TO REDUCE FRACTIONS TO A COMMON DENOMINATION. 35 all the denominators of these fractions. Here we proceed according to the following rule: TO REDUCE FRACTIONS TO A COMMON DENOMINATION. RULE.-Multiply each numerator into every denominator except its own for a new numerator, and multiply all the denominators together for a common denominator. When this operation has been performed, it will be found that the numerator and denominator of each fraction have been multiplied by the same quantity, and consequently that the fractions retain the same value, while they are at the same time brought to a common denomination. tion. Example. Reduce 1, 3, 4, †, and %, to a common denomina 60 and 60÷2=30 90 and 90÷2=45 1 × 3 × 4× 5 × 6=360 and 360÷6 2×3×4×5×6=720 and 720÷6=120 and 120÷÷2=60 Here, then, we first multiply 1, which is the numerator of the fraction, by the denominators of all the other fractions in succession. We next multiply the number 2, which is the numerator of the fraction, by the denominators of all the other fractions-excepting always its own denominator—and we proceed in this manner through all the fractions whatever their number may be. We next multiply all the denominators together for the common denominator. Proceeding in this way we find the first numerator to be 360, the second 480, the third 540, the fourth 576, and the fifth 600; while the new denomiinator we find to be 720. It is clear, however, that these fractions are not in their lowest terms, and that the numerator and denominator of each may be divided by some common number without leaving a remainder. We may try 6 as such a divisor, and we shall find that the numerators will then become 60, 80, 90, 96, and 100, and the denominator 120. These numbers. however, are still divisible by 2, and performing the division the numerators become 30, 40, 45, 48, and 50, and the denominator becomes 60. The same result would have been attained if we had divided at once by 12. And as we cannot effect any further division upon all of the numbers by one common number, without leaving a remainder in the case of some of them, the fractions, we must conclude, are now in their lowest common terms. To add together these fractions we have only to add together the numerators, and place the common denominator under the sum. Performing this addition we find that in this case we have 213, and as 88 are equal to 1, it follows that 213 are equal to 3 and 83, or 3 11. 33 which have been substituted for them. Dividing numerator and denominator of the first term by 30 we obtain ; dividing numerator and denominator of the second term by 20 we obtain ; 15 is the divisor in the case of the third term when we obtain ; 12 is the divisor in the case of the fourth term when we obtain the fraction ; and 10 is the divisor in the last case when we obtain the fraction. Dividing the numerator and denominator of each of the transformed fractions, therefore, by the greatest number that will divide both without a remainder, we get the fractions 1, 3, 4, 4, and § which, it will be seen, are the fractions with which we set out, and they are now in their lowest terms, but are no longer of one common denomination. The lowest terms with a common denominator are as determined above. 30 40 45, 48, and 50 60 60 601 601 The subtraction of fractions from one another is accomplished by reducing them to a common denomination as for ad As whenever the numerator of a fraction is a la than the denominator, the value of the fraction is unity, and is equal to unity when numerator and is the same, we have only to divide the numerato nominator to find the number of integers which contains. So in subtracting a fraction from a wh we must break one or more integers up into frac same denomination as that which has to be subtra if we have to take 8 from 1, we must instead of 60, and 28 taken therefrom obviously leaves 8. 40 If add together such sums as 33 and 23, we see at or whole numbers when added will be 5, and the equ tions under a common denominator will be 3 and 4 is 1, so that the total quantity will be 61. 809 78963.874 83952.2 364.003 10000-997 The addition and subtraction of decimal fracti formed in precisely the same way as the addition tion of whole numbers-the only precaution necess place the decimal point in the proper place. Thus 83952 2+364-003+10000·997 are added together as Here, beginning as in the additio numbers with the first column to the ri that 7 and 3 are 10 and 4 are 14. W the 4 beneath the column and carry 1 column. Adding up the next column, two significant figures in it, and we sa 9 makes 10, which added to 7 makes 17. We set do carry the 1 as before to the next column, which wh we find to be 20. This means 20 tenths, and we se 0 and carry the 2 to the next column just as in simp So likewise in subtraction, if we take 2.25 from 4.7 173281-074 will be 2.50; or if we take 1.79 from 3, the result is 1 21. In such a case we write the 3 thus: 3:00 1.79 1.21 Here we write the 3 with a decimal point after it, and we add as many ciphers after the decimal point as there are decimal figures to be subtracted, or we suppose those ciphers to be added. This does not alter the value of the 3, as 3 with no fractions added to it is just 3. Performing the subtraction we say 9 from 10 leaves 1, and 8 taken from 10 leaves 2, and 2 from 3 leaves 1, just as in simple subtraction. MULTIPLICATION AND DIVISION OF FRACTIONS. If we wish to multiply a fraction any number of times, it is clear that it is only the numerator we must multiply. Thus if we multiply of an inch by 3, it is obvious that we shall get of an inch as the product of the multiplication, or repeated 3 times. We have already seen that to multiply both terms of a fraction by any number does not alter the value of the fraction, and if we were to multiply the numerator and denominator of the fraction by 3 we should get, which is just the same as . Thus also 3 times makes & or 13. 3 times makes or 1. 3 times makes or 1. 4 times makes 9 or 1 or 1}. Instead, however, of multiplying the numerator, we may attain the same end by dividing the denominator, and this is a preferable practice when it can be carried out, as it shortens the arithmetical operation. Thus multiplied by 2 is or. But by dividing the denominator of by 2, we obtain the same quantity of at one operation. So also if we have to multiply by 3 we obtain 24, or §. But if, instead of multiplying the numerator, we divide the denominator, we obtain the § at one operation. In the same way 13 multiplied by 6 is equal 13, or 31. Where the integer with which the multiplication is performed is exactly equal to the denominator of the fraction, the product will be equal to the numerator. Thus མཨཔཔ་ v མ་ = plying the denominator. It is clear that if we inch into two parts, each of these parts will be and we divide quarter of an inch into two parts, parts will be of an inch, so that ÷2-1 and quantities we obtain by successively multiplying tors. We may accomplish the same object by di merator where it is divisible without a remaindo vided by 2 is clearly, and divided by 3 is 4. 1 divided by 2 gives, divided by 3 gives, 1 divided by 4 gives 35. When the numerator is not divisible by the d a remainder, the fraction may be put into some eq when the division may be effected. Thus if we by 2, we might turn it into the equivalent frac divided by 2, gives 3. But the same number is iently found by multiplying the denominator ins viding the numerator. We have next to consider the case where one be multiplied by another. Thus if the fraction & tiplied by the fraction, we have first to rememb pression means 2 divided by 3, and we may first which produces, and then divide by 5, which Hence, in multiplying a fraction by a fraction, w numerators together for the new numerator, and ors together for the new denominator. Thus, |