Having now said all that I think necessary for the learner's instruction, I would advise him to prove the sums of multiplication and division. He may do so in one afternoon, and he will thus perfect himself in the knowledge of integers, or whole numbers, before he enters on the next lesson, which is of fractions or broken numbers. 36 rem. ANSWERS TO DIVISION. (1.) 242985 ... 1 rem. (20.) 36 280 rem. (2.) 14549. (21.) 252 (3.) 11648 1 rem. 275 rem. (4.) 580694 3 rem. (23.) 2568 300 rem. (5.) 96376. (24.) 103 2319 rem. (6.) 54495 3 rem. 54 rem. (7.) 36841 3 rem. (26.) 192268 miles...340rem. (8.) 88471 8 rem. (27.) 72 ... 22 rem. miles per (9.) 74580 6 rem. day. 3 m. per hour. (10.) 12658 9 rem. (28.) 59 ...115 rem. years. (11.) 41303 7 rem. (29.) 34 boy's share, 38 girl's (12.) 4135 44 rem. share. (13.) 1767 22 rem. (30.) 68040 miles an hour, (14.) 2207 21 rem. 1360 rem.; 1134 miles (15.) 845 47 rem. a minute. (16.) 395 78 rem. (31.) 1084 41600 (17.) 1094 ... 370 rem. years. (18.) 17 8762 rem. (32.) 1311 dividend. (19.) 829 ... 891 rem. rem. FRACTIONS. The words Integer and Fraction are derived from the Latin ; Integer signifying entire, whole ; Fraction signifying broken. If we imagine the figure b on the right to be a pasteboard card, and wish to express it in figures, we would write 1, and the unit would express it. But if we cut the card at the line a a which divides it in two equal parts, and we wished to express one of these parts by figures, we could not then use a whole number; if we did, it would not convey our thoughts. But we have an equally simple mode of doing so, which is fractionally. Thus, į; the one above the line numerates exactly the part I wish to express, and from its office is called the numerator; the two below the line denoting the number of parts into which the card was divided, and is therefore called the denominator. If we cut the card again at the line b b, it divides the card into four equal parts. Here the denominator, or lower figure, would be four, denoting the four parts into which the card was divided, while the numerator would alter from one to three, according to the number of parts you wish to express, & a fourth part, 4 two fourths, three fourths, 4 four fourths, a term not used, as four fourths represent the whole card; and it is to be borne in mind that whenever the numerator and denominator of a fraction are of the same amount, the sum represents a whole number; and that the nearer these two numbers approach one another in amount, the higher the value of the fraction is. But let us again cut the card at the lines c c and d d, and we find it then divided into eight equal parts. To express these the denominator is eight, and the numerator alters from one to seven. § one eighth, s two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, which is within one eighth of representing the whole card. All numbers possess this quality, namely, that if any two are multiplied by another number, their proportion or value to one another is not altered; and also that any two numbers can be divided by another number, and provided such number divides both evenly, their relative value one to another is not altered. If we take the 4written above, we know that 4 is the half of 8; if we multiply these two figures by 6, thuswe see the product of the 4 is 24, and the product of the 8 is 48. Now we 승 6 = 중 know as well that 24 is the one half of 48, as we did that 4 was the half of 8. Again, if we divide the 24 by any number that they both will contain evenly, we find the same result. You see on the right :$ 24 - ] divided by 24, and the quotient is one half. Now let us inquire into the truth of the above. In the first place, the card when whole represented one. You eventually divide this one card into eight equal parts. If you take four of these parts in your hand, four remain on the table, consequently you know you hold half the card in your hand; and wishing to express the value of these four parts in figures, you write f ; the eight representing the number of parts into which you cut the card, and the four numerating the parts you hold in your hand. You also see, that however we might multiply or divide this , provided we multiplied or divided both by the same numbers, the value would remain the same; and bearing this in mind, the working of fractions becomes easy. Having partially explained what Fractions are, we will 24 • 24 now proceed to consider how they so often arise in our calculations. One reason is, the complicated manner in which our currency, weights, and measures, are kept. This causes fractions frequently to occur in the terms of our propositions; as, What would 464 pounds of meat cost at 60 per pound ? Here we have two, the one expressive of three fourths of a penny, the other of a quarter of a pound. If fractions only occurred as ., or I, they could be easily managed. But they as frequently arise from the operations of division, and sometimes we have four or five figures expressive of a numerator, and four or five more expressive of a denominator ; and these, although expressive of hundreds fractionally, we know to be less than a unit. Yet, however small their amount may be singly, in calculations where they are expressed perhaps a hundred times, their amount becomes considerable ; as, At 6 d per pound, what would be the cost of 200 lb. ? Here you perceive, that although by casting away the farthing for one pound you lose only the fourth of a penny ; yet to cast it out in your calculation of the price of two hundred pounds, you would lose four shillings and twopence. Such, therefore, being their importance in our calculations, we have adopted the two following methods of preparing them for our purposes; namely, the first, for reducing fractions from long and inconvenient terms to terms shorter and more easily expressed; the second, for reducing fractions of different denominators to one, or a common, denominator ; thereby preparing them for addition or subtraction, whichever of these rules we are required to perform. We will first consider the method of reducing a fraction from long and inconvenient terms to shorter and more convenient ones; and to enable us fully to consider this process, let us examine how they arise. Divide 3948136 pence amongst 2764 persons. Here then we have a sum 2764) 3948136 (1428 of long division, and we find 2764• • from the quotient that each 11841 person would receive 1428 11056 pence, and that there is a remainder of 1144. If we were 7853 to multiply this by four, we 5528 should find that the product would be 4576, and that the 23256 22112 divisor would be contained in this sum once, and leave a re 1144 mainder of 1812 ; thus, by lengthening our process, and 2764 reducing the remainder to farthings, we find we should have the į of a penny, and not only that, but still a remainder of tā, making 2764 parts of a farthing, and keeping 1812 parts out of these ; or that the remainder is something more than half a farthing. Now this is a thing that must never be attempted, because it produces the fraction of a fraction. You will always, therefore, when dividing, reduce to the lowest denomination that can be expressed by a whole number. After dividing this lowest denomination, should there be a remainder, you will draw a line, and under this line write your divisor, which then becomes the denominator of your fraction, as the remainder will become the numerator. Now in the above fraction we have 2764 parts made of a penny, out of which we retain 1144 parts. This fraction would be very inconvenient for any purposes of calculation, and therefore we reduce it to a |