The greatest common divisor of two or more numbers is the product of their common prime factors. PRINCIPLES. — A common divisor of two numbers is a divisor of their sum, and also of their difference. A divisor of a number is a divisor of every multiple of that number; and a common divisor of two or more numbers is a divisor of any of their multiples. To find the Common Prime Factors of Two or More Numbers. RULE. — Divide the numbers by any common prime factors, and the quotients in like manner, until they have no common factor; the several divisors are the common prime factors. To find the Greatest Common Divisor of Numbers that are Easily Factored. RULE. — Separate the numbers into their prime factors; the product of those that are common is the greatest common divisor. To find the Greatest Common Divisor of Numbers that are not Easily Factored. RULE. — Divide the greater number by the less; then divide the last divisor by the last remainder, continuing until there is no remainder. The last divisor is the greatest common divisor. If there are more than two numbers, find the greatest common divisor of two of them; then of that divisor and another of the numbers until all of the numbers have been used. The last divisor is the greatest common divisor. LEAST COMMON MULTIPLE A Multiple of a number is a number that exactly contains that number. A Common Multiple of two or more numbers is a number that is a multiple of each of them. The Least Common Multiple of two or more numbers is the smallest number that is a common multiple of them. PRINCIPLES. — A multiple of a number contains all the prime factors of that number. A common multiple of two or more numbers contains each of the prime factors of those numbers. The Least Common Multiple of two or more numbers contains only the prime factors of each of the numbers. To find the Least Common Multiple of Two or More Numbers. RULE. — Divide by any prime number that is an exact divisor of two or more of the numbers, and write the quotients and undivided numbers below. Divide these numbers in like manner, continuing until no two of the remaining numbers have a common factor. The product of the divisors and remaining numbers is the least common multiple. FRACTIONS A Fraction is one or more of the equal parts of anything. The Unit of a Fraction is the number or thing that is divided into equal parts. A Fractional Unit is one of the equal parts into which the number or thing is divided. The Terms of a Fraction are its numerator and its denominator. The Denominator of a fraction shows into how many parts the unit is divided. The Numerator of a fraction shows how many of the parts are taken. A fraction indicates division; the numerator being the dividend and the denominator the divisor. The Value of a Fraction is the quotient of the numerator divided by the denominator. Fractions are divided into two classes – Common and Decimal. A Common Fraction is one in which the unit is divided into any number of equal parts. A common fraction is expressed by writing the numerator above the denominator with a dividing line between. Common fractions consist of three principal classes – Simple, Compound, and Complex, A Simple Fraction is one whose terms are whole numbers. A Proper Fraction is a simple fraction whose numerator is less than its denominator. An Improper Fraction is a simple fraction whose numerator equals or exceeds its denominator. A Compound Fraction is a fraction of a fraction. A Complex Fraction is one having a fraction in its numerator, or in its denominator, or in both. A Mixed Number is a whole number and a fraction written together. The Reciprocal of a Number is one divided by that number. The Reciprocal of a Fraction is one divided by the fraction, or the fraction inverted. PRINCIPLES. — Multiplying the numerator or dividing the denominator multiplies the fraction. Dividing the numerator or multiplying the denominator divides the fraction. Multiplying or dividing both terms of a fraction by the same number does not alter the value of the fraction. Reduction of fractions is changing their terms without altering their value. To reduce a Fraction to Higher Terms. RULE. — Multiply both numerator and denominator by the same number. To reduce a Fraction to its Lowest Terms. RULE. — Divide both terms of the fraction by their greatest common divisor. A fraction is in its lowest terms when the numerator and the denominator are prime to each other. To reduce a Mixed Number to an Improper Fraction. RULE. — Multiply the whole number by the denominator; to the product add the numerator; and place the sum over the denominator. To reduce an Improper Fraction to a Whole or to a Mixed Number. RULE. — Divide the numerator by the denominator. A Common Denominator is a denominator common to two or more fractions. The Least Common Denominator is the smallest denominator common to two or more fractions. To reduce Fractions to their Least Common Denominator. RULE. — Find the least common multiple of all the denominators for the least common denominator. Divide this multiple by the denominator of each fraction, and multiply the numerator by the quotient. ADDITION OF FRACTIONS PRINCIPLE. -- Only like fractions can be added. RULE. — Reduce the fractions, if necessary, to a common denominator, and over it write the sum of the numerators. If there are mixed numbers, add the fractions and the whole numbers separately, and unite the results. SUBTRACTION OF FRACTIONS PRINCIPLE. — Only like fractions can be subtracted. RULE. — Reduce the fractions, if necessary, to a common denominator, and over it write the difference between the numerators. If there are mixed numbers subtract the fractions and the whole numbers separately, and unite the results. MULTIPLICATION OF FRACTIONS RULE. — Reduce whole and mixed numbers to improper fractions ; cancel the factors common to numerators and denominators, and write the product of the remaining factors in the numerators over the product of the remaining factors in the denominators. DIVISION OF FRACTIONS RULES. — I. Reduce whole and mixed numbers to improper fractions. Reduce the fractions to a common denominator. Divide the numerator of the dividend by the numerator of the divisor. II. Invert the divisor and proceed as in multiplication of fractions. To reduce a Complex Fraction to a Simple One. RULES. — I. Multiply the numerator of the complex fraction by its denominator inverted. II. Multiply both terms by the least common multiple of the denominators. DECIMALS A Decimal Fraction is one in which the unit is divided into tenths, hundredths, thousandths, etc. A Decimal is a decimal fraction whose denomination is indicated by the number of places at the right of the decimal point. The Decimal Point is the mark used to locate units. A Mixed Decimal is a whole number and a decimal written together. A Complex Decimal is a decimal with a common fraction written at its right. To write Decimals. RULE. — Write the numerator; and from the right, point off as many decimal places as there are ciphers in the denominator, prefixing ciphers, if necessary, to make the required number. To read Decimals. RULE. — Read the numerator, and give the name of the righthand order. PRINCIPLES. — Prefixing ciphers to a decimal diminishes its value. |