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Note, If the first and third terms consist of different denominations, reduce them both to the same; and if the second term be a compound number, it is mostly convenient to reduce it to the lowest denomination mentioned. If, after division, there be any remainder, reduce it to the next lower denomination, and divide by the same divisor as before, and the quotient will be of this last denomination. Proceed in the same manner with all the remainders, till they be reduced to the lowest denomination which the second term admits of, and the several quotients taken together will be the answer required.

Note also, The reason for the foregoing Rules will appear when we come to treat of the nature of Proportions. Sometimes also two or more statings are necessary, which may always be known from the nature of the question.

An engineer having raised 100 yards of a certain work in 24 days with 5 men, how many men must he employ to finish a like quantity of work in 15 days?

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COMPOUND PROPORTION teaches how to resolve such questions as require two or more statings by Simple Proportion; and that, whether they be Direct or Inverse.

In these questions, there is always given an odd number of terms, either five, or seven, or nine, &c. These are distinguished into terms of supposition and terms of demand, there being always one term more of the former than of the latter, which is of the same kind with the answer sought.

RULE.-Set down in the middle place that term of supposition which is of the same kind with the answer sought. Take one of the other terms of supposition, and one of the demanding terms which is of the same kind with it; then place one of them for a first term, and the other for a third, according to the directions given in the Rule of Three. Do the same with another term of supposition, and its corresponding demanding term; and so on if there be more terms of each kind; setting the numbers under each other which fall all on the left-hand side of the middle term, and the same for the others on the right-hand side. Then to work.

By several Operations.-Take the two upper terms and the middle term, in the same order as they stand, for the first Rule of Three question to be worked, whence will be found a fourth term. Then take this fourth number, so found, for the middle term of a second Rule of Three question, and the next two under terms in the general stating, in the same order as they stand, finding a fourth

term from them; and so on, as far as there are any numbers in the general stating, making always the fourth number resulting from each simple stating to be the second term of the next following one. So shall the last resulting number be the answer to the question.

By one Operation. Multiply together all the terms standing under each other, on the left-hand side of the middle term; and, in like manner, multiply together all those on the right-hand side of it. Then multiply the middle term by the latter product, and divide the result by the former product, so shall the quotient be the answer sought.

How many men can complete a trench of 135 yards long in 8 days, when 16 men can dig 54 yards in 6 days?

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A FRACTION, or broken number, is an expression of a part, or some parts, of something considered as a whole.

It is denoted by two numbers, placed one below the other, with a line between them:

3 numerator thus,

4 denominator

}

which is named three-fourths.

The Denominator, or number placed below the line, shows how many equal parts the whole quantity is divided into; and represents the Divisor in Division. And the Numerator, or number set above the line, shows how many of those parts are expressed by the Fraction; being the remainder after division. Also, both these numbers are, in general, named the Terms of the Fractions.

Fractions are either Proper, Improper, Simple, Compound, or Mixed.

A Proper Fraction is when the numerator is less than the denominator; as, or, or, &c.

An Improper Fraction is when the numerator is equal to, or exceeds, the denominator; as, or, or , &c.

A Simple Fraction is a single expression denoting any number, of parts of the integer; as, or 3.

A Compound Fraction is the fraction of a fraction, or several fractions connected with the word of between them; as of, or of of 3, &c.

A Mixed Number is composed of a whole number and a fraction together; as 31, or 124, &c.

A whole or integer number may be expressed like a fraction, by writing 1 below it, as a denominator; so 3 is, or 4 is 1, &c.

A fraction denotes division; and its value is equal to the quotient obtained by dividing the numerator by the denominator; so 12 is equal to 3, and 20 is equal to 4.

Hence, then, if the numerator be less than the denominator, the value of the fraction is less than 1. If the numerator be the same as the denominator, the fraction is just equal to 1. And if the numerator be greater than the denominator, the fraction is greater than 1.

