fractional unit is 2 times the value of the fractional unit. += }· Hence,

161. A fraction is multiplied by multiplying its numerator or dividing its denominator, by a number greater than unity.

If we divide the numerator of by 2, we obtain . The fractional unit in and is the same, has only one half as many fractional units as . + * = = 1. Again,

If we multiply the denominator of 4 by 2, we obtain }. The number of fractional units in and is the same. But the value of the fractional unit is only one half the value of the fractional unit †. x=. Hence,

162. A fraction is divided by dividing its numerator or multiplying its denominator, by a number greater than unity.

If we multiply both terms of by 2, we obtain . The number of fractional units in is twice as many as in, but the value of each fractional unit is only one half as much. That is, the increase in the number of parts taken equals the decrease in the size of the parts. 1. Again,

=

If we divide both terms of by 2 we obtain . The number of fractional units in is only one half as many as in g, but the value of the fractional units is twice as much. That is, the decrease in the number of parts taken equals the increase in the size of the parts. Hence,

=+

163. The value of a fraction is not changed by either multiplying or dividing both terms by the same number.

Fractions primarily arise from performing the operation of division, when the division is not exact (99, 1).

That of which the fraction expresses a part is called the Unit of the Fraction; as in the expression $, one dollar is the Unit of the Fraction.

The Unit of the Fraction is not always a single thing; it may be a collection of things taken as a whole; as in the expression of fifty men, fifty men is the unit of the fraction.

SECTION II.

REDUCTION

164. Reduction of Fractions is changing them into other equivalent expressions. It also includes the changing of whole and mixed numbers to the form of a fraction.

Before proceeding with Reduction of Fractions it is necessary to understand something of Measures and Multiples.

165. A Divisor, or Measure, of a number is a number which will divide that number without a remainder. Thus, 3 is a divisor, or measure, of 6.

66

Measure" is here used in the limited sense of exact divisor (99, 1). Any number may be used as the measure of any other number of the same kind (191, 259).

166. The Divisors or Factors of a number are the numbers which multiplied together will produce it. Thus, 2 and 3 are factors of 6; 2, 6, and 5, of 60.

An Even Number is one that is exactly divisible by 2. The lefthand-page numbers of this book are even.

An Odd Number is one that is not exactly divisible by 2. The righthand-page numbers of this book are odd.

A Prime Number is one that can not be separated into integral factors; as 2, 3, 5, 7, 11, 13.

The number 1 is not regarded as a factor.

167. The Prime Factors or Prime Divisors of a number are the prime numbers which multiplied together

will produce it. Thus, 2, 6, and 5, or 2 and 30, or 5 and 12 are factors of 60; but 2, 2, 3, and 5 are the prime factors of 60, because they are the prime numbers whose product. is 60.

When the numbers are small, we can find the prime factors by inspection; but when they are large, we have generally to obtain them by trial.

In making trial a knowledge of the following facts may be of service :

FIRST.-Any number ending in 0, 2, 4, 6, or 8 has 2 for an exact divisor.

SECOND. Any number the sum of whose digits is divisible by 3, has 3 for a divisor.

THIRD.-Any number ending in 5, or 0, has 5 for an exact

divisor.

EXAMPLE. Find the prime factors of 2310.

SOLUTION.

2) 2310

3) 1155

5) 385

7)77

11

EXPLANATION.-From the 1st Fact, we see that 2 is an exact divisor of 2310. From the 2d Fact, we see that 3 is an exact divisor of the quotient 1155.

From the 3d Fact, we see that 5 is an exact divisor of the quotient 385.

None of these Facts aid us in obtaining the next factor; but since 7 is a prime number and a divisor of 77, we divide by Ans. 2 × 3 × 5 × 7 × 11. it and obtain 11, a prime number, for the

quotient.

The product of the divisors 2, 3, 5, 7, and the last quotient, 11, all of which are prime numbers, is 2310. Hence they are the prime factors of that number (167).

RULE.-Divide by any prime number except 1 that is an exact divisor; divide the quotient in the same manner; and so proceed until the quotient is a prime number.

The several divisors and the last quotient are the prime factors.

A composite number is one that can be separated into integral factors; as 4, 6, 8, 9, 10, 12.

168. The Composite Factors or Divisors of a number are the composite numbers which multiplied together will produce it. Thus, 4 and 15 are the composite factors of 60, because they are the composite numbers whose product is 60.

If a number is an exact divisor of two or more numbers, it is said to be a divisor, or measure, common to them.

169. A Common Divisor or Measure of two or more numbers is an exact divisor of each of them. Thus, 2 is a divisor common to 2, 4, 6, 8, 10, &c., and is called a common divisor.

Numbers are prime to each other when 1 is their only exact divisor. 170. The Greatest Common Divisor or Measure of two or more numbers is the greatest number that is an exact divisor of each of them. Thus, 6 is the greatest

number that will exactly divide 12 and 18, hence it is their greatest common divisor.

5 x 7, therefore, is the greatest common divisor.

SOLUTION.

105) 175 (1 105

SECOND METHOD.

70) 105 (1
70

Ans. 35) 70 (2
70

EXPLANATION. -105 is the greatest measure of itself. If it also measured 175 it would be the greatest divisor of both numbers.

But it does not measure 175, because 70 remains. That is, 175 = 105 + 70. Now, as 70 is the greatest measure of itself, if it also measures 105 it will measure 175.

But it does not measure 105, because 35 remains. That is, if 175 = 105 +70, then 175 35 + 70 + 70. Now, as 35 is the greatest measure of itself, if it measures 70 it will measure 105 (which 3570), and 175 (which = 35+ 70 + 70). Since 35 does measure 70, it will also measure 105 and 175. It must be the greatest common divisor, because it is the greatest number contained in 105 and in the difference between 105 and 175.

=

RULE I.-Find the product of all the prime factors common to the given numbers.

OR, II.--Divide the greater number by the less, and if there is a remainder, divide the preceding divisor by it. So continue dividing the last divisor by the last remainder, till nothing remains. The last divisor will be the greatest common divisor of the two numbers.