therefore, are to have the same denominator, that denominator must be a multiple of 9, 12, and 18. Thus, But it is, moreover, convenient that this 'denominator should be as small as possible, and hence the least common multiple of 9, 12, and 18 is chosen. This we know to be 36. To reduce to a fraction with denominator 36 we must multiply both its numerator and denominator by 4 making 18. In the case of we must multiply by 3, making 1; and in the case of 13 by 2, making 3. Therefore This operation is termed reducing fractions to their Least Common Denominator, and the general rule is the following: Find the L. C. M. of the denominators, take this as the common denominator, and reduce the fractions to equivalent fractions having that denominator. To add fractions together, Reduce them to their least common denominator, and add together their numerators, retaining their common denominator. Thus— To subtract one fraction from another, Reduce them to their Least Common Denominator, and subtract one numerator from the other, retaining their common denominator. Thus The following remarks relate to the application of these rules: (1) When whole and mixed numbers occur among the fractions to be added, the whole numbers and fractions should be added separately. (2) Compound fractions should be reduced to simple fractions, and improper fractions to mixed numbers, before the application of the rule. 105+341+13 of of +6+31 +13+7 16 = 9+ 80 72 405369+ 130 + 630 720 (3) In subtraction of fractions also, compound fractions must be reduced to simple fractions, and improper fractions should be reduced to mixed numbers before the application of the rule, and whole numbers and fractions ought to be subtracted separately. As there arise two or three cases which are best expressed in somewhat different forms, an instance of each shall be given. Where the second fraction is less than the first. Where the second fraction is greater than the first. In each of the two latter cases it has been necessary to consider 1 from the whole number replaced by an equivalent fraction having the common denominator. Thus in one case 2-4=1+24-74 since 24 is the same as 1. And 1+ 24-2-144. 7 In the other case 8-311=7+48-311=437. When, however, the reason of the process is once seen, there is no need to put down the intermediate step. Every operation in Arithmetic can only be a combination of the four processes, Addition, Subtraction, Multiplication, and Division; and the application of these to fractions has been considered. Before, however, proceeding to the next subject in Arithmetic, there is one minor point to which it is necessary to call attention. 13 3 Let three fractions 3, 4, and be given, and let it be required to determine which is the greatest, and which the least. Such examples as these, involving any number of fractions, may be at once worked by reducing the fractions to a common denominator. Thus And 13 is the least, and the greatest of the fractions. 25. 12 of 7-1 of 34+13-127. 26. of 11 of 3- of of 73+35-23 of 4. 35 27. Which is the greatest, and which the least, of the fractions 7 71 105' 17 19 259 28 71 CHAPTER IV. DECIMALS; AND THE APPLICATION OF FRACTIONS AND DECIMALS TO CONCRETE QUANTITIES. SINCE proper fractions have the numerator less than the denominator, their values must in all cases be less than one, and by means of fractions any quantity less than one may be either exactly expressed, or approximated to as nearly as we please. This is not, however, the only method of expressing such quantities. Another way of doing so is by using an extension of our ordinary system of notation. It will be remembered that ordinary numbers are expressed by a system in which the value of the separate figures increases tenfold as we pass from figure to figure towards the left, and consequently diminishes similarly towards the right. Thus, 2436 means 2 thousands, 4 hundreds, 3 tens, and 6 units. Hitherto this system has been supposed to stop at units, no smaller value being given to any figures. But we might if we pleased consider it extended indefinitely, following the same law, in which case tenths would follow units, hundredths would come after tenths, and so on. Such an extension gives rise to the quantities called decimals, and a dot coming after the place of units, separating the whole numbers from the decimals, is called the decimal point. For example, 2436.1024 means 2 thousands, 4 hundreds, 3 tens, 6 units, 1 tenth, hundredths, 2 thousandths, 4 ten thousandths. It will be best to consider first the nature and properties of decimals, reserving till afterwards the explanation of their relations to fractions. 28.75 means 2 tens, 8 units, 7 tenths, 5 hundredths. Let the same figures be retained, and let the decimal point be moved one place to the left. Then 2.875 means 2 units, 8 tenths, 7 hundredths, 5 thousandths. Every figure has thus been made to represent one tenth of its former value, and the whole quantity has therefore been divided by ten. Hence 28.75÷10=2·875. Had the decimal point been moved two places to the left, the expression would have been changed to 2875, or 2 tenths, 8 hundredths, 7 thousandths, 5 ten thousandths. Here each figure has been made to represent one hundredth of its former value, and 28.75÷100 =2875. Similarly, had the decimal point been moved one place to the right, the quantity would have been multiplied by ten, if two places, by a hundred: 28·75 × 10=287·5; 28·75 x100=2875. The decimal point may be moved any number of places to either the left or right by the employment of ciphers. Thus, in moving it 8 places to the left, and 7 places to the right, we should have respectively— 28-751000000000000002875 and 28.75 x 10000000= 287500000. As it is very important to understand the effect produced by the change of place of the decimal point, some examples are given to illustrate it :— 730 29 x 1000-730290. 730 29 x 10=7302⚫9. 730-29 x 100000 73029000. 730 291000='73029. The following particulars must be borne in mind :— 1. After whole numbers, or integers, as they are called, we omit the decimal points. Thus, 16 means 16'; 80000 means 80000. If this decimal point be expressed, any number of ciphers afterwards added make no difference. Thus 16 is the same as 16.000; 80000 is the same as 80000-00000. 2. With respect to figures to the left of the decimal point, ciphers before the first significant figure (that is, figure other than 0), make no difference, ciphers after the last significant figure increase the value. For instance, 000819 is the same as 819', but 819000819 × 1000. |