REDUCTION OF FRACTIONS.

REDUCTION OF FRACTIONS is the bringing them out of one form or denomination into another, commonly to prepare them for the operations of Addition, Subtraction, &c., of which there are several cases.

To find the greatest common measure of two or more numbers. The Common Measure of two or more numbers is that number which will divide them both without a remainder: so 3 is a common measure of 18 and 24; the quotient of the former being 6, and of the latter 8. And the greatest number that will do this, is the greatest common measure: so 6 is the greatest common measure of 18 and 24; the quotient of the former being 3, and of the latter 4, which will not both divide farther.

RULE.-If there be two numbers only, divide the greater by the less; then divide the divisor by the remainder; and so on, dividing always the last divisor by the last remainder, till nothing remains; then shall the last divisor of all be the greatest common measure sought.

When there are more than two numbers, find the greatest common measure of two of them, as before; then do the same for that common measure and another of the numbers; and so on, through all the numbers; then will the greatest common measure last found be the answer.

If it happen that the common measure thus found is 1, then the numbers are said to be incommensurable, or to have no common

measure.

To find the greatest common measure of 1998, 918, and 522.
So 54 is the greatest common measure

918) 1998 (2
1836

of 1998 and 918.

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So that 18 is the answer required.

To abbreviate or reduce fractions to their lowest terms. RULE.-Divide the terms of the given fraction by any number that will divide them without a remainder; then divide these quotients again in the same manner; and so on, till it appears that there is no number greater than 1 which will divide them; then the fraction will be in its lowest terms.

Or, divide both the terms of the fraction by their greatest common measure, and the quotients will be the terms of the fraction required, of the same value as at first.

That dividing both the terms of the fraction by the same number, whatever it be, will give another fraction equal to the former, is evident. And when those divisions are performed as often as can be done, or when the common divisor is the greatest possible, the terms of the resulting fraction must be the least possible.

1. Any number ending with an even number, or a cipher, is divisible, or can be divided by 2.

2. Any number ending with 5, or 0, is divisible by 5.

3. If the right-hand place of any number be 0, the whole is divisible by 10; if there be 2 ciphers, it is divisible by 100; if 3 ciphers, by 1000; and so on, which is only cutting off those ciphers.

4. If the two right-hand figures of any number be divisible by 4, the whole is divisible by 4. And if the three right-hand figures be divisible by 8, the whole is divisible by 8; and so on.

5. If the sum of the digits in any number be divisible by 3, or by 9, the whole is divisible by 3, or by 9.

6. If the right-hand digit be even, and the sum of all the digits be divisible by 6, then the whole will be divisible by 6.

7. A number is divisible by 11 when the sum of the 1st, 3d, 5th, &c., or of all the odd places, is equal to the sum of the 2d, 4th, 6th, &c., or of all the even places of digits.

8. If a number cannot be divided by some quantity less than the square of the same, that number is a prime, or cannot be divided by any number whatever.

9. All prime numbers, except 2 and 5, have either 1, 3, 7, or 9, in the place of units; and all other numbers are composite, or can be divided.

10. When numbers, with a sign of addition or subtraction between them, are to be divided by any number, then each of those num10 + 8 4 2

bers must be divided by it. Thus,

=5+42=7.

11. But if the numbers have the sign of multiplication between

10 × 8 × 3

them, only one of them must be divided. Thus, 6 × 2

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144) 240 (1 Therefore 48 is the greatest common measure, and

g the answer, the same as before.

144

48) 244

144 =

96) 144 (1

96

48) 96 (2

96

To reduce a mixed number to its equivalent improper fraction. RULE.-Multiply the whole number by the denominator of the fraction, and add the numerator to the product; then set that sum above the denominator for the fraction required.

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To reduce an improper fraction to its equivalent whole or mixed

number.

RULE.—Divide the numerator by the denominator, and the quotient will be the whole or mixed number sought.

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