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43. The capacities of two cubical cisterns are to each other as 1 to 8; but if 2 feet were added to each of their dimensions, their capacities would be to each other as 27 to 125. Required the capacity of each.

44. The sum of the squares of three numbers is 29, the sum of the products of them taken two together is 26, and the first is 5 less than the sum of the other two. Find the numbers.

45. A general drew up his army in the form of a square and found he had 615 men over; he then increased the side of the square by 5 men, and lacked 60 men to complete the square. How many men were in the army?

46. A man has a farm of 150 acres, in the form of a rectangle, whose length is to its breadth as 5 to 3. A road of uniform width, containing 3340 acres, surrounds the farm, and is a part of it. How wide is the road?

47. The fore-wheel of a carriage makes 88 revolutions more in going a mile than the hind-wheel, but if the circumference of the fore-wheel were diminished 2 feet the fore-wheel would make 220 revolutions more than the hind-wheel. Required the circumference of each wheel.

48. A merchant gains annually 20 per cent of his capital; of this he spends $1000, and adds the balance to his capital for the next year; at the end of 4 years his stock is $25,736. What was his original stock?

49. A man starts at the foot of a mountain to walk to its top. During the first half of the distance he walks 1⁄2 a mile an hour faster than during the last half, and he reaches the top in 4 hours 24 minutes. Returning, he walks 1⁄2 a mile an hour faster than during the latter half of his ascent, and completes the descent in 4 hours. Find the distance to the top of the mountain.

50. A lump of gold 22 carats fine contains 36 ounces of alloy. How many ounces of alloy in a lump of the same weight only 16 carats fine?

51. In a mile walk, A gives B a start of 1 minute and overtakes him at the mile-post. In a second trial, A gives B a start of 60 yards, and beats him 10 seconds. At the rate of how many miles an hour does cach walk?

52. If the cost of an article had been 8% less, the gain would have been 10% more. Find the gain per cent.

53. A railway-train, after traveling for 1 hour, has an accident which delays it 60 minutes, after which it proceeds at 3% of its former speed, and arrives at its destination 3 hours behind time. Now, had the accident occurred 50 miles farther on, the train would have arrived 134 hour sooner. What is the length of the line?

54. Two men, A and B, engaged to work for a certain number of days at different rates. At the end of the time, A, who had been idle 4 days, received 75 shillings; but B, who had been idle 7 days, received only 48 shillings. Now, had B been idle only 4 days, and A 7 days, they would have received the same sum. For how many days were they engaged?

55. Three pipes, A, B, and C, can fill a cistern in one hour. B delivers twice as much water per minute as A. C alone will fill it in one hour less than B alone. long will it take each to fill it?

How

General Definitions.

1. Quantity is anything that may be increased, diminished, and measured.

2. Quantity is estimated by assuming some definite portion of it as a standard of measure, and finding how many times it contains this standard.

3. Any definite portion of quantity assumed as a standard of measure is a unit.

4. Number is that which denotes how many units a quantity contains.

5. A quantity that contains a definite number of units is a specific quantity; as, five pounds.

6. A quantity that may contain any number of units is a general quantity; as, a flock.

7. The number of units in a specific quantity is expressed by one or more of the figures of arithmetic.

8. The number of units in a general quantity is expressed by one or more of the letters of the alphabet, or by both figures and letters.

9. By a figure of speech, the representation of the number of units in a quantity, by figures or letters, is also called a quantity.

10. When the number of units in a quantity is denoted by figures, the expression is called a numerical quantity.

11. When the number of units in a quantity is represented wholly or partially by letters, the expression is called a literal quantity.

12. Quantities which are opposed to each other in character - that is, which tend to destroy each other when combined—are positive and negative quantities.

13. Of two opposite quantities, it does not matter which is considered positive and which negative, if consistency is maintained throughout the operation or investigation into which they enter.

14. A positive quantity is characterized by placing before it the symbol + (plus); and a negative quantity, by placing before it the symbol (minus). This peculiar notation gives rise to symbolized numbers.

15. Arithmetic is the science of numbers, irrespective of their character as positive or negative. Arithmetic based on the literal notation is Literal Arithmetic.

16. Algebra is the science of symbolized numbers as the representatives of positive and negative quantities.

Principles.

1. The algebraic sum of two or more similar terms with like signs equals their arithmetical sum with the same sign (page 16).

2. The algebraic sum of two similar terms with unlike signs equals their arithmetical difference with the sign of the greater (page 16).

3. The algebraic sum of two or more dissimilar terms equals a polynomial composed of those terms (page 17).

4. The algebraic difference of two quantities equals the algebraic sum obtained by adding to the minuend the subtrahend with the sign changed (page 23).

5. The product of two quantities with like signs is positive (page 27).

6. The product of two quantities with unlike signs is negative (page 27).

7. The exponent of a factor in the product equals the sum of its exponents in the multiplicand and multiplier (page 28).

8. Multiplying one factor of a quantity multiplies the quantity (page 28).

9. Multiplying every term of a quantity multiplies the quantity (page 31).

10. The quotient of two quantities with like signs is positive (page 33).

11. The quotient of two quantities with unlike signs is negative (page 33).

12. The exponent of a factor in the quotient equals the difference of the exponents of the factor in the dividend and divisor (page 34).

13. Any quantity with an exponent of zero equals unity (page 34).

14. Dividing one factor of a quantity divides the quantity (page 35).

15. Dividing every term of a quantity divides the quantity (page 37).

16. If the same quantity or equal quantities be added to equal quantities, the results will be equal (page 39).

17. If the same or equal quantities be subtracted from equal quantities, the results will be equal (page 39).

18. If equal quantities be multiplied by the same or equal quantities, the results will be equal (page 39).

19. If equal quantities be divided by the same quantity or equal quantities, the results will be equal (page 40).

20. A term may be taken from one member of an equation to the other, if its sign be changed (page 40).

21. If both members of a fractional equation be multiplied by a common denominator of its terms, it will be cleared of fractions (page 40).

22. If the sign of every term of an equation be changed, the members will still be equal (page 41).

23. If a number of terms are inclosed by a parenthesis preceded by plus, the symbol and the sign before it may be removed without altering the value of the expression (page 50).

24. If a number of terms are inclosed by a parenthesis preceded by minus, the symbol and the sign before it may be removed, if the sign of every term inclosed be changed (page 51).

25. Any number of terms may be inclosed by a parenthesis and preceded by plus, without changing the value of the expression (page 51).

26. Any number of terms may be inclosed by a parenthesis and preceded by minus, if the sign of every term inclosed be changed (page 51).

27. An even power of a positive or a negative quantity is positive (page 62).

28. An odd power of a quantity has the same sign as the quantity (page 63).

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