a number of mules, at 75 dollars a head, or a number of horses, at 125 dollars a head? and what number of each could I purchase for that sum ?

Ans. 375 dollars; 5 mules, or 3 horses. 17. A has 413 dollars, B 531 dollars, and C 590 dollars; and they agree to purchase oxen at the same price per head, provided each man can thus invest all his money. How many oxen can each man purchase?

Ans. A, 7; B, 9; and C, 10. 18. For what sum could I hire workmen, for one month, at 15 dollars, 21 dollars, or 24 dollars each, allowing the whole sum to be thus expended?

Ans. 840, or 1680 dollars, &c.

19. Three hundred and eighty-five Irishmen, 455 Frenchmen, and 700 Germans are to be ferried over a river. What are the largest equal companies into which they may all be divided, so that those of the same nation shall go over together? Ans. Companies of 35.

20. A, B, C, and D start together, and travel the same way around an island which is 600 miles in circuit; A goes 20 miles per day, B 30, C 25, and D 40. How long must their journeyings continue, in order that they may all come together again? Ans. 120 days.

CHAPTER IV.

FRACTIONS.

(57.) A FRACTION is one or more of the equal parts into which any quantity may be supposed to be divided.

One half is one of the two equal parts of any quantity; two thirds are two of the three equal parts of any quantity; and so on.

Any quantity consists of how many halves of that quantity? Of how many thirds? Of how many fourths? Of how many tenths? Of how many hundredths?

Numerator and Denominator.

(58.) The numerator of a Fraction is the number of equal parts in the fraction: the denominator shows the number of such parts in the whole quantity divided.

Thus in, three fourths, 3 is the numerator, and 4 the denominator.

In the fraction, which number is the numerator? and what does it show? Which the denominator? and what does it show? In ? In ? The numerator and denominator are called the terms of the fraction

Fractions express Division.

(59.) Every fraction is equal to its numerator divided by its denominator.

Thus the fraction is the quotient of 4 divided by 9, since it expresses the part that 4 is of 9, (34). Hence also

Every fraction is equal to that part of its numerator which is denoted by the reciprocal of its denominator (35).

is equal to of 4; four ninths of any quantity is of 4 such quantities.

ğ of 1 is what part of 5?

of 1 is what part of 7 ?

of 1 is what part of 3 ?

of 1 is what part of 5 ?

Proper and Improper Fractions.

(60.) A proper fraction is one whose numerator is less than its denominator; and whose value is therefore less than a unit or whole one.

2 3

,,, &c. are proper fractions.

(61.) An improper fraction is one whose numerator is equal to, or greater than, its denominator. Its value in units is found by dividing its numerator by its denominator.

Thus is an improper fraction; and its value in units is 2 (59).

What is the value in units of ğ?
What is the value in units of 17?
What is the value in units of 35?

2

What is the value of §1
What is the value of 24?
What is the value of 42?

Integral and Mixed Numbers.

(62.) An integral number, or simply an integer, is a number in which there is no fraction; as 1, 3, 5, &c.

A mixed number consists of an integer and a fraction: as 51.

(63.) An integer is made an improper fraction by taking any number for a denominator, and multiplying the integer by that number for a numerator.

6 9

Thus the integer 3 is equal to 3,,, or 12, &c.

The integer 4 is equal to how many halves?—how many thirds? The integer 5 is equal to how many thirds?-how many sixths?

(64.) A mixed number is made an improper fraction, by multiplying its integer by the denominator annexed, and adding the numerator to the product, for a numerator to be placed over said denominator.

Thus 33 is equal to 17; the numerator 17 being 5 × 3+2.

Constant value of à Fraction.

(65.) The value of a Fraction remains the same, when its numerator and denominator are both multiplied, or both divided, by the same number.

For if any quantity were divided into 4 fourths, each one of these fourths, divided into two equal parts, would make 2 eighths of the quantity; then 3 fourths would make 6 eighths.

is equal to how many 10ths?-how do you prove it?
is equal to how many 18ths?—how do you prove it?

The preceding principle also follows from regarding a Fraction as the quotient of its numerator divided by its denominator (59) (38).

REDUCTION OF FRACTIONS.

(66.) The reduction of a quantity, in general, consists in changing its expression, without altering its value.

Thus the mixed number 33 may be reduced to the improper fraction 17. (64).

A fraction is reduced to its lowest term when its numerator and denominator are made the smallest that will express the value of the given fraction.

Thus reduced to its lowest terms is 3,-found by dividing 27 and 36 both by 9, which is their greatest common measure (65).

RULE XII.

(67.) To reduce a Fraction to its Lowest Terms.

1. Divide both terms of the fraction by their greatest common measure; the quotients will be the lowest terms of the given fraction. Or

2. Divide both terms by any common measure, and the quotients by any common measure, and so on, until the quotients become prime to each other.

EXAMPLE.

To reduce to its lowest terms.

90

The greatest common measure of 90 and 120 is 30 (53), by which we divide both terms of the given fraction. This does not alter the value of the fraction (65).

Or, we might divide, successively, by 10 and 3 (44).