765 No. of yds. purchased. 523 No. of yds. sold.

242 No. of yds. unsold.

We first write down the greater number, then we write the less number under the greater, placing units under units, tens under tens, We first take 3 units from

and hundreds under hundreds. 5 units, and 2 units remain, which we write directly under the units. Then 2 tens from 6 tens, and 4 tens remain. Lastly, 5 hundred from 7 hundred, and 2 hundred remain. The whole remainder, or difference, is 242.

Art. 16. When any figure of the less number is larger than the corresponding figure of the greater number, the following question and its illustration will show the method of performing the operation.

A man borrowed 94 dollars, and has paid 46 dollars; how many dollars remain unpaid?

948 tens and 14 units. 464 tens and 6 units.

48: 4 tens and 8 units.

9 tens and added to the 4 units, 8 tens and 14 units.

We cannot take 6 units from 4 units, but we can take 46 from 94. If 1 of the tens, which is equal to 10 units, be taken from the 94 will be decomposed into

Decomposing 46 into tens and units, we have 4 tens and 6 units. We now take 6 units from 14 units, and 8 units remain; 4 tens from 8 tens, and 4 tens remain. The whole remainder is 4 tens and 8 units, or 48, the number of dollars unpaid.

Art. 17. If two unequal numbers be equally increased, their difference will not be altered. As 10 units of any lower order are equal to 1 unit of the next higher order, whenever a figure of the less number is larger than the corresponding figure of the greater number, if we add 10 to the upper figure, and then add 1 to the next lower figure, the two numbers will be equally increased, yet their difference will remain unaltered.

As 94 is equal to 9 tens and 4 units, if we add 10 units to the 4 units, the amount will be 9 tens and 14 units; - 46 is equal to 4 tens and 6 units,—if we add 1 ten to 4 tens, the amount will be 5 tens and 6 units; — and if we take 5 tens and 6 units from 9 tens and 14 units, the remainder, or difference, will be 4 tens and 8 units, or 48, the number of dollars unpaid.

From the above examples and illustrations, we deduce the following rule for subtraction of simple numbers.

RULE. Write down the greater number, then write the less number under the greater, placing units under units, tens under tens, hundreds under hundreds, and draw a line underneath.

Begin with the units, and subtract each figure of the less number, in succession, from the figure over it, and write the remainder underneath.

Whenever a figure of the less number is greater than the figure over it, add ten to the upper figure, subtract the lower figure from the amount, then add one to the next lower figure before it is subtracted.

METHOD OF PROOF. Add the remainder to the less number; their sum will be equal to the greater number, if the operation has been correctly performed.

49. America was discovered by Christopher Columbus in 1492. How many years since to the present time, 1847? Ans. 355 years. 50. The United States were declared independent July 4th, 1776. How many years since to July 4th, 1847 ? Ans. 71 years. 51. The first settlement at Boston was made in 1630. How many years since to the present time, 1847? Ans. 217 years. 52. The greater of two numbers is seven hundred and fifty million; the less number is five hundred and forty thousand; what is their difference? Ans. 749460000.

53. The mariner's compass was invented in the year 1302; how many years was this before the discovery of America by Columbus, in 1492? Ans. 190 years.

54. A is worth ten thousand dollars; B is worth seven thousand nine hundred and twenty-five dollars. How many dollars is A worth more than B? Ans. 2075 dollars. 55. The planet Saturn is 890000000 miles from the Sun; the Earth is 95000000 miles from the Sun. How many miles further from the sun is Saturn than the Earth?

Ans. 795000000 miles. 56. The diameter of Jupiter is eighty-nine thousand miles; the diameter of Venus is seven thousand seven hundred miles. What is the difference of their diameters? Ans. 81300 miles. 57. Massachusetts contains seven thousand five hundred square miles; Connecticut, four thousand seven hundred and sixty-four square miles. How many more square miles does Massachusetts contain than Connecticut ?

Ans. 2736 square miles.

58. The number of inhabitants in France is 33500000; the number of inhabitants in the United States is 17069453. How many more inhabitants are there in France than in the United States? Ans. 16430547 inhabitants.

59. The library of Harvard University contains 49000 volumes; the library of Yale College contains 35000 volumes. How many more volumes does the former contain than the latter? Ans. 14000 volumes.

60. The first settlement in the United States was made by the English, at Jamestown, in Virginia, in 1607; the United States were declared independent in 1776. How many years between these two events? Ans. 169 years.

61. The population of the United States in 1840 was 17069453, of which 2487355 were slaves. How many free persons were there in the United States in 1840?

Ans. 14582098 free persons.

62. A merchant owns property to the amount of thirty thousand five hundred and seventy-five dollars; but there are demands against him to the amount of seventeen thousand and sixty-five dollars. What number of dollars will he have left after paying his debts? Ans. 13510 dollars.

63. The population of Boston, in 1790, was 18038; in 1800, it was 24927; in 1810, it was 32250; in 1820, it was 43298; in 1830, it was 61392; in 1840, it was 93383. What was the increase in each successive period of 10 years? What was the increase in the whole period of 50 years?

