tions, having the least common denominator. By inspection (§ 39) the L. C. M. must contain the factors § 51. To compare the value of different fractions, i.e. to find out which is the greatest and which the least. Bring the given fractions to others of the same value, having a common denominator; then the respective values of the fractions will depend upon their numerators, that fraction being greatest which has the greatest numerator. and the prime factors included in the L.C.M. being 2, 2, 3, 2, 2, 2, we have Hence the values of the given fractions are in the order of their equivalents as follows: The comparison of fractions may sometimes be more concisely performed by transforming the given fractions to others having a common numerator: the fraction that has then the least denominator is the greatest: thus § 52. Proper fractions are increased and improper fractions are diminished by adding the same quantity to both numerator and denominator. We may observe in general, that the higher a number is, the less, relatively to another number, is its increase made by the addition of 1. Thus 2 is double of 1; but 3 is not double of 2; still less is 4 double of 3; or 100 double of 99. So that by adding an unit, or any number of units to each of two numbers, the increase to the smaller will be more in proportion than the increase to the larger. Hence, if a fraction be proper, i.e. if its numerator be less than its denominator, by adding the same quantity to both, the increase to the numerator will be more in proportion than the increase of the (1) of the signs > and < here used for "greater than" and "less than," it may help the memory to observe that the sign for "less than" is turned the same way as the letter L. denominator, and the value of the fraction will be increased; while conversely, if the fraction be improper, the increase to the denominator will be less than the increase to the numerator, and the value of the fraction will be diminished. 1 2 3 4 5 6 Now let us take the proper fractions 2' 3 4' 5 where each successive fraction is made by adding 1 to the numerator and denominator of the fraction preceding it. Reducing these to equivalent fractions having the same common 30 40 45 denominator, they are respectively equal to 60 60' 60 48 , 50 60 60 ; and as of these fractions the first is the least and the last the greatest, we see that by adding the same quantity to the numerator and denominator of proper fractions, their value is continually increased. as of these fractions the first is the greatest and the last the least, we see that by adding the same quantity to both numerator and denominator of improper fractions, their value is continually diminished. Whence we conclude that we cannot add the same quantity to the numerator and denominator of any fraction, without thereby altering its value. Conversely, proper fractions are diminished and improper fractions increased by subtracting the same quantity from both numerator and denominator; whence we conclude that we cannot subtract the same quantity from the numerator and denominator of any fraction without thereby altering its value. [Obs. It is of great importance to remember from this, in reducing fractions, that although we may divide both numerator and denominator by the same quantity, we may not take away the same quantity from both numerator and denominator by subtraction.] G 82 CHAPTER VII. THE ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF VULGAR FRACTIONS. § 53. The addition of two or more fractions is effected by finding some single fraction which shall express the sum of all the given fractions. It is however impossible to find such a fraction unless all the given fractions be first expressed with a common denominator: for, since the denominator of a fraction expresses the number of equal parts into which the unit is divided, it follows that in two fractions which have not a common denominator the unit is not divided into the same number of equal parts: therefore in endeavouring to 3 add together two such fractions, for example and so as 4' 2 to express their sum by a single fraction, if we did not first bring them to equivalent fractions with a common denominator we should have to seek for a new denominator which would express that the unit was to be divided into three equal parts and four equal parts, while the new numerator must express that two of the three equal parts and three of the four equal parts were to be taken; but no single numbers could express this; and the process could only be represented symbolically thus, + but if we reduce the fractions 23 3 2 3 3 and others of the same value having a common denomi 4 8 9 nator (§ 50), they become and respectively; and the 12 12 first fraction is made up of eight of the twelve equal parts into which the unit is now divided, while the second fraction is made up of nine of those parts; the sum of the two fractions must therefore contain eight and nine, or seventeen, 2 3 8 9 17 of these twelfth parts; therefore + -= + Hence the Rule for the Addition of Fractions: Transform the fractions to be added to equivalent fractions having the least common denominator; take the sum of the ་ |