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36 Tae Marine REVIEW } SCIENTIFIC LAKE NAVIGATION. By Clarence E. Long. PROPER, OR COMMON FRACTIONS. PART IV, FACTORS AND MULTIPLES. The factors of a number are the integers (whole numbers) whose product makes the number. Note.--An integer is any whole number. 2 and 3 are factors of 6, because 6 is divided exactly by 2 and by 3. _ 2,3 and 5 are the factors of 30, because 30 is divided exactly aby 2, by 3 and Dy-5. A number that contains another number an exact number of times is a multiple of that number. 24 is the multiple of 12; 36, 48, etc. are also multiples -of 12. 30 is a multiple of 2, 3, 5, 6, I0, 15. 95 is the multiple of what two numbers? Ans. 5 and 19, because 95 contains both 5 and 19 an equal number of times. Give the two factors of 51. Ans. 3 and 17, because they are the whole numbers, which, multiplied together, produce it; 51 is the multiple of 3 and 17 because it is exactly divisible by 3 and by 17. PRIME NUMBERS, A number that has no factors, except 1 and itself, is a prime number. Note.---1 is not considered a factor. I,.2, 3, 5, 7, étc., ate prune numbers. - Name the prime numbers between Io and 20. Ans. II, 13, 17 and 19. An integer that will divide a number without having a re- mainder is called an exact divisor of the number. Thus, 2, 3, 4 and 6 are exact divisors of 12. The factors of a number are exact divisors of it. A number that has exact divisors besides itself and I is called a composite number. Hence, a composite number is al- ways the product of two or more factors. Thus, 12, 18, 21, 40, etc., are composite numbers. _A number that is exactly divisible by 2 is called an even number. Thus, 10, 12, 16, 18, etc., are even numbers. A number that is not exactly divisible by 2 is called an odd number. Thus, 3, 5,. 9, 11, 13,. ete., are odd numbers. Although proper, or common fractions are not used as much as decimal fractions in navigation, the student should ac- quaint himself with them as they will help him to learn deci- mals. There are many cases in which they are used to advantage, as will be presently seen. When anything is divided into two equal parts what is each part called? Ans. %. Into three equal parts? Ans. 1-3. Into eight equal parts? Ans.-%. [nto mine equal parts? Ans. 1-9. Into fifteen equal parts? Ans. 1-15. _ How many halves are there in anything? How many thirds? How many fifths? How many tenths? How many fifteenths? How many twentieths? «There are always two halves in anything; 3 thirds; 5 fifths; 10 tenths; 15 fifteenths; 20 twentieths. What part of a vessel will each man own when it is divided equally among 8 men? Ans. 1%. 8 shows that the ownership of the vessel is divided among 8 people, hence, 1 man's share is equal to I part of the whole of 8, or %. A fraction is one or more of the equal parts of a unit. Thus, 1 half and 2 thirds are fractions. Two numbers, written one above the other with a line between them, are used to express a fraction. A fractional unit is one of the equal parts into which any unit is divided; thus, 1 fourth and 1 fifth are fractional units of fourths and fifths. Fractional units take their name and their value from the number of parts into which the whole number is divided. The terms of a fraction are its numerator and denominator; thus, 6 and 7 are the terms of the fraction 6-7; or, the numer- ator and denominator together are called the terms of the fraction, In fractions the numbers above the line are called nume- rators; the numbers below the line are called denominators. The denominator is written below the line. Thus, in the fraction 34, 4 is the denominator. It shows that something has been divided into 4 equal parts. The numerator is written above the line and shows how many parts form the fraction. Thus, in the fraction 7%, 7 is the numerator. It shows that the fraction contains 7 of the 8 equal parts. Fractions are proper or improper. A proper fraction is one whose numerator is less than its denominator. Its value is less than a unit. Thus, 1%, 1-3, 3%, 34 and 2-3 are proper fractions. These are also called com- mon fractions. 'The value of a proper fraction is, there- fore, less than one. An improper fraction is a fraction whose numerator equals or exceeds its denominator. Its value is equal to or is greater than a unit. Thus, 6-6, 10-4, 21-9 are improper fractions. 6-6 equals one whole one; 10-4 equals 214; and 21-9 equals 2 I-3. The value of. an improper fraction is, therefore, 1 or more than I. A mixed number is a whole number and a fraction written together. Thus 12 5-9 is equivalent to 12 + 5-9. The unit which is divided into equal parts is called the unit of the fraction. A fraction whose unit has been divided into any number of equal parts is called a common fraction. A fraction whose unit has been divided into tenths, hun- dredths, thousandths, etc., is called a decimal fraction. To analyze the fraction 7: 8 is the denominator and shows that the unit is divided into 8 equal parts; % is the fractional unit since it is one of the eight equal parts into which the unit is divided; 7 is the numerator and shows that seven of these equal parts are taken; 7 and 8 are the terms of the fraction. It is a proper fraction, since the numerator is less than the denominator. - Five-ninths expressed. by figures is 5-9. Seven-twenty-fifths, 7-25. Nine-eighteenths, 9-18. Twelve- twentieths, 12-20. Twenty-six forty-eighths, 26-48. To reduce fractions to higher or lower terms. Example, % is equal to how many fourths? Since 1 is equal to 4 fourths, ¥ is equal to one-half of 4 fourths, or 2 fourths, 2-4. One-third of a mile is how many sixths of a mile? Answer 2-6 of a mile. One-half. of a dollar is how many fourths of a dollar? Answer 2-4 of a dollar. Name some equivalent fractions for halves. Answer 2-4, 3-6, 4-8, and 5-10. Name some equivalent fractions for thirds. Answer 2-6, 3-9, 5-15, 8-24. Express 2-3 in terms 3 times as great; 4 times as great. Answer 6-9; 8-12. Multiply both terms of 34 by 3, and show that the value of the fraction is not changed. Answer 9-12. Name the equivalent for 4-5; for %; for 34: Answer 8-10; 2-16; 6-16,