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dividend is the product of its divisor and quotient, it must contain the same factors as its divisor and quotient.

COR. 1.-The same is true if one or both divisor and quotient be fractional; for when reduced to a common denominator, their numerators may be regarded as integral.

COR. 2.—Every factor of the divisor will cancel the same factor in the dividend.

CoR. 3.—The factors which are not canceled by those of the divisor will be the factors of the quotient.

COR. 4.-Canceling a factor in the dividend divides the quotient by the same factor.

Cor. 5.-Canceling a factor in the divisor multiplies the quotient by the same factor.

PROBLEMS. 1. Divide 648 by 36. 648 2, 2, 2, 3, 3, 8, $.

= 18, Ans. 36

2, 2, 3, 3. 2. Divide 625 by 125.

625 $,$,$, 5.
125

= 5, Ans.

$, $,$. 3. Divide 500 by 100.

500

5, Ans.

100 4. A man bought 30 yards of cloth at $5 a yard; he then exchanged it for other cloth at $3 a yard. How many yards of the latter did he get ?

PRACTICAL EXAMPLES.

1. A man bought 30 cows at 25 dollars each; he then exchanged the cows for horses at 50 dollars each ; how many horses did he get ?

Ans. 15 horses. 2. A farmer sold 1500 bushels of wheat at 125 cents per bushel, and received in return barley at 75 cents per bushel. How many bushels of barley did he get ?

Ans. 2500 bu. barley. 3. How many acres of land, at 25 dollars per acre, can be obtained for 5 houses and lots at 750 dollars each ?

Ans. 150 acres. 4. How many yards of flannel, three-quarters of a yard wide, will line a coat made of 3 yards of cloth six-quarters wide ?

Ans. 6 yards. 5. Sold 320 acres of land at 60 dollars per acre, and invested the proceeds in other land at 40 dollars per acre; how many acres did I get ?

Ans. 480 acres. 6. Exchanged 432 pieces of cloth at 18 dollars each, for linen at 6 dollars a piece. How many pieces of linen did I get ?

Ans. 1296 pieces. 7. Sold a farm of 477 acres at 48 dollars per acre, and invested the returns of sale in another farm at $36 per acre. How many acres did I buy? Ans. 636 acres.

8. Sold 15 horses at 732 dollars each, and bought sheep for the proceeds at 4 dollars each. How many sheep did I buy?

Ans. 2745. 9. Multiply the following numbers: 12, 15, 27, 28, 32, and divide the product by 2, 3, 4, 5, 6, 7, 8, and 9.

Ans. 12.

FRACTIONS.

DEF. 1.- If a unit or any other number is divided into equal parts, one or more of these parts is a fraction of the whole, and all the parts constitute the whole. If a unit is divided into two equal parts, each part is called one-half, and is written t; and the two halves constitute the whole; thus, t =1. If a unit is divided into three equal parts, each part is one-third (}); two of the parts, f; and the three thirds constitute the whole; thus,

= 1. If 5 is divided into two equal parts, each part is five-halves (); two of the parts, 10 = 5, etc.; if 6 is divided into three equal parts, each part is f= 2; and two-thirds of 6 is 4, etc. Fourths, fifths, sixths, etc., are similarly constructed.

2. When a unit is divided into equal parts, any number of the parts less than the whole, expressed fractionally, is called a Proper Fraction; as, 1, $, , , etc. The quotient is the same in division when the dividend is less than the divisor; as, t, 1t, etc.; but when the divisor is less than the dividend, the quotient is called an Improper Fraction; as, 1, 5,

, etc.

3. When the division indicated by an Improper Fraction is performed, and the divisor is not contained an exact number of times in the dividend, the quotient is partly integral and partly fractional, and is termed a

Mixed Number; thus, = 11; 50 = 8; and 2,5 = 10%.

COR.—The denominator expresses the number of parts into which a unit or any other number is divided, and the numerator expresses the number of parts of a unit taken, or the number divided.

EXEMPLIFICATION.-If each half is divided into two equal parts, the whole number of parts is four, and the one-half has made two of those parts; hence, =*; if each half is divided into three equal parts, the whole number of parts is six, and f =*;.: 1=*=== Po, etc., and f=*=f=t=1s, etc.; hence, if both terms of a fraction are multiplied by the same number, the value of the fraction is not changed, and by this principle fractions are reduced to a Common Denominator.

PROBLEMS.

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1. Reduce j and to a common denominator.

3x3, and 2. Reduce 3, 5, and x to a common denominator.

3x8, 40, and 3x3s. 3. Reduce 4 and to a common denominator.

4213, and . COR. 1.- The least common multiple of all the denominators is the least common denominator.

COR. 2.-Fractions are reduced to a common denominator thus: Multiply both terms of each fraction by the quotient obtained by dividing its denominator into the least common denominator ; or, when all the denom

inators are prime to each other, multiply both terms of each fraction by all the other denominators.

COR. 3.-An integral number is reduced to a fraction by multiplying it by the denominator of the fraction.

EXAMPLES. 1. Reduce }, }, and to a common denominator.

Ans. 48, 4, and 48. 2. Reduce 1, /, and to a common denominator.

Ans. , , and 3. Reduce , }, and s to a common denominator.

Ans. 244, 444, and 4 4. Reduce }, }, }, and is to a common denominator.

Ans. 416, 447, 347, and 39:40 5. Reduce 1, 2, Tło, and Toon to a common denominator.

Ans. 125, 14%, 17%, and Tobo. EXEMPLIFICATION.—Since ==== P, etc., and f=*=*=*=, etc., and 1, 31, 1951, etc., and 3, 1, 243}, etc.; hence, if both terms of a fraction are divided by the same number, the value of the fraction is not changed.

COR.-When both terms of a fraction have a common factor, it may be canceled, and the fraction is thereby reduced to its lowest terms.

REM.—The common factor may be either a prime or a composite number, and it is the greatest common divisor of the terms.

EXAMPLES. 1. Reduce as to its lowest terms. Ans. ** 2. Reduce }} to its lowest terms.

Ans. 1. 3. Reduce 18 to its lowest terms.

Ans. 131

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