of ftating the queftion; as may be partly feen in the fourth and fixth examples, which, though quite different queftions from each other, yet confift of the very fame figures, and may be stated in the fame manner; but are here varied for the information of the learner, and admit of ftill greater variety, as the learner may prove at his leifure.

All the caution that is neceffary in linking the numbers together, is, that of every two numbers that are linked together, one must be greater and the other lefs than the rate or price of the mixture. Therefore, the first example in this rule, having but one number lefs than the rate of the mixture, admits of no other method of ftating than that defcribed.

SECT. XIII.

OF VULGAR FRACTIONS.

Reduction, Subtraction, Multiplication, Divifion, and the Rule of Three.

A FRACTION is any part or parts of an integer or unit, and (in vulgar fractions) is reprefented by two numbers placed one above the other, with a line drawn between them.

The number below the line is called the denominator, and fhews how many parts the integer is divided into; the num ber above the line is called the numerator, and shews how many of those parts the fraction represents.

Thus the fraction to reprefent three farthings is thus written Numerator and is properly called three-fourths of a

Denominator

penny-a penny being the integer, or unit, of which the fraction is a part.

Vulgar

Vulgar fractions are either proper or improper, fingle or compound.

A proper fraction has its numerator less than its denomi nator: as, two fixths, three fourths, three ninths; and always ftands for lefs than the integer it is taken from.

An improper fraction has the numerator equal to or greater than the denominator, as,,, and always reprefents as much or more than the integer.

A fingle fraction is only a fingle expreffion of any affigned parts of an integer.

A compound fraction confifts of more than one fraction," and is a fraction of a fraction, and they have the particle of placed between them, as of, of 3, &c.'

There are also mixed numbers, which are whole numbers united with fractions, as 84, 127, &c.

The common measure of two or more numbers, is that number which will divide each of them without a remainder ; thus 4 is the common measure of 12 and 16.

A number that can be exactly measured by two or more numbers is called their common multiple; and if it be the least number that can be measured, it is called their least common multiple: thus, 45 and 60 are the common multiples of both 3 and 5, but their least common multiple is 15.

Before the learner can proceed to reduction of fractions, it is neceffary that he be able to folve the two following Problems.

PROBLEM I.

To find the greatest common Measure of two or more. Numbers.

Rule. If there be only two numbers, divide the greater by the lefs and if there be no remainder, the divifor is the greatest common measure; but if there be a remainder, the divifor is to be divided by fuch remainder, and if there still

be

be a remainder, the last divisor is ftill to be divided by the laft remainder till there be no remainder; then the last divisor is the greatest common measure.

But if there be more than two numbers after having found the greatest common measure of two of them, the common measure of that common measure and one of the other numbers is to be found in the fame manner; and proceed in this manner through all the numbers: then the last common measure will be the common measure of each of them.

Example 1. What is the greatest common measure of 624, 3126, and 336?

Thus 6 is the greater common measure of 624, 3126, and 336.

2. What is the greatest common measure of 81 and 63 ? -Anfwer 9.

3. What is the greateft common measure of 720, 336, d 1736?-Answer 8.

PROBLEM II.

To find the leaf common Multiple of two or more Numbers.

Rule. Divide the numbers by any number that will divide two or more of them without a remainder, and set the quotient of each number under the dividend to which it belongs; bring down the undivided numbers, or those which cannot be divided, into the fame line with the quotient. Then

divide the second line, in the fame manner, by any number that will divide two or more of them, and bring down the quotients and undivided numbers, as before. Proceed in this manner as long as there can be found any two numbers that can be exactly divided by one number.

When the numbers are so far divided that there are no two numbers that can be divided by one, then multiply all the divifors and the quotients, continually together, and the product is the least common multiple.

Example 1. What is the least common multiple of 7, 8, 4, and 2?

Here I divide the numbers firft by 2, as that number exactly measures the 8, 4, and 2; the quotients I place under their dividends; I then divide the next line by 2, as that measures 4 and 2; the quotient I place below, and the I bring down.

• Then, as there are no two others which admit of a common measure, I multiply the divisors and quotients continually together, and the product 56 is the leaft common multiple

of them all.

Before fractions can be wrought by any of the rules in arithmetic, it is neceffary to reduce them from compound to fimple ones, and to bring them into feveral other forms; for which purpose there are eight ways of altering the form of fractions, without changing their value, as follow:

Cafe 1. To reduce a mixed number to an improper frac

tion.

Rule. Multiply the integer by the denominator of the fraction, and add thereto the numerator, and the product

will form a numerator to a fraction, whofe dénominator is the denominator of the former fraction.

Example 1. Reduce 10 gallons to an improper fraction.

2. Reduce 56 to an improper fraction.-Anftver 1245. To reduce an improper fraction to a whole or mixed number.

Cafe 2.

This is the reverfe of the former cafe.

Rule. Divide the numerator by the denominator, and the quotient will be the whole number, and the remainder (if any) will be the numerator to a fraction, whose denominator is the divifor.

Thus, to reverse the firft example in the former cafe, Example 1. Reduce 10 to its equivalent, whole, or mixed number.

22

2. Reduce the fecond example in the former cafe 124 to its equivalent, whole, or mixed number.—Answer 5611. Cafe 3. To reduce a fraction to its loweft terms.

Rule. Divide the two terms of a fraction by any number that will exactly divide them without a remainder, and then divide the quotients by a number that will exactly divide them, and fo on till there can be found no number that will exactly divide the last quotients; then thefe quotients will be the fraction reduced to its lowest terms.

Or, 2dly, find the greatest common measure of the numerator and denominator, as taught in the first problem in this fection, and divide them thereby; then the quoilents will be the fraction reduced to its leaft terms.