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16. How many days from the birth of Christ to Christmas, 1834, allowing the year to contain 365 days 6 hours, or 365 days? Ans. 669868d. 2=1=12h. 17. From March 2d to November 19th following, inclusive how many days? Ans. 262. 18. Suppose a rail-way car to run a mile in 31⁄2 minutes, how long would it be at that rate in running 1280 miles? Ans. 3d. 2h. 40m. 19. How many strokes does a regular clock strike in 365 days, or a year? Ans. 56940. 20. How long would it take to count the national debt of England, which is not less than $1900,000,000, at the rate of 50 dollars a minute, reckoning, without intermission, 12 hours a day, and 365 days to the year?

4 rods.

Ans. 144 years 217d. 9h. 20m. 21. There is a certain piece of land 4 rods square; how many rods does it contain? Ans. 16 sq. rods. Illustration.-A piece of land 1 rod long and 1 rod wide, contains just 1x1=1 square rod. A piece 2 rods long and 1 rod wide contains 2x12 sq. rods, &c.; and a piece 2 rods long and 2 rods wide contains 2 x2=4sq. rods. So a piece of land 4 rods long and 4 rods wide contains just 4×4=16 square rods. the annexed figure.

4 rods.

See

Therefore, to find the number of square feet, rods, acres, miles, &c., in any figure that has four right angles, (corners,) multiply the length and breadth together, and the product will be the answer.

22. How many square feet in a floor that is 14ft. long and 12 feet wide? Ans. 14 X 12=168. 23. How many acres in a piece of land 40 rods long and 20 rods wide? Ans. 800 rods=5 acres. 24. How many square feet in a floor that is 4 yards long and 4 yards wide, or 4 yards square? Ans. 144 sq. ft. Note. From the foregoing it will be seen that a square foot is a space a foot long and a foot wide, but of no thickA solid or cubic foot, is a foot long, a foot wide and

ness.

a foot thick. Thus, a block 2 feet long, 2 feet wide and 2 feet thick, contains 2×2×2=8 cubic feet, &c.

25. How many solid feet in a pile of wood 8 feet long, 4 feet wide and 4 feet high? Ans. 8X4×4=128. 26. How many cords of wood in a pile 8 feet long, 8 feet wide and 8 feet high? Ans. 4 cords. 27. How many cords of bark in a pile 9 feet long, 8 feet wide and 4 feet high ?—Thus, 9×8×4=288 feet, and 288 solid feet-2 cords 32 feet. Ans.

28. How many cubic inches are contained in a room 4 yards long, 4 yards wide, and 3 yards high? Ans. 2239488. 29. How much wood is contained in a load 8 feet long, 4 feet 6 inches high, and 3 feet 9 inches wide?

Ans. 1 cord 7 feet.

FRACTIONS.

I. Fractions or broken numbers are expressions for any part of a unit or whole number.

All fractions arise from the division of an integer or a unit into parts; thus, when a whole thing is divided into 2 equal parts, these parts are halves; when into 3 equal parts, they are thirds; when into 4 equal parts, they are fourths; when into 5 equal parts, they are fifths; when into 6 equal parts, they are sixths; when into 7 equal parts, they are sevenths; when into 8 equal parts, they are eighths; when divided into 9 equal parts, they are ninths, and so on: that is, the fraction takes its name or denomination from the number of parts into which the unit is divided. Thus, if the unit be divided into 15 equal parts, they are fifteenths; if into 20 equal parts, they are twentieths, &c.

Fractions are of two kinds, Vulgar, and Decimal.

II. A Vulgar Fraction is represented by two numbers placed one above the other, with a line drawn between them. 1 one 3 three 5 five 3 three 11 eleven Thus, 2 half. 4-fourths. 6-sixths. 8 eighths. 12 twelfths. III. The number below the line is called the denominator, because it denominates or shows how many equal parts the integer or whole is divided into.

IV. The number above the line is called the numerator, because it numerates or shows how many of those equal parts are expressed by the fraction.

In division, the denominator is the divisor; and the numerator the dividend, thus :

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1. If we divide an apple into 2 equal parts, what part of the whole apple will one of those parts be? If an apple be cut into 4 equal parts, what will one of those parts be? What part of an apple will 2 of those parts be?—will 3 be?

