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COMPOUND PROPORTION.

COMPOUND PROPORTION is a rule by means of which the student may resolve such questions as require two or more statings in simple proportion.

The general rule for questions of this kind may be exhibited in the following precepts: viz.

1. Set down the terms that express the conditions of the question in one line.

2. Under each conditional term, set its corresponding one, in another line, putting the letter a in the (otherwise) blank place of the term required.

3. Multiply the producing terms of one line, and the produced terms of the other line, continually, and take the result for a dividend.

4. Multiply the remaining terms continually, and let the product be a divisor.

5. The quotient of this division will be a, the term required.*

Note. By producing terms are here meant whatever necessarily and jointly produce any effect; as the cause and the time; length, breadth, and depth; buyer and his money; things carried, and their distance, &c. all necessarily inseparable in producing their several effects.

In a question where a term is only understood, and not expressed, that term may always be expressed by unity.

A quotient is represented by the dividend put above a line, and the divisor put below it.

EXAMPLES.

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

*This rule, which is as applicable to Simple as to Compound Proportion, was given, in 1706, by W. Jones, Esq. F.R.S., the father of the late Sir W. Jones.

Here 16 men and 6 days, are the producing terms of the first line, and 135 yards, the produced term of the other. Therefore, by the rule,

If a garrison of 3600 men have bread for 35 days, at 24 oz each a day: How much a day must be allowed to 4800 men, each for 45 days, that the same quantity of bread may serve ?

AN EXAMPLE IN SIMPLE PROPORTION.

If 14 yards of cloth cost 211, how many yards may be bought for 737 10s?

yds.
14

= of 73=49 yards, Answer.

2. If 100 in one year gain 51 interest, what will be the interest of 7501 for seven years? Ans. 2621 10s. 3. If a family of 8 persons expend 2001 in 9 months; how much will serve a family of 18 people 12 months?

Ans. 6007.

4. If 27s be the wages of 4 men for 7 days; what will be the wages of 14 men for 10 days?

If a footman travel 130 miles in 3 days, are 12 hours long; in how many days, of may he travel 360 miles?

Ans. 61 15s. when the days

10 hours each, Ans. 93 days.

6. If 120 bushels of corn can serve 14 horses 56 days; how many days will 94 bushels serve 6 horses?

Ans. 1021 days.

7. If 3000 lbs of beef serve 340 men 15 days; how many lbs will serve 120 men for 25 days? Ans. 1764 lb 11

oz. 8. If a barrel of beer be sufficient to last a family of 8 persons 12 days; how many barrels will be drank by 16 persons in the space of a year? Ans. 60 barrels. 9. If 180 men, in six days, of 10 hours each, can dig a trench 200 yards long, 3 wide, and 2 deep; in how many days of 8 hours long, will 100 men dig a trench of 360 yards long, 4 wide, and 3 deep? Ans. 483 days.

OF VULGAR FRACTIONS.

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 :

Thus,

3 numerator

which is named 3-fourths.

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

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

A Proper Fraction, is when the numerator is less than the denominator; as,, or, or 3, &c.

An Improper Fraction, is when the numerator is equal to, or exceeds, the denominator; as, 3, or, or 7, &c. In these cases the fraction is called Improper, because it is equal to, or exceeds unity.

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

A Compound Fraction, is the fraction of a fraction, or two or more 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 Complex Fraction, is one that has a fraction or a mixed number for its numerator, or its denominator, or both ;

REDUCTION OF VULGAR FRACTIONS.

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 +, &c.

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

Hence then, if the numerator be less than the denominator, the value of the fraction is less than 1. But 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 VULGAR FRACTIONS.

REDUCTION of Vulgar 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.

PROBLEM.

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 all without 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 further.

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; so 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; so 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 not to have any common measure, or they are said to be prime to each other.

EXAMPLES.

1. To find the greatest common measure of 1908, 936, and 630.

Hence 18 is the answer required.

2. What is the greatest common measure of 246 and 372 ?

Ans. 6.

3. What is the greatest common measure of 324, 612, and 1032? Ans. 12.

CASE I.

To Abbreviate or Reduce Fractions to their Lowest Terms.

* DIVIDE the terms of the given fraction by any number that will divide them without a remainder; then divide these

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 these 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

Note. 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 two ciphers, it is divisible by 100; if three ciphers, by 1000: and so on; which is only cutting off those ciphers.