Here the numerator and denominator of the complex fraction being expressed as mixed numbers, they are reduced to improper fractions, and thus we have numerator 4= 4, and denominator 6; and if we divide by 19 we shall evidently divide by a number which is 3 times too great, and the answer will consequently be 3 times too small, and will therefore require to be multiplied by 3 to bring it to its required value. Now can be divided by 19 by simply multiplying its denominator by 19, thus can be multiplied by 3 by simply multiplying its numerator by 3, thus A consideration of these two examples enables us to lay down the following rule for reducing a complex fraction to a simple one. RULE XXXII. Express its numerator and denominator as fractions, keeping the denominator below the numerator; then multiply together the extremes (i.e., the top and bottom numbers) for the numerator of the answer, and the means (i.e., the middle numbers) for its denominator. The process of cancelling is thus applied in such cases, Since the answer divisible by the same mon measure, thus, the product of the means, number as will divide one of the means, we may cancel by this com What are the simple fractions equal to the complex fractions 96. To compare the value of different fractions, i.e., to find out which is the greatest and which is the least. RULE XXXIII. 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. L Hence, the values of the given fractions are in the following order :-83, 83, 88, and . When fractions have a common numerator the greater has the less denominator. Thus Ex. is greater than 10; 1 greater than 1}. 97. It is a general rule that if we add the numerators of a number of fractions, and also the denominators the fraction given by lies between the greatest and least of the original fractions. sum of numerators sum of denominators 98. Proper fractions aro increased and improper fractions are decreased 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 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 to the 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. Now, let us take the proper fractions,,,,, where each successive fraction is made by adding 1 to the numerator and denominator of the fraction preceding it. 30 40 45 48 50 Reducing these to equivalent fractions having the same common denominator, they are respectively equal to 38, 48, 38, 43, 88; and of these fractions the first is the least and the last the greatest; we see that by adding the same quantity to both numerator and denominator of proper fractions, their value is continually increased. 80 75 72 But, if the improper fractions,,,, g, be taken, these are respectively equal to 120, 20 80, 88, 88, 38, 38; and 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. EXAMPLES FOR PRACTICE. I. Find which is the greater fraction in each of the following pairs: 2. I. and. 2.1 and 18. 4 X 5 3. Of,,, and, find the greatest and the least. 4. Which is the greatest and which the least, fa, 15, 25. 5. Arrange the following fractions in order of magnitude- 19 28 4. 3 and 8. 6. In the fractions,,, 1, 48, and 1, show that the between the greatest and least of the original fractions. CHAPTER V. THE ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF VULGAR FRACTIONS. 99. 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 add together two such fractions (for example, and ), so as 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 four equal parts and six equal parts, while the new numerator must express that three of the four equal parts, and five of the six equal parts, were to be taken; but no single numbers could express this, and the process could only be expressed symbolically, thus +; but if we reduce the fractions and to others of the same value having a common denominator (§ 92) they become and 19, respectively, and the first fraction is made up of nine of the twelve parts into which the unit is now divided while the second fraction is made up of ten of these parts; the sum of the two fractions must therefore contain nine and ten, or nineteen, of these twelve parts; therefore, += {+};= 1;. We may show how to add fractions together by the following illustration. Taking the foregoing example, viz., Let AB, BC, each represent unity. Divide AB, BC, each into 4 equal parts in a, b, c, d, e, f. Then Ac represents . Again, divide AB, BC, each into 6 equal parts in g, h, k, l, m, n, o, p, q, r. Then Am represents. The divisions at b, k, coincide, because Ab being 2 parts out of 4 is half AB, and so also is Ak, being 3 parts out of 6. Now the parts Aa, ab, bo, &c., are greater than the parts Ag, gh, hk, &c., so that the 3 parts of the former cannot be added immediately to the 5 parts of the latter. Now the L.C.M. of 4 and 6 is 12. Divide each of the parts Aa, ab. bc, &c., into 3 equal parts, that is, the units AB, BC, cach into 12 parts, and each of the parts Ag, gh, hk, &c., into 2 parts, that is, the units AB, BC, each into 12 parts. Then these latter divisions will all coincide with the former divisions, and Ac contains 9 such parts, and Am contains 10 such parts, and Ac, Am, together are equal to Az, and contain 19 such parts, that is,— #+# = &+}=}}· In the same manner we should proceed if we had several fractions. Hence we have the following rule: RULE XXXIV. If the fractions have a common denominator : Add the numerators together and place the sum over the common denominator. But if the fractions have different denominators, Find the L.C.M. of all the denominators, and reduce each of the fractions to equivalent fractions having the L.C.M. as their denominators, and then add the numerators thus found together for the new numerator, the new denominator being the L.C.M. EXAMPLES. Ex. 1. Add together §, §, §, and 7 (i.e., find the value of † + § + § + 7). Ex. 2. Add together,,, 11, and 18. Since we can only add together quantities of the same kind, we must express these given fractions as equivalent ones having one common denominator, before we can find their sum, The L.C.M. of all the denominators = 360, and, reducing each of the fractions to equivalent ones having the L.C.M. as their denominators, we have 1061 }+&+*+13 + 18 = 338 + 388 + 388+ 380 + $13 = 400 = 2}}} Ans. (1.) If any of the given quantities are whole or mixed numbers it is best to take separately the sum of the integral and fractional parts, then add the two results together. EXAMPLE. Ex. Find the sum of 24, 36, 281, and 1127. Adding together the whole numbers separately we have 24 +36 +28+ 112 = 200, and, adding together the fractions separately, we have }+1+1+}=1+#+#+}=Y=2} therefore, 24+ 363 + 284 + 1127 = 200 + 24 = 202} Ans. (2.) If some of the given fractions are improper, they should be reduced to mixed numbers, and the integral parts of the mixed numbers should be added together separately, as follows: NOTE. In this example, to save labour, instead of writing the new denominator 240 under each of the new numerators, these last have all been written together at once, since they have to be afterwards added together. |