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54. If a man can walk 35 miles in a day, how far could he walk, at that rate, in a year, or 365 days? Ans. 12775 miles.

55. A has 340 acres of land worth 18 dollars an acre; and B has 239 acres worth 22 dollars an acre. How many acres have the two together, and what is the value of the whole. Ans. 579 acres; and 11378 dollars. 56. A merchant bought 475 barrels of flour, at 15 dollars a barrel. He sold 280 barrels of it, at 16 dollars, and the rest at 14 dollars a barrel; what did he gain or lose?

Ans. gained 85 dollars.

57. One manufacturer exported 234 bales of cotton cloth,— each bale containing 2400 yards; another exported 370 bales, each containing 1050 yards. Which of them exported the greater quantity, and by how many yards?

Ans. The first, 173100 yards.

58. Farmer A had in wheat 205 acres, which produced 27 bushels per acre. Farmer B had 320 acres, which produced 19 bushels per acre. What quantity of wheat was raised by them both. Ans. 11615 bushels. 59. A speculator bought 150 head of cattle, and 47 mules. He made a profit of 13 dollars a head on the former, and 17 on the latter; what was gained by the speculation?

Ans. 2749 dollars.

60. Bought 360 acres of land, at 35 dollars per acre; and at another time double that quantity, at double the price per acre. What was the whole quantity of land purchased, and the sum paid for it? Ans. 1080 acres; and 63000 dollars.

61. Two persons start together from the same place, and travel in the same direction. One proceeds at the rate of 29 miles per day, and the other at the rate of 31 miles per day. What distance will be between them at the end of 25 days? Ans. 50 miles.

62. A merchant bought 18 bales of linen, each containing 22 pieces, and each piece containing 40 yards. How many pieces and how many yards did he buy?

Ans. 396 pieces; and 15840 yards. 63. In a certain orchard there are 30 rows of apple trees, with 44 trees in each row. Allowing 2500 apples to each tree, what number of apples would there be in the orchard? Ans. 3300000 apples.

64. A farmer bought three tracts of land. The first and second contained each 280 acres, and the third as many as both the other two; how many acres did the farmer purchase, and what did the whole amount to, at 33 dollars per acre?

Ans. 1120 acres; and 36960 dollars.

65. Allowing a person's annual income to be 5000 dollars, and his daily expenses 3 dollars, what would be the amount of his annual saving,—there being 365 days in a year?

Ans. 3905 dollars. 66. If 327 head of cattle were purchased at 13 dollars a head, and 405 were purchased at 11 dollars a head, what would be the profit or loss on the whole at 12 dollars a head?

Ans. Profit, 78 dollars.

67. A planter sold 139 bales of cotton, at an average of 32 dollars per bale, and out of the proceeds bought 29 mules, at 49 dollars each, and 4 pair of oxen, at 52 dollars a pair; what sum had he left from the sale of his cotton?

Ans. 2819 dollars.

68. A sends 209 tons of coal to New York city; B sends as much as A, wanting 10 tons; and C sends as much as A and B together. What was each man's proceeds of sale, at 13 dollars per ton? Ans. A's 2717 dollars; B's 2587; C's 5304.

69. The President of the United States receives a salary of 25 thousand dollars a year. To what sum does his salary amount in 4 years, or one presidential term?

Ans. 100000 dollars.

70. The circumference of the Earth is about 25 thousand miles, and the distance to the Sun is 3 thousand 8 hundred times the Earth's circumference. What then is the distance to the Sun ? Ans. 95000000 miles.

71. The velocity of light is 192 thousand 500 miles per second. Through what distance then does light move in one minute, which is 60 seconds? Ans. 11550000 miles.

72. The Earth turns around its axis once in every 24 hours, and moves 68 thousand miles an hour in its orbit around the Sun. How far then are we carried along the Earth's orbit during one revolution of the Earth on its axis?

Ans. 1632000 miles.

DIVISION.

$ 46. DIVISION consists in finding how many times a greater number contains a less, or what part a less number is of a greater.

The number to be divided is called the dividend; the dividing number the divisor; and the number or part found, the quotient.

One half is one of the two equal parts,-two thirds are two of the three equal parts,—and so on, into which any quantity may be divided.

What is meant by one third? By one fifth? By two fifths?

By one fourth? By three fourths?
By one tenth? By five ninths?

When we say 2 is contained in 6, 3 times, we divide 6 by 2; 6 is the dividend, 2 the divisor, and 3 the quotient.

Also, 2 is one third of 6, because if 6 were divided into three equal parts, each part would be 2.

3 is what part of 6? 4 is what part of 12? 5 is what part of 20?

How many times is 3 contained in 6? How many times is 4 contained in 12? How many times is 5 contained in 20? If the dividend be 24, and the divisor 4, what will the quotient be? If the dividend be 35, and the divisor 5? If the dividend be 42, and the divisor 7? If the dividend be 56, and the divisor 8?

