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PROBLEMS.

1. What is the least number of oranges that can be divided equally among 21, 24 or 30 boys ?

2. Find the least number of acres in a farm that can be divided exactly into lots of 12, 15 or 18 acres.

3. What is the smallest sum of money I can consume in paying workmen 12, 14, 16, 18 or 20 dollars a week ?

4. A man desires to purchase a piece of cloth that can be cut without waste into parts 5, 6, or 8 yards long. How many yards must the piece contain ?

5. How many bushels does the smallest bin contain that can be emptied by taking out 7 bushels, 10 bushels, or 30 bushels at a time?

6. If 5 boys start together and run around a square in 12, 15, 16, 18 and 20 minutes respectively, in how many minutes will they all meet at the starting-point if they continue their course around the square?

7. How many quarts does the smallest vessel hold that can be filled by using a 3-quart measure, a 4-quart measure, a 5-quart measure or a 6-quart measure ?

8. What is the smallest sum of money that can be wholly expended in buying horses at $75, cows at $50, or sheep at $9?

9. What is the shortest piece of cord that can be cut into pieces 10, 12, 15, 16 or 18 feet long?

10. A heap of pebbles can be made up into groups of 25; but when made up into groups of 18, 27 or 32 there is in each case a remainder of 11. Find the least number of pebbles such heap can contain.

11. What is the smallest number that can be divided by 360, 460, 636, and 748, respectively, and leave a remainder of 260 ?

REVIEW.

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1. Divide 5 X 15 X 80 X 56 X 91 by 10 X 5 X 16 X 78.

2. Divide 18 X 15 X 90 by 12 x 27 x 25.

3. Find the greatest number that will divide each of the two numbers 849 and 1132, and explain the process.

4. What is the L. C. Dd. of 4, 9, and 29?

5. How many barrels of sugar, 240 pounds each, at 5 cents a pound, can be exchanged for 8 pieces of sheeting, of 45 yards each, at 10 cents a yard ?

6. How much does the L. C. Dd. of 1751 and 2369 exceed their G. C. D.?

7. Find the difference between the G. C. D. of 1, 3, 5, 7, 9, and the L. C. Dd. of 2, 4, 6, 8, 10.

8. Find the least number of oranges that, arranged in groups of 6, 7, 8, or 9, have just 5 over in each case.

9. I have just money enough to buy a whole number of dozens of oranges at $.40 a dozen, or a whole number of baskets of peaches at $1.25. How much money have I ?

10. Four boys start together to run around a square; the first can run around in 12 minutes, the second in 15 minutes, the third in 16 minutes, and the fourth in 18 minutes. How long will it be before they all meet at the starting point ? 11. Define the following terms: 1. Integer.

9. Factoring 2. Fraction.

10. Cancellation. 3. Factor.

11. Common divisor. 4. Exact divisor. 12. Greatest common divisor. 5. Prime number. 13. Dividend. 6. Composite number. 14. Common dividend. 7. Even number. 15. Least common dividend. 8. Odd number. 16. Prime to one another.

12. Repeat the principles pertaining to Prime and Composite numbers; to the G. C. D.; to the L. C. Dd.

13. Repeat the rules for finding the L. C. Dd.

14. Invent 5 problems that will involve the use of the L. C. Dd.

FRACTIONS.

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INDUCTIVE STEPS. 1. Divide 5 apples into 2 equal parts.

5+2=2, with one apple remaining undivided. To indicate that one apple remains to be divided into 2 parts, we write it thus, ], and call the expression one-half. The exact quotient, therefore, of 5 • 2 is 21, read “two and one-half." 喜 1}, read “one and two-thirds." All the remainders,

" resulting from division, are commonly written over their divisors, and thus form numerical expressions called Fractions, because they denote parts of a unit.

2. It is the divisor that names the fraction. If the divisor is 2, one part (1) is called one-half; two parts (3) two halves. If the divisor is 3, one part () is called one-third ; two parts (3) two-thirds.

3. In like manner we have fourths, fifths, sixths, serenths, eighths, ninths, tenths, etc.

is read “five-fifths," and is equal to one. Any quantity divided by itself gives one for quotient.

4. į may denote one-half of a whole line. may denote

One-half. two-thirds of a line.

2 In this way we might illustrate fourths, fifths, sixths, etc.

Ž, Ž, 4, etc., are called fractional units.

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5. denotes how many fractional units ? 6. What is the value of į? 7. Read the following fractions: , , 5, 10, 11, 11, 18. Which has a decimal divisor? Which is equal to one?

8. Write two-thirds, four-ninths, seven-twelfths, ten-seventeenths. Write a fraction whose value is one.

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DEFINITIONS.
1. An Integral unit is a whole or undivided unit.

2. An Integer is a whole unit or a collection of whole units.

3. A Fractional unit is one of the equal parts into which the unit is divided.

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

As we have seen, a fraction is the expression of a division that cannot always be performed, and is written with the number to be divided (dividend) above a horizontal line, and the divisor below that line.

5. The number below the line is called Denominator, because, as we have seen, it names the fractional unit.

6. The number above the line is called Numerator, because it numbers the fractional units.

7. The Numerator and Denominator are called the Terms of the Fraction.

8. A Common fraction is expressed by writing both numerator and denominator, as in }, {}, 31.

9. A Decimal fraction is usually expressed by simply writing a point before the numerator, as in .5, .37, .25.

10. A Proper fraction has the numerator less than the denominator, as 2, 3, 1o, etc.

11. An Improper fraction has the numerator equal to or greater than the denominator, as 7, 11, 13, etc.

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12. A Mixed Number consists of an integer and a fraction, as 2}, 73, etc., read, “two and one-third, seven and three-fifths."

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EXERCISES.

Analysis examines the separate parts of a subject and their connection with one another.

1. Analyze 7.

9 is the denominator, is a divisor, makes and names the fractional unit. 7 is the numerator, is a dividend, and numbers the fractional units.

is a proper fraction, its terms are 7 and 9, and its value is less than

one.

5

11

8

5

5

4

19

3

9
32

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2. Analyze : , 3, 4, 4, , 15, 1, $, 1, 5, 13, 184, 13, 31, ot, 135, 14

3. Name the proper and improper fractions and mixed numbers among these : }, 37, 187, 435, 5, 7, 2, 113, 315, 130, 12, 13, 1814, . 1, 27100, 3, 4, 18, 31. 4. Write with figures : 1. Five-ninths.

11. Sixty-five hundredths.
2. Ten-elevenths. 12. 110 ninetieths.
3. Seven-twenty-firsts. 13. 211 eighths.
4. Six and two-thirds. 14. Thirty and seven-eighths.
5. Seventy-eightieths. 15. Five twenty-fifths.
6. Ninety-one ninetieths. 16. Fifteen-sixteenths.
7. 314-tenths.

17. Nine thirtieths.
8. 1898-millionths. 18. Four and two-fifilis.
9. Ten and five-sixths. 19. Threc-thirds.
10. Ten-tenths.

20. Two and twelve-twentieths. 5. Write : 1. A common fraction.

3. A proper fraction. 2. A decimal fraction. 4. An improper fraction.

5. An improper fraction equal to one. Point out the terms of the fractions you have written.

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