3. Reduce and and and to a common denominator. A. 1, 18. to a common denominator. A. › 18. to a common denominator. A. 118, 128, 126. 6. Reduce 3, and to a common denominator. Compound fractions must be reduced to simple fractions before finding the common denominator; also the fractional parts of mixed numbers may first be reduced to a common denominator, and then annexed to the whole numbers. 9. Reduce 108 and of to a common denominator. A. 1036 › $8. 10. Reduce 8 and 144 to a common denominator. A. 8777, 1424. Notwithstanding the preceding rule finds a common denominator, it does not always find the least common denominator. But, since the common denominator is the product of all the given denominators into each other, it is plain, that this product ( XLII.) is a common multiple of all these several denominators; consequently, the least common multiple found by ¶ XLII. will be the least common denominator. 11. What is the least common denominator of, and ? OPERATION. 3)3.6.2 2)1.2 2 1 1 1 Ans. 2X3=6 Now, as the denominator of each fraction is 6ths., it is evident that the numerator must be proportionably increased; that is, we must find how many 6ths each fraction is; and, to do this, we can take ,, and of the 6ths., thus: of 64, the new numerator, written over the 6, =†. Hence, to find the least common denominator of several fractions, find the least common multiple of the denominators, for the common denominator, which, multiplied by each fraction, will give the new numerator for said fraction. 14. Reduce 14 and 13 to the least common denominator. A. 1412, 1312. Fractions may be reduced to a common, and even to the least common denominator, by a method much shorter than either of the preceding, by multiplying both the terms of a fraction by any number, that will make its denominator like the other denominators, for a common denominator; or by dividing both the terms of a fraction by any numbers that will make the denominators alike, for a common denominator. This method oftentimes will be found a very convenient one in practice. Reduce and to a common, and to a least common, denominator. X2; then & and common denominator, A. 2); then and least common denominator, A. In this example both the terms of one fraction are multiplied, and both the terms of the other divided, by the same number; consequently, (T XXXVII.,) the value is not altered. Reduce and to the least common denominator. A. 12, 12. Reduce and to the least common denominator. 500 ¶ XLIV. 1. A father gave money to his sons as follows,. to William of a dollar, to Thomas, and to Rufus ; how much is the amount of the whole How much are,, and 3, added together? 2. A mother divides a pie into 6 equal pieces, or parts, and gives to her son, and to her daughter'; how much did she give away in all? How much are and added together? 3. How much are +3 +3 ? 4. How much are f++? 5. How much are 75 +75 +15? 6. How much are 2%+2%+2%? When fractions like the above have a common denominator expressing parts of a whole of the same size, or value, it is plain, that their numerators, being like parts of the same whole, may be added as in whole numbers; but sometimes we shall meet with fractions, whose denominators are unlike, as, for example, to add and together. These we cannot add as they stand; but, by reducing their denominators to a common denominator, by ¶ XLIII., they make and, which, added together as before, make §, Ans. 1. Bought 3 loads of hay, the first weighing 194 cwt., the second 20 cwt. and the third 22 cwt.: what was the weight of the whole? ,,, reduced to a common denominator, are equal to ‡ž, 8 and 8: these, joined to their respective whole numbers, give the following expressions, viz. give 14, Ans. By adding together all the 60ths, viz. 45, 12 and 40, we have 87=187; then writing the down, and carrying the whole number, 1, to the amount of the column of whole numbers, makes 62, which, joined with EJ, makes 6287, Ans. 2. How much is of, and, added §; then and , reduced to a common deand, which, added together as before, From these illustrations we derive the following RULE. I. How do you prepare fractions to add them? A. Reduce compound fractions to simple ones, then all the fractions to a common or least common denominator. II. How do you proceed to add? A. Add their numerators. More Exercises for the Slate. 3. What is the amount of 16 yards, 17 yds. and 3 yards? ¶ XLV. 1. William, having Thomas; how much had he left? leave? of an orange, gave to How much does from 2. Harry had of a dollar, and Rufus §; what part of a dollar has Rufus more than Harry? How much does from § leave? 3. How much does 18 from 1 leave? From the foregoing examples, it appears that fractions may be subtracted by subtracting their numerators, as well as added, and for the same reason. 1. Bought 20 yards of cloth, and sold 15 yards; how much remained unsold? In this example, we cannot take fz from, but, by borrowing 1 (unit), which is 1, we can proceed thus: and are, from which taking, or 9 parts from 20 parts, leaves 11 parts, that is, t; then, carrying 1 (unit, for that which I borrowed) to 15, makes 16; then, 16 from 20 leaves 4, which, joined with, makes 411, Ans. 2. From take. and, reduced to a common denomina tor, give 13 and 3o; then, 3 from 18 leaves 36, Ans, From these illustrations we derive the following RULE. 1. What is the rule? A. Prepare the fractions as in addition; then, the difference of the numerators, written over the denom inator, will give the difference required. More Exercises for the Slate. ¶ XLVI. To divide a Whole Number by a Fraction. Lest you may be surprised, sometimes, to find in the following examples a quotient very considerably larger than the dividend, it may here be remarked, by way of illustration, that 4 is contained in 12, 3 times, 2 in 12, 6 times, 1 in 12, 12 times; and a half (1) is evidently contained twice as many times as 1 whole; that is, 24 times. Hence, when the divisor is 1 (unit), the quotient will be the same as the dividend; when the divisor is more than 1 (unit), the quotient will be less than the dividend; and when the divisor is less than 1 (unit), the quotient will be more than the dividend. 1. At of a dollar a yard, how many yards of cloth can you buy for 6 dollars? 1 dollar is, and 6 dollars are 6 times, that is, 24; then or 3 parts are contained in 2, or 24 parts, as many times as 3 is contained in 24; that is, 8 times. A. 8 yards. |