$47. The Quotient of a less number divided by a greater, is the part that the less is of the greater; and is denoted by the less over the greater, with a line between them.

Thus 1 divided by 2 is one half, because 1 is one half of 2.

How much is 1 divided by 3;
How much is 1 divided by 4?
How much is 2 divided by 3;
How much is 2 divided by 5?

that is, 1 is what part of 3?
1 divided by 5? 1 divided by 6?
that is, 2 is what part of 3?

2 divided by 7? 3 divided by 5?

Subtraction and Division.

§ 48. The subtraction of a less number from a greater, repeatedly, is equivalent to dividing the greater by the less, because it shows how many times the greater contains the less.

Thus 5 from 15 leaves 10, 5 from 10 leaves 5, and 5 from 5 leaves 0; so that 5 may be subtracted 3 times from 15, or is contained 3 times in 15.

How many times may 5 be subtracted from 20? How many times may 6 be subtracted from 24? 7 from 35? 8 from 48?

Multiplication and Division.

§ 49. Multiplication and Division are the reverse of each other.

In Multiplication, two numbers or factors are given, to find their product; in Division, a product and one of its factors are given, to find the other factor.

The product being 15, and one factor 3, what is the other factor? The product being 30, and one factor 5, what is the other factor? The product being 36, and one factor 9, what is the other factor? The product being 63, and one factor 7, what is the other factor?

Reciprocal of a Number.

§ 50. The reciprocal of a number is a unit or 1 divided by that number.

Thus the reciprocal of 2 is, and of 3 is }.

What is the reciprocal of 4? Of 5? Of 6?

Of 10 Of 20?

The Quotient as a Part of the Dividend.

§ 51. The Quotient is always such a part of the dividend as is expressed by the reciprocal of the divisor.

Thus 15 divided by 5 gives 3, and 3 is } of 15.

Again, if we divide 2 by 3, the quotient will be } (§ 47); and since two thirds of any quantity is one third of two such quantities, is equal to of 2; such a part of the dividend 2 as is expressed by the reciprocal of the divisor 3.

of 1 cent is what part of 3 cents?
of 1 pint is what part of 2 pints?
of 1 mile is what part of 5 miles?
of 5 cents is what part of 1 cent?

3 of 1 is what part of 3?

4

of 1 is what part of 2? of 1 is what part of 5? of 5 is what part of 1?

of 7 pints is what part of 1 pint?of 7 is what part of 1? of 8 miles is what part of 1 mile?

How would you find of any number?

of 8 is what part of 1?

of any number?

any number? of any number? of any number?

Sign of Division

!of

$ 52. The sign, called by, placed between two numbers, signifies that the first of them is to be divided by the second. Thus 36÷9, 36 by 9, signifies that 36 is to be divided by 9.

[blocks in formation]

Division is also denoted by the dividend over the divisor with

a line between them.

Thus 63 denotes the same as 63-9.

Quotient of Concrete Numbers.

$ 53. When the dividend and divisor are similar concrete numbers, the quotient is the number of times the dividend contains the divisor, or the part the dividend is of the divisor.

Thus 12 cents-3 cents gives 4; and 12 cents 13 cents gives 13 (§ 47).

How many times 4 miles in 12 miles? 5 pounds in 30 pounds? 6 inches in 42 inches? 7 yards in 77 yards? 12 dollars in 96 dollars What part is 3 days of 7 days? What part is 9 ounces of 20 ounces? What part is 20 feet of 49 feet?

§ 54. When the dividend and divisor are dissimilar concrete numbers, the quotient is such a part of the dividend as is expressed by the reciprocal of the divisor taken abstractly.

For example, if 5 pencils cost 20 cents, one pencil will cost 20 cents-5; that is, of 20 cents, which is 4 cents.

If 3 slates cost 36 cents, what will one slate cost? If 4 hats cost 16 dollars, what will one hat cost? If in 9 hours a stage runs 54 miles, at what rate does it run per hour?

Remainder in Division.

§ 55. A remainder, in Division, is an overplus or excess of the dividend above so many times the divisor as it is contained in the dividend.

Thus 5 is contained in 17, 3 times, with 2 over, since 3 times 5 is 15; then 2 is the remainder of the dividend.

If the divisor be 6, and the dividend 27, what will the quotient and the remainder be? If the divisor be 8, and the dividend 45? If the divisor be 9, and the dividend 70 ?

§ 56. The remainder divided by the divisor, and so annexed to the quotient, completes the quotient.

Thus 175 gives quotient 3, and remainder 2. This 2÷5 gives (47); the complete quotient is then 33, three and twofifths.

In like manner find the quotient of 9÷2. Of 13÷4. Of 21÷5. Of 276. Of 307. Of 418. Of 100÷9.

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