COMMON FRACTIONS. 109. If anything, as a line, is divided into two equal parts, each of these equal parts is called one half, and is written 110. If the unit is divided into three equal parts, each of these parts is called one third, written ; two of these parts are called two thirds, written If the unit is divided into four equal parts, each of these parts is called one fourth, written; two of these parts are called two fourths, writ ten; three, three fourths, ; and so on. 111. These expressions, 1, 1, 3, 4, etc., are called Fractions; and to distinguish them from Decimal Fractions, they are called Common Fractions. 112. A Fraction is one or more of the equal parts of a unit. 113. A Common Fraction is expressed by two numbers, one above and the other below a line; thus, (two fifths), etc. (one half), 114. The number below the line shows into how many equal parts the unit is divided, and is called the Denominator, because it denominates or gives name to the parts; thus, if a unit is divided into three equal parts, each part is one third; if into eight, each part is one eighth; etc. 115. The number above the line is called the Numerator, because it numerates or numbers the parts taken. 116. The numerator and the denominator are the Terms of the fraction. 117. To distinguish them from fractions whole numbers are called Integers, or Integral Numbers. 118. A fraction is only an expression of division (Art. 68), the numerator being the dividend, and the denominator the divisor. Hence, (a.) The value of a fraction is the quotient of the numerator divided by the denominator; thus, 12-12÷4=3. (b.) Any change in the numerator causes a like change in the value of the fraction, and any change in the denominator causes an opposite change in the value of the fraction. (Art. 79.) (c.) Multiplying or dividing both numerator and denomi nator of a fraction by the same number does not change the value of the fraction. (Art. 80.) 119. A Proper Fraction is one whose numerator is less than the denominator; as, 120. An Improper Fraction is one whose numerator equals or exceeds its denominator; as, 4, 8. An improper fraction equals or exceeds a unit; hence its name, improper fraction. 121. A Simple Fraction is one whose terms are integers; as, 8, 12. 122. A Mixed Number is a whole number and a fraction united; as, 34, 208. 1. Write in figures two fifths; three sevenths; five elevenths; seventeen thirty-thirds. Write a proper fraction; an improper fraction; a mixed number. 2. How many thirds make a unit? how many fifths? 3. Which is greater, a half or a third of the same thing? a third or a fifth? REDUCTION OF FRACTIONS. 124. Reduction of Fractions is changing the terms without changing the value of a fraction. 125. To reduce a fraction to its lowest terms. A fraction is in its lowest terms when its terms are the least integers that will express its value. terms are mutually prime. 1. Reduce to its lowest terms. In this case its The principle is the same as in cancellation (Art. 107). 2. Reduce to their lowest terms; ; ; 3 99 4 TO 8 3. Reduce to their lowest terms fo; ; ; 14; 14; 1 ¦ ¦ 14; † ; †%; 15. 69 4. Reduce to their lowest terms 1; 1; fe; fe; fe; fe; 18; 18; 18; fs. 5. Reduce to their lowest terms ;; 21; 28; 33; 18; 9 4 6. Reduce to their lowest terms 2; ; ft; da; fz; }; ; ; ; 3%. 127. Hence, to reduce fractions to their lowest terms, Rule. Cancel all the factors common to both numerator and denominator. Or, Divide both terms by their greatest common divisor. 129. To reduce an integer or a mixed number to an improper fraction. 23. In 3 oranges how many fourths of an orange are there? Solution. In one orange there are 4 fourths; therefore, in 3 oranges there are 3 times 4 fourths, or 12 fourths; 42, Ans. 24. How many sixths are there in 2? Solution. As 6 sixths make a unit, in 2 units there are 2 times 6 sixths, or 12 sixths; 12 sixths and 5 sixths are 17 sixths; 17, Ans. 130. Oral Exercises. 25. How many fifths in 6 apples? 26. Reduce 7 to fourths; 9 to sevenths; 6 to elevenths; 5 to eighths; 8 to fifths; 4 to ninths; 9 to fourths. 27. How many quarters in $8? 28. Reduce 12 to a fraction whose denominator is 7. 29. Reduce 9 to a fraction whose denominator is 8. 30. Reduce 7 to a fraction whose denominator is 1. Ans. }. NOTE. Any integer may be expressed as a fraction by writing 1 under it as its denominator. 31. Reduce 16 to a fraction whose denominator is 1. 32. Reduce to improper fractions in their lowest terms 2; 83; 94; 34; 8; 6; 74; 11; 5; 9FT; 54; 83; 65; 94; 57; 93; 111⁄2; 12§; 83; 97. 131. Written Exercises. 33. Reduce 7143 to an improper fraction. 7148 3573 Ans. As there are 5 fifths in a unit, in 714 there are 714 times 5 fifths, or 3570 fifths; 3570 fifths and 3 fifths are 3573 fifths; or 3573, Ans. As the product is numerically the same whether we multiply 5 by 714, or 714 by 5, and the denominator of the fractional part is usually the smaller number, in practice we use the denominator as the multiplier. Hence, To reduce an integer, or a mixed number, to an improper fraction, Rule. Multiply the whole number by the denominator of the fraction, to the product add the numerator, and under the result write the denominator. 34. Reduce 87 to a fraction whose denominator is 87. Reduce to improper fractions in their lowest terms: 132. To reduce an improper fraction to an integer or mixed number. 47. In 8 fourths of a dollar how many dollars are there? Solution. As 4 fourths make a unit, there are as many dollars in 8 fourths of a dollar as the number of times 4 fourths is contained in 8 fourths, that is, 2, Ans. See also Art. 118 a. |