79. It is clear, from what has been said, that every integer may be considered as a fraction whose denominator is 1; thus, 5, for the unit is divided into 1 part, comprising the whole unit, and 5 of such parts, that is, 5 units, are taken. 80. To show that of 1 = of 2, or the third part of 2. of 1 is 2 third parts of unity. Now I is 3 third parts of unity. Therefore, 2 is 6 third parts of unity. Therefore, But of 1 of 2 is of 6 third parts of unity, i.e., is 2 third parts of unity. of 2, i.e., =23. Or, we may show that A of 1 of 2 by the following illustration. I Let AB, BC each represent unity; then AC represents 2. Divide AB, BC each into 3 equal parts in ab, cd. Then AC is divided into 3 equal parts in bc. Again, since AC represents 2, Ab represents of 2. Hence we may consider the fraction to represent either "unity divided into 3 equal parts, of which 2 parts are taken," or else "the integer 2 divided into 3 equal parts, of which one is taken." 81. The following definition of a fraction will now be easily understood. A Fraction is a simple means of expressing the division of the numerator by the denominator, e.g., The fraction = (1) three sevenths of one. = = (2) one seventh of three, which is found by dividing 3 by 7, i.e., the numerator by the denominator. Hence we see that one of the definitions above involves the other. 82. From the notion attached to the words numerator and denominator, it is evident that 1.—A fraction is increased by increasing the numerator. 4. A fraction is decreased by increasing the denominator. For, by increasing or diminishing the numerator we take more or fewer of the parts of the unit; by diminishing the denominator the magnitude of the parts is increased, while the same number of parts are taken. Also, if the denominator be increased, the magnitude of the parts is diminished and the fraction is diminished. Thus the fraction— is three times greater than ; is the double of . is three times greater than; is the double of. is the third part of; are the half of . is less than ; are less than *. 83. To express a whole number as a fraction with a given denominator. RULE XXIII. Multiply the number by the given denominator, and the result will be the numerator of the required fraction. For since I contains 7 seventh parts, 4 contains 4 X 7 or 28 seventh parts, that is, 4 = 2. EXAMPLES FOR PRACTICE. Reduce the following numbers to fractions having a common denominator. I. 5; 6; 7; 8; 9; 10; 11; 12, to denominators 7 and 15. 2. 7 to denominators 11, 13, and 143; 909 to denominators 11 and 101. 84. To multiply any given fraction by any given integer. Here we want to make the quotient, found by dividing 5 by 12, three times greater. Now, any quotient may be made three times greater in either of two ways, viz. :— Ist. By making the dividend or numerator (§ 82), three times greater; or, 2nd. By making the divisor or denominator (§ 82), three times less. 1st. By keeping the same denominator and multiplying its numerator by 7. Thus, ' X 35x3==t, or 12 2nd. By keeping the same numerator, and dividing its denominator by 3. We may show how to multiply a fraction by a whole number by the following illustration. Let equal lines AB, BC, CD, each represent unity; divide each of them into 5 equal parts. Then Aa represents . Now Aaab = be, because they each represent 4 of the smaller divisor; hence, AC represents 3 times Aa, or X 3. But Ac represents 2 since it contains 12 smaller divisions. Hence we deduce the following rule for multiplying any given fraction by any given integer. RULE XXIV. Multiply the numerator of the fraction by the integer, or divide the denominator by it. 85. It sometimes happens that when a fraction has been multiplied by a whole number, it can afterwards be reduced to lower terms; this will be the case when the denominator and the multiplier have some common factor; this factor may be rejected from both the multiplier and denominator before multiplication, instead of from the new humerator and denominator after multiplication. Thus, X 8 is better worked thus, 2 7 12 ×8=7×2=, striking out, or dividing by, the factor 4, which is common to 8 and 12 before multiplication, instead of from 56 and 12 after multiplication. Here the 12 and 8 are divided by 4, leaving only 2 in the multiplier and 3 in the denominator. 2 Again, X 14, or 7 X 2 = × 1 = × 2, striking out or dividing by the 77 II factor 7, which is common to 77 and 14. 7 goes in 14, 2 times, write 2 above the 14; 7 goes in 77, 11 times, write 11 under 77. 86. To divide any given fraction by any given integer. Here we want to make the quotient, found by dividing 14 by 16, seven times less. Now, any quotient may be made seven times less in either of two ways, viz. : 1st. By making the dividend seven times smaller; or, 2nd. By making the divisor seven times greater. So that may be divided by 7, either 1st. By keeping the same denominator, and dividing its numerator by 7. Operations containing fractions with very high numbers are of little practical value, decimal fractions being preferred. But as exercises of arithmetical accuracy we shall give among the rest, a few cases of high numbers. K 2nd. By keeping the same numerator, and multiplying its denominator by 7. Hence we deduce the following rule for dividing any given fraction by any given integer. RULE XXV. Divide the numerator of the fraction by the given integer, or multiply the denominator by it. Ex. 1. Divide 12 by 3. EXAMPLES. = (by keeping the same denominator and dividing the numerator). Ex. 2. Divide by 3. 4 ÷ ÷ 3 = 5 + 3 = 1 (dividing the numerator is impossible, therefore, multiply the denominator). 86. In the same manner as in multiplying a fraction by a whole number, if there is a factor common to the numerator and the divisor, this factor may be struck out before the denominator is multiplied. Thus, 6 is better worked thus, ÷ 6= 2 Dividing numerator 4 and divisor 6 by the common factor 2. ÷8=1÷3=&. 6; 2. 5; 10; 28; 216; 18. 3;÷7;&÷11; 4÷969; 1840; 3. Divide by 146; 10091 by 146; 73 by 10001; 1001 by 77; 4. Divide 10 by 24; by 110; 2 by 300; 298277 by 5117; 9 107 1001 1001. by 28; } by 8. by 804. 87. To represent an improper fraction as a whole or a mixed number. Let equal lines AB, BC, CD, DE, EF, each represent unity. Divide them each into 3 equal parts; then Ag contains 14 such parts, that is, Ag = 1. But AE is 4 units, and Eg is 3, .'. Ag = 43, that is, = 4. Hence the rule RULE XXVI. Divide the numerator by the denominator; the quotient will be a whole number, and the remainder (if any) will be the numerator of the fractional part of the mixed number, and the divisor its denominator. If there be no remainder the improper fraction is equal to the whole number thus found. 1071 194 357 N.B.-All improper fractions which occur in any answer should (except the contrary be desired) be expressed as a whole or mixed number. 88. To reduce a mixed number to its equivalent improper fraction. A quantity made up of two others, one of which is an integer and the other a fraction, may be represented in the form of a fraction alone. Let us take 3, which is called a mixed quantity and is intended to express the integer 3 and the fraction, taken together, or 3+, and is read three and four-fifths. Multiply the integer by the denominator of the fraction, and to the product add the numerator, this sum placed over the denominator will form the improper fraction required. |