$30 more; and the chaise cost twice the price of the horse. What did he give for each ?

Ans. For the harness $50; horse $130; chaise $260. 14. Two men talking of their ages, the first says your age is 18 years more than mine, and twice your age is equal to three times mine. What is the age of each?

Ans. Youngest 36 years. Eldest 54 years.

15. A boy had 41 apples which he wished to divide among three companions as follows; to the second, twice as many as to the first, and 3 apples more; and to the third, three times as many as to the second, and 2 apples more. How many did he give to each?

Ans. To the first 3; second 9; third 29.

16. How many gallons of wine, at 9 shillings a gallon, must be mixed with 20 gallons at 13 shillings, so that the mixture may be worth 10 shillings a gallon?

Ans. 60 gallons.

17. Two persons, A and B, have each an annual income of $400. A spends, every year, $40 more than B; and, at the end of 4 years, they both together save a sum equal to the income of either. What do they spend annually? Ans. A $370; B $330.

FRACTIONS.

$65. All the division which the pupil has as yet performed, has been the division either of numeral quantities, or of the numeral co-efficients. But in Algebra, it is frequently necessary to divide literal quantities. For example, after having made a to stand for an unknown quantity, we may wish to find the half of x, or the third of x, or the fourth of x, &c.

§66. In common arithmetic, if we wish to divide 1 by 2, we do it by writing 2 under the 1; thus, . So if we wish to divide 2 by 3, we write 3 under the 2; thus, . In the same manner, 2 divided by 5 is written ; 34 is written; 6÷7 is written. The quantities that are obtained by dividing in this manner, are called fractions.

§67. In Algebra, we most generally make use of this method of dividing; especially when we divide literal quantities. Or, in other words, we divide a literal quantity by writing the divisor under the dividend, with a straight line between them; thus, a divided by 2, is written ; and is read, x-half. x÷3, is written; and is read, x-third; x÷4, is written, and is read x-fourth; 3x÷4, and is read, 3x-fourth.

§68. The two separate numbers that we employ in writing a fraction, are called terms. The upper term is called the numerator, and the lower term is called the denominator. Thus, in the fraction, we call x the numerator, and 3 the denominator.

§69. If the one-third of x is, it is evident that

2x

is two times as much; that is

If of x is

of x, then 3 of

3x

x is Whence the rule to multiply a whole number by a fraction, is, to multiply the whole number by the numerator, and divide by the denominator; as of x is; of y is 34; 3 of a is 24.

3y

3

9

Examples. The pupil may multiply a, x, and y, each of them by; and then by ; and then by 3, 3, 4, 5, 4, 4, 흑, successively.

§70. As we can multiply a number of parts as well as a number of wholes, and as the denominator is nothing more than the name of the parts; it is plain, that to multiply a fraction, we multiply the numerator, and retain the denominator without alteration. Thus, 2 times is ; 3 times is ; 2 times is 2; 4 times 24 is ; &c.

15

2x

2

8a

Examples. Multiply each of the following fractions by 2, then by 3, and then by 4. 3, 3, 3, 4, 1, 2, 3, 3, 3, 6 ཙོ, 7, (,2, 3,,『, 2x 3x 2x 3x, 4x.

3' 4' 5' 5' 7

=

§71. As we know that 2 halves = a whole, we readily conclude that 4 halves = 2 wholes; and that 6 halves = = 3 wholes; &c. Likewise, because 3 thirds a whole, 6 thirds must equal 2 wholes; and 9 thirds must equal 3 wholes. In the same manner 8 fourths =2 wholes; 20 fifths = 4 wholes; 18 thirds = 6 wholes; &c. Such fractions are called improper fractions.

§72. Hence, in order to find how many whole ones there are in any number of halves, we have only to see how many times two halves are contained in that number. Thus, in 10 halves there are as many whole ones as there are 2 halves contained in 10 halves; which is 5. In the same manner, in 12 thirds there are as many whole ones as there are 3 thirds contained in 12 thirds; which is 4.

$73. Thus we have the rule, to change an improper

fraction to a whole number, divide the numerator by the denominator.

When the answer consists of an integer and a fraction, it is called a mixed number.

Examples, 1. How many whole ones in.

Ans. 8÷3-23.

2. How many whole ones in ? 12 ? 14 ? 20 ? 25 ?

3. How many whole ones in

4. How many whole x's in

5. How many whole x's in

2

20? 26 ? 29? 36 ? 40?

3

4

3

10

2

r 30.x

3

4

6. How many whole x's in 3 times 2?

7. How many whole x's in 4 times?

3x

8. How many whole x's in 5 times ?

5

§74. If we have the quantity, we know that, as it takes 5 fifths to make a whole one, it will take 5 times this quantity to make a whole x. Therefore, if we multiply by 5, we shall obtain 5*, or exactly x. If we multiply by 3, we shall obtain , or, which is the same, x. multiply by 4, we shall obtain 4

three times as much, or 3x. As 3 times is equal to x;

2r

3.x

so 3 times must be twice as much, or 2x. So 5 times 5 must be 3 times as much as 5 times; and therefore is 3x.

$76. Any fraction when multiplied by the number which is the same as the denominator, will produce a quantity

which is the same as the numerator. Thus,

We shall be able to make use of this principle in the solution of many equations, if we operate in accordance with the following axiom or self-evident truth.

$77. If equals be multiplied by the same, their products will be equal. Thus, if x=10, then 2x=20; 4x=40;

&c.

EQUIVALENT FRACTIONS.

§78. It is evident [§71,] that each of the following fractions, 1, 3, 4, 5, 6, 7, 8, 3, &c., is equal to 1. Therefore, they must be equal to one another. Also, each of the following fractions,,,,,, &c., is equal to 2; and consequently they are all equal to one another. In the same manner, we may make many fractions that will equal 3; and so of any other number.

§79. Let us take from the first set of the above fractions, and which are equal to one another. We see that both 444 the numerator and denominator in the last fraction are twice as much as in the first. We see the same fact in the equal fractions and&; and also in the equal fractions 4 and §. We find the same, by taking from the second set, the equal fractions and; and also 1⁄2 and &; and also § and 12.

§80. Again in the equal fractions, and, we find each term in the last fraction three times as great as the correspondent term in the first fraction. The same may be observed in the fractions and; and also in 2 and §; and also in and.

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