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66. If a man receive $1200 salary, and pay $364 for board, $275 for clothing, $150 for books, and $187 for other expenses, how much can he save in 5 years?

67. A speculator gained $5760, and afterwards lost $2746; at another time he gained $3575, and then lost $4632. How much did his gains exceed his losses?

68. A mechanic earns $75 a month, but his necessary expenses are $50 a month. How long will it take him to pay for a farm of 60 acres worth $40 an acre?

69. A man worth $12750 received a legacy of $8000; he spent $456 in traveling, and gave $1000 to a college. How much money had he left?

70. A farmer bought a lot of sheep and hogs, of each an equal number, for $1276. He gave $4 a head for the sheep, and $7 a head for the hogs; what was the whole number purchased, and what was the difference in the total cost of each ?

71. The population of ten of the largest cities in the United States was as follows:

1870.

Cities.
1860.
Cities.
New York.....944,129 805,658 Baltimore.
Philadelphia...674,022 565,529 Boston....
Brooklyn.......396,300 266,661 Cincinnati
St. Louis.......
Chicago

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.310,864 160,773 New Orleans...191,322 168,675 .298,983 109,260 San Francisco..149,482 56,802 What was the total increase of these ten cities in ten years?

72. What was the average increase per year?

73. What was the increase of each city during the time? 74. How much larger was New York in 1870 than each of the other cities?

75. The area of the Pacific Ocean is 65,630,000 square miles; of the Atlantic Ocean, 34,780,000; of the Indian Ocean, 30,590,000. Which is the larger, the Pacific or the other two combined; and how much?

76. What is the area of a floor 27 feet long and 18 feet wide? What will it cost to cover the floor with carpeting a yard wide, at $2.25 a yard?

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78. If I sell a lady a set of furs for $45.50; 10 yd. of silk at $2.50 per yd. ; a bonnet for $7.50; a pair of gloves for $1.75; and an umbrella for $3.50, what is the amount of the bill?

79. Make out bill for the following: Jas. Smith bought of Johnson & Co., 25 lb. of coffee sugar at 11; 5 lb. of tea at 75; a box of crackers containing 25 lb. at 6; a box of soap for $1; and 3 lb. of coffee at 35.

80. If you go to the store of Jones & Co., and buy 5 yd. of cloth at $1.25 a yd.; 32 yd. of flannel at 60 cents a yard; 40 yd. of muslin at 11 cents; 1 pair of gloves for $2; 20 yd. of carpet at $1.10, what is the amount of your bill?

FRACTIONS.

224. A Fraction is one or more of the equal parts of a unit, or of anything regarded as a whole.

Review from Page 54 to Page 57 inclusive.

225. A Fractional Unit is one of the equal parts of a unit, or of anything regarded as a whole.

Fractions are classified as Proper or Improper. 226. A Proper Fraction is a fraction whose numerator is less than its denominator. Its value is less than a unit. Thus, t,, and are proper fractions.

227. An Improper Fraction is a fraction whose numerator equals or exceeds its denominator. Its value is equal to, or greater than a unit. Thus, &, 10, and 1 are improper fractions.

228. The Value of a fraction is the quotient of its numerator divided by its denominator; thus

= 3.

229. Since a fraction may indicate division, all changes in the terms of a fraction will affect the value of that fraction according to the laws of division (187), and may be embraced in the following

GENERAL LAW.

230. A change in the NUMERATOR produces a LIKE change in the value of the fraction; but a change in the DENOMINATOR produces an OPPOSITE change in the value of the fraction. (See Page 137.)

REDUCTION.

231. Reduction of Fractions is the process of changing their form without changing their value.

232. To reduce fractions to higher terms.

A fraction is reduced to Higher Terms, when the nu merator and denominator are expressed in larger numbers of the same comparative value.

233. PRINCIPLE.-Multiplying both terms of a fraction by the same number does not change the value of the fraction.

WRITTEN EXERCISES.

1. Reduce to sixteenths.

OPERATION.

16÷8= = 2

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ANALYSIS.

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Since the given denominator 8 is contained in the required denominator 2 times, multiply both terms of the fraction by 2. (PRIN.) Hence § = 18.

2. Change to a fraction whose denominator is 30. 3. Change to a fraction having 45 for its denominator. 4. Reduce to a fraction having 40 for its denominator.

RULE.-Divide the required denominator by the denominator of the given fraction, and multiply the terms of the fraction by the quotient.

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11. Change to a fraction, whose denominator is 120.

234. To reduce fractions to lower terms.

A fraction is reduced to its Lowest Terms, when its numerator and denominator are prime to each other.

235. PRINCIPLES.-Dividing both terms of a fraction by the same number does not change its value.

1. How many fourths are equal to ? ? 18? 2. Reduce to fifths; to eighths.

4. Change

3. What are the factors common to the terms of 1 ? ? to a fraction having lower terms; 14; 11. to a fraction having a denominator as

5. Change

great; as great; as great.

WRITTEN EXERCISES.

1. Reduce to its lowest terms.

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38 +12
= 1

36+12

ANALYSIS.-Dividing both terms of the given fraction by 4 the result is. (PRIN.) Again dividing both terms of by 3 the result is . Since the terms of are prime to

each other, the lowest terms of ag are 3.

The same result may be obtained more directly by dividing the terms of the given fraction by their greatest common divisor 12.

RULE.- Reject all factors common to the terms of the given fraction. Or,

Divide the terms of the given fraction by their greatest common divisor.

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