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EXAMPLES.

1. 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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2. If 1007. in one year gain 57. interest, what will be the interest of 7501. for 7 years?

Ans. 262/. 10s.

3. If a family of 9 persons expend 1207. in 8 months; how much will serve a family of 24 people 16 months? Ans. 6407. 4. If 278. be the wages of 4 men for 7 days; what will be the wages of 14 men for 10 days?

Ans. 6l. 15s.

12 hours long ; Ans. 983 days.

5. If a footman travel 130 miles in 3 days, when the days are in how many days, of 10 hours each, may he travel 360 miles? 6. If 120 bushels of corn can serve 14 horses 56 days; how many days will 94 bushels serve 6 horses? Ans 1024 days.

7. If 3000 lb. of beef serve 340 men 15 days; how many lbs. will serve 120 men for 25 days? Ans. 1764 lb. 111⁄2 oz. 8. If a barrel of beer be sufficient to last a family of 7 persons 12 days; how many barrels will be drunk by 14 persons in the space of a year?

Ans. 60g barrels.

9. If 240 men, in 5 days, of 11 hours each, can dig a trench 230 yards long, 3 wide, and 2 deep; in how many days, of 9-hours long, will 24 men dig a trench of 420 yards long, 5 wide, and 3 deep? Ans. 278 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;

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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 Fraction.

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

A Proper Fraction, is when the numerator is less than the denominator; as , or 3, or 4, &c.

An Improper Fraction, is when the numerator is equal to, or exceeds, the denominator; as 3, or 4, 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 34, or 12, &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 4, &c.

A fraction denotes division; and its value is equal to the quotient obtained by dividing the numerator by the denominator; so 2 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 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 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

EXAMPLES.

1. 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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2. If 100%. in one year gain 5l. interest, what will be the interest of 7507. for 7 years?

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3. If a family of 9 persons expend 1204. in 8 months; how much will serve a family of 24 people 16 months?

Ans. 6401. 4. If 278. be the wages of 4 men for 7 days; what will be the wages of 14 men for 10 days? Ans. 61. 158. 5. If a footman travel 130 miles in 3 days, when the days are 12 hours long; in how many days, of 10 hours each, may he travel 360 miles? Ans. 983 days. 6. If 120 bushels of corn can serve 14 horses 56 days; how many days will 94 bushels serve 6 horses? Ans 1024 days.

7. If 3000 lb. of beef serve 340 men 15 days; how many " men for 25 days?

8. If a barrel of beer be sufficient

many barrels will be drunk by

9. If 240 men, in 5 days.

3 wide, and 2 deep; in trench of 420 yards lon

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;

3 numerator

thus, 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 Fraction.

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 3; &c.

A Simple Fraction, is a single expression denoting any number of parts of the integer; as, or 2.

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 34, 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 4, &c.

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

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REDUCTION of Vulgar Fractions, is the bringing them out of one form or

e them for the operations of al cases.

Do or more numbers.

s, is that number which will mmon measure of 18 and 24; r 8. And the greatest number

hat 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.

EXAMPLES.

1. To find the greatest common measure of 1998, 918, and 522.

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2. What is the greatest common measure of 246 and 372 ?
3. What is the greatest common measure of 336, 720, and 1736?

Ans. 6.
Ans. 8.

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CASE I.

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 man

* 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 ran 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 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.

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