2. How many halves are equal to the whole of any thing? how many thirds? how many fourths? how many fifths? how many sixths? how many sevenths? how many eighths? how many ninths? how many twelfths? &c.

3. If 4 apples be equally divided among 6 boys, what part of 1 apple is each boy's share?

1 apple among 6 boys would be of an apple apiece; and 4 apples would be 4 times as much, or 4, the answer. 4. What part of 5 dollars is 1 dollar? 5. What part of 8 dollars is 5 dollars?

Ans. .

Ans.

6. If you cut an orange into 8 equal parts, what part of

the whole orange will 1 of those parts be?

7. If I divide an orange into 8 equal parts, and give 5 of those parts to Henry, and the other three to Eliza, what part of whole orange will each have ?

How are five eighths expressed?
How are three eighths expressed?
Which fraction is larger, or ?

V. Fractions are either proper, improper, single, compound, or mixed.

1. A simple or proper fraction is when the numerator is less than the denominator, and the fraction is less than a whole thing or unit; as,,,, 11, &c.

2. An improper fraction is when the numerator is equal to, or greater than the denominator, and the fraction is equal to, or greater than a unit or whole 1; thus, 3, 1, 7, Y.

Obs. 3 thirds of any thing are equal to the whole, and 4 halves, 7 fifths, and 12 fourths, are each greater than unity or 1; consequently these are called improper fractions.

3. A compound fraction is the fraction of a fraction, coupled by the word, of, thus; of, of of 5.

4. A mixed number is composed of a whole number and a fraction joined together, thus; 8, 163, 193, &c.

A whole number may be expressed as a fraction by drawing a line under it and putting 1 for a denominator; thus, 6, and 12=Y, &c.

PROBLEM I.

To reduce Fractions to their lowest terms.

The numerator and denominator together are called terms of a fraction; and may often be changed without altering the value of a fraction; thus, take an orange, or any thing, and divide it into 2 equal parts, and 1 of these parts will be of the orange again, if we divide it into 4 equal parts, it is evident that 2 of those parts (2) will be just of the orange and if we divide it into 8 equal parts, 4 of these parts (4) will be just equal to of the orange and the fractions,, and, are all equal in value, but expressed in different terms. Hence the terms of fractions may be changed without altering the value of the fraction; for if we multiply both the terms of the fraction by 2 it becomes 2, which is equal to : again, if we divide the terms by 2, the fraction will be, which is expressed in its lowest terms possible.

RULE.

1. Divide the terms of the fraction by any number that will divide them both without a remainder.

2. Divide these quotients again in the same manner, and so on, until no number greater than 1 will divide them.

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Thus 9), and 5)15=3, the answer.

2. Reduce 132 to its lowest terms.

3. Reduce

4. Reduce

264

432 to its lowest terms.

576

147 to its lowest terms.

168

Ans.

Ans.

Ans.

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49 to its lowest terms.

Ans.

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168 to its lowest terms.

Ans.

77

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84

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To change a Whole or Mixed Number to an Improper

Fraction.

RULE.

Multiply the whole number by the denominator of the fraction and to the product add the numerator; this sum written over the denominator will form the fraction required.

EXAMPLES.

1. In 27 dollars how many fourths of a dollar? Operation.

27 +4-fourths in 1 dollar. 108-fourths in 27 dollars. +3=fourths in .

111-fourths Ans. 111.

=

$1 4 fourths of a dollar, and 27 dollars=27 times 4 or 108 fourths, and 3 fourths added to 108 fourths make 111 fourths11 the Ans.

2. In 365 dollars, how many eighths of a dollar?

3. Reduce 45 to ninths.

Ans. 293
Ans. 405.

8

Ans. 53. Ans. 100.

4. Reduce 85 to an improper fraction, that is, reduce it to sixths.

5. Reduce 33 to an improper fraction. 6. Reduce 28 to a fraction having 12 for a denominator,

that is, reduce it to twelfths.

7. Reduce 45 to fifths.

8. Reduce 61956 to an improper fraction. 9. Reduce 84 to an improper fraction. 10. What improper fraction is equal to 5618 ?

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11. What improper fraction is equal to 1489 ?

12. What improper fraction is equal to 225 ?

Ans. 1189.

Ans. 1803,

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