10. If a mechanic receives 784 dollars a year for labor, and his expenses are 426 dollars a year, how much can he save in 6 years? Ans. 2148 dollars.

11. A farmer sold 40 bushels of wheat at 2 dollars a bushel, and 16 cords of wood at 3 dollars a cord. He received 15 yards of cloth at 4 dollars a yard, and the rcmainder in money; how much money did he receive?

Ans. 68 dollars.

12. How many pounds of cheese worth 10 cents a pound, can be bought for 22 pounds of butter worth 15 cents a pound? Ans. 33 pounds. 13. If 56 yards of cloth cost 336 dollars, how much will

12 yards cost, at the same rate?

14. If 100 barrels of flour cost 350 barrels cost, at the same rate?

Ans. 72 dollars. 600 dollars, what will

Ans. 2100 dollars.

15. How long can 60 men subsist on an amount of food Ans. 127 days.

that will last 1 man 7620 days?

16. If I buy 225 barrels of flour for 1125 dollars, and sell the same for 1800 dollars, how much do I gain on each barrel? Ans. 3 dollars. 17. A man sold his house and lot for 5670 dollars, and took his pay in bank stock at 90 dollars a share; how many shares did he receive? Ans. 63 shares.

18. How many pounds of tea worth 75 cents a pound, ought a man to receive in exchange for 27 bushels of oats, worth 50 cents a bushel? Ans. 18 pounds.

19. The quotient of one number divided by another is 40, the divisor is 364, and the remainder 120; what is the dividend? Ans. 14680. 20. How many tons of hay at 12 dollars a ton, must be given for 21 cows at 24 dollars apiece? Ans. 42 tons.

21. Bought 150 barrels of flour for 1050 dollars, and sold 107 barrels of it at 9 dollars a barrel, and the remainder at 7 dollars a barrel; did I gain or lose, and how much?

Ans. gained 214 dollars.

22. A mechanic earns 45 dollars a month, and his necessary expenses are 27 dollars a month. How long will it take him to pay for a farm of 50 acres, at 27 dollars an acre? Ans. 75 months. 23. How many barrels of flour at 7 dollars a barrel, will pay for 30 tons of coal, at 4 dollars a ton, and 44 cords of wood, at 3 dollars a cord? Ans. 36 barrels.

PROBLEMS IN SIMPLE INTEGRAL NUMBERS.

74. The four operations that have now been considered, viz., Addition, Subtraction, Multiplication, and Division, are all the operations that can be performed upon numbers, and hence they are called the Fundamental Rules.

75. In all cases, the numbers operated upon and the re sults obtained, sustain to each other the relation of a whole to its parts. Thus,

I. In Addition, the numbers added are the parts, and the sum or amount is the whole.

II. In Subtraction, the subtrahend and remainder are the parts, and the minuend is the whole.

III. In Multiplication, the multiplicand denotes the value of one part, the multiplier the number of parts, and the product the total value of the whole number of parts.

IV. In Division, the dividend denotes the total value of the whole number of parts, the divisor the value of one part, and the quotient the number of parts; or the divisor the number of parts, and the quotient the value of one part.

76. Let the pupil be required to illustrate the following problems by original examples.

Problem 1. Given, several numbers, to find their sum. Prob. 2. Given, the sum of several numbers and all of them but one, to find that one.

Prob. 3. Given, two numbers, to find their difference. Prob. 4. Given, the minuend and subtrahend, to find the remainder.

Prob. 5. Given, the minuend and remainder, to find the subtrahend.

Prob. 6. Given, the subtrahend and remainder, to find the minuend.

Prob. 7. Given, two or more numbers, to find their prod

uct.

Prob. 8. Given, the multiplicand and multiplier, to find the product.

Prob. 9. Given, the product and multiplicand, to find the multiplier.

Prob. 10. Given, the product and multiplier, to find the multiplicand.

Prob. 11. Given, two numbers, to find their quotients. Prob. 12. Given, the divisor and dividend, to find the quotient.

Prob. 13. Given, the divisor and quotient, to find the dividend.

Prob. 14. Given, the dividend and quotient, to find the divisor.

Prob. 15. Given, the divisor, quotient, and remainder, to find the dividend.

Prob. 16. Given, the dividend, quotient, and remainder to find the divisor.

FRACTIONS.

DEFINITIONS, NOTATION, AND NUMERATION.

77. If a unit be divided into 2 equal parts, one of the parts is called one half.

If a unit be divided into 3 equal parts, one of the parts is called one third, two of the parts two thirds.

If a unit be divided into 4 equal parts, one of the parts is called one fourth, two of the parts two fourths, three of the parts three fourths.

If a unit be divided into 5 equal parts, one of the parts is called one fifth, two of the parts two fifths, three of the parts three fifths, &c.

And since one half, one third, one fourth, and all other equal parts of an integer or whole thing, are each in themselves entire and complete, the parts of a unit thus used are called fractional units; and the numbers formed from them, fractional numbers. Hence

78 A Fractional Unit is one of the equal parts of an integral unit.

79. A Fraction is a fractional unit, or a collection of fractional units.

80. Fractional units take their name, and their value, from the number of parts into which the integral unit is divided. Thus, if we divide an orange into 2 equal parts, the parts are called halves; if into 3 equal parts, thirds; if into 4 equal parts, fourths, &c.; and each third is less in value than each half, and each fourth less than each third; and the greater the number of parts, the less their value.

The parts of a fraction are expressed by figures; thus,

One half is written

One third

Eight tenths

To write a fraction, therefore, two integers are required, one written above the other with a line between them.

81. The Denominator of a fraction is the number below the line. It shows into how many parts the integer or unit is divided, and determines the value of the fractional unit.

82. The Numerator is the number above the line. It numbers the fractional units, and shows how many are taken.

83. Thus, if one dollar be divided into 4 equal parts, the parts are called fourths, the fractional unit being one fourth, and three of these parts are called three fourths of a dollar, and may be written

3 the number of parts or fractional units taken,

4 the number of parts or fractional units into which the dollar is divided.

84. The Terms of a fraction are the numerator and denominator, taken together.

85. Fractions indicate division, the numerator answering to the dividend, and the denominator to the divisor. Hence,

86. The Value of a fraction is the quotient of the numerator divided by the denominator.

Thus; the quotient of 4 divided by 5 is , or expresses the quotient of which ( 4 is the dividend. {

5 is the divisor.