplication.

Q. What is multiplication? What is the number called which is to be repeated? What does the multiplier denote? What is the product? In the case of the two oranges, which is the multiplicand? Which is the multiplier? Which is the product? In the case of three oranges, which is the multiplicand, which the multiplier, and which the product? What are the multiplicand and multiplier called? How do you denote that two numbers are to be multiplied together? What is the sign called.