64. The population of New York, in 1790, was 33131; in 1800, it was 60489; in 1810, it was 96373; in 1820, it was 123706; in 1830, it was 202589; in 1840, it was 312710. What was the increase in each successive period of 10 years? What was the increase in the whole period of 50 years?

65. The population of the northern district of New York in 1840, was one million six hundred eighty-three thousand and sixty-eight, and the population of the southern district was seven hundred forty-five thousand eight hundred and fifty-three. How many more inhabitants were there in the northern than in the southern district?

66. The population of the United States, in 1790, was 3929827; in 1800, it was 5305925; in 1810, it was 7239814; in 1820, it was 9638131; in 1830, it was 12866020; in 1840, it was 17069453. What was the increase in each successive period of 10 years? What was the increase in the whole period of 50 years?

PRACTICAL QUESTIONS IN ADDITION AND SUBTRACTION OF SIMPLE NUMBERS.

Art. 18. 1. A merchant purchased 5000 bushels of salt; he has since sold 1500 bushels to one man, and 750 bushels to another. What number of bushels has he remaining unsold?

2. A man's property is worth 10650 dollars. He owes A, 1800 dollars; B, 1260 dollars; C, 750 dollars; and D, 500 dollars. What will he have remaining, after paying his debts?

3. A gentleman purchased 15840 acres of new land. He afterwards sold 2350 acres to one individual, 4500 acres to another, and 3225 to a third. What number of acres has he left?

4. A merchant purchased 4500 yards of cloth; he has since sold 1750 yards to one of his customers, 1275 yards to another, and 950 yards to a third. What number of yards has he remaining unsold?

5. A man died leaving an estate amounting to 12650 dollars, which he bequeathed as follows: 2500 dollars to each of his three daughters, and the remainder to his son. What was the son's share?

6. A flour merchant purchased 1250 barrels of flour for 6250 dollars. He has since sold 500 barrels to A, for 2625, dollars, and 450 barrels to B, for 2475 dollars. How many barrels remain unsold? What was the whole cost of the flour above the amount of sales?

7. A merchant deposited 1200 dollars in the Suffolk Bank. He has since given A a check for 4500 dollars, B a check for 2750 dollars, and C a check for 1675 dollars. What amount has the merchant remaining in the bank?

8. A merchant purchased 12750 yards of cloth of one manufacturer, 10675 yards of another, and 7425 yards of a third. He has since sold 8450 yards to one of his customers, 6745 yards to another, and 5925 yards to a third. How many yards did he purchase? How many yards has he sold? What number of yards has he remaining unsold? 9. A gentleman purchased a house for 12500 dollars, a carriage for 750 dollars, and a span of horses for 500 dollars. He has paid 6500 dollars at one time, 2750 dollars at another time, and 1250 dollars at a third time. What was the whole amount of his purchases? What was the whole amount of payments? And what amount remains unpaid?

MULTIPLICATION OF SIMPLE

NUMBERS.

Art. 19. Multiplication is the method of repeating any given number any required number of times.

The terms used in multiplication are the multiplicand, multiplier, and product. The multiplicand is the number to be multiplied or repeated. The multiplier is the number to multiply by, and it expresses the number of times the multiplicand is to be repeated. The product is the number produced by repeating the multiplicand the required number of times. The multiplicand and multiplier are called the factors of the product.

ILLUSTRATION. If a yard of cloth be worth 10 cents, how many cents are 5 yards worth?

If one yard is worth 10 cents, 5 yards must be worth 5 times 10 cents, or 50 cents. In the above question, 10 cents is the multiplicand, 5 is the multiplier, and 50 cents is the product.

Multiplication is indicated by an inclined cross; thus, 6×5=30; and is read 6 multiplied by 5 is equal to 30?

1. If a quart of milk is worth 5 cents, what are 2 quarts worth? 3. How many are 2 times 5? 5. There are 3 feet in one yard; how many feet are there in 5 yards?

7. How many are 5 times 3? 9. There are 6 shillings in 1 dollar; how many shillings are there in 6 dollars?

11. How many are 6 times 6? 13. James has 8 marbles, and Henry has 5 times as many; how many has Henry?

15. How many are 5 times 8? 17. If one pound of sugar is worth 10 cents, what are 8 pounds worth?

19. How many are 8 times 10? 21. There are 7 days in one week; how many days are there in 11 weeks?

23. How many are 11 times 7? 25. In one shilling there are 12 pence; how many pence are there in 9 shillings?

2. If 1 orange cost 6 cents, how many cents will 3 oranges cost? 4. How many are 3 times 6?

6. There are 4 farthings in one penny; how many farthings are there in 4 pence?

8. How many are 4 times 4? 10. If one quart of cherries is worth 7 cents, how many cents are 7 quarts worth?

12. How many are 7 times 7? 14. If one yard of ribbon cost 9 cents, how many cents will 7 yards cost?

16. How many are 7 times 9? 18. There are 10 cents in one dime; how many cents are there in 10 dimes?

20. How many are 10 times 10? 22. How many yards of cloth are there in 10 pieces, each piece measuring 11 yards?

24. How many are 10 times 11? 26. If a mechanic can earn 12 dollars in one week, how many dollars can he earn in 12 weeks?