A shorter mode of finding numerators, may often be employed. Thus, 10. Reduce and 3|12, 27, to the least com. den. A. and Process for the least com. mult. Then 3X4X9=108 is the least com. mult. 3 | 4, 9, and 4 (the first two of these factors) multiplied, make 12, the first given denominator. The remaining factor, being 9, shows that the com. den. is 9 times the given den. 12. Of course the corresponding numerator, in the answer, ought likewise, to be 9 times the given numerator 5. Hence, 9X5-45 the first numer. ator. It will be seen that this is obtained by excluding from the fac. tors of the com. den. just enough to produce 12, the given den., and multiplying the numerator 5, by the remaining factor 9. The process is similar for the other numerator. Thus, 3×9-27 the second den. Therefore, exclude 3 and 9, and multiply 4 the second numerator by 4, the remaining factor. Then 4X4-16 the second required numerator. When seeking the least com. den. we must of course take care first to bring the given Fractions to their lowest terms. 30 40 135 13. Reduce 1, 28, 77, and 14 to the least com. den. A., 135, 18, and 84. 14. Reduce, 24, 75, 81, 44 and 32 to the least com. den. Sometimes the denominator of one of the given Fractions may be a measure of that of another. In this case, by multiplying both terms of the former, it may be brought to the same denominator with the latter: or, if the numerator of the latter admits of being divided, we may bring this one to the same denom. inator with the former. Common denominators, and sometimes a least common denominator may often be readily found in this way. Sometimes both Division and Multiplication may be used, but we leave the pupil, in this respect, to exercise his own ingenuity. and. Muland, having a com. and, having a 15. Take, for example the Fractions tiplying the former by 2 gives us den. Dividing the latter by 2 gives least com. den. A. 16. Reduce and 11⁄2 to a com. den. 17. Reduce and to a com. den. A. To a least con. den. A. § and 3. to a least com. den. A. 12 and . and 19. Reduce and 4 and 2 to a least com. den. and to a least com. den. ADDITION. MENTAL EXERCISES. XLVI. 1. George gave of an orange to his sister, to his brother, and then had himself. How many fourths had he at first? 2. William bought of a quart of chesnuts at one place, at another, and at another. How many sixths did he buy in all? 3. A man has his garden, and has he in all? 4. A man gave to another, and give away? of an acre of ground in his yard, in in his corn lot. How many fourths of a bushel of rye to one person, to another. How many fifths did he 5. How many eighths in 1+2+3? In 3+8+}+{? 6. How many twelfths in ++? In t 3? 12+12 +3? 2 T2 In the above examples, the denominators are alike, and the process of Addition is very easy, since we only have to add the numerators. But when the denominators are different, we cannot add in this way. But, by § XLIV, we can make the denominators alike; that is, we can reduce the Fractions to a common denominator, 7. Add and . and . and. and . 8. Add and. and ¦. and. and 7. The following are to be written. It is always best to reduce Fractions to their lowest terms before adding, The sum may often be reduced to a whole or mixed number. When this cannot be done, it may often be reduced to lower terms. 9. Add, and . A. =). 10. Add, and 8. A. 2. 3,1 3 60 96 11. Add 97%, 14 and 38. A. 21113. 120 49 12. Add 1,5, 18, 33 and 32. A. 21. When mixed numbers occur, it is plain that we must add the whole, and the Fractional parts separately. The two sums must then be united into one, 13. Add 13 and 53 13+5=18. 1+3=18=1'. 18 +1=19 Á. 14. Add 17 and 16. A. 33. 15. Add 15, 143, 3 and 3. A. 30. 16. Add 7, 194, 14, 9, 71 252 and 15. A. 28319! 76 17. Add 38, 18, 259, 5478, 3293%, 54 and 31⁄2ʊ. A. 1,194. 18. Add 13, 17, 30, 99, 1017, 2018, 19, and 15,144. A. 216. 1728 19. Add 1373, 26, and 243. A. 407%. In this example, reducing to its lowest terms, it becomes which added to makes, or 1. It is therefore unnecessary to reduce and to the same denominator with 4. Such artifices may often be employed. 20. Add 33, 411, 311, 942, 161 13 108 A. 102. 21. Add 184, 1, 172, 11, 6,3 and 5. A. 59. 22. Add 1313, 17, 233, 138, 14 and 531. 23. Add 4, 3, 1050 737, 6,27, 844, 14. 27, 4, 5 and 853. A. 609. 19 MULTIPLICATION. MENTAL EXERCISES. 34 1049 221 § XLVII. 1. If 1 dollar will buy of a bushel of wheat how many thirds will 2 dollars buy? 2. If1 cent will buy of an orange, how much will 5 cents buy? 3. If 1 bushel of oats cost of a dollar, what will 2 bushels cost? 4. If 1 apple cost of a cent, what will 3 apples cost? 5. A boy gave of a pine-apple to one of his companions, and 3 times as much to another, how much did he give the last? 6. 4 times 7. 3 times 8. 10 times 9. 5 times are how many 6ths? 7 times ? In the above examples, a Fraction is the multiplicand, and a whole number the multiplier; in other words, it is required to mul. tiply a Fraction by a whole number. As Multiplication is only a repeated Addition of the multiplicand, (§ x1.) it is plain, that, in this case, it may be performed like Addition of Fractions, in the last section. Thus, 10. A man had of a dollar in each pocket, and he had 4 pockets. How many fourths of a dollar had he? Repeat, 4 times by Addition, thus, 2+2+3+3=4 Ans. Hence it appears, that Multiplication of a Fraction by a whole number, may be performed by a repeated Addition of the Numerator of the multiplicand. But this repeated Addition is Multiplication of the Numerator. (§ XI.) Hence, to multiply a Fraction by a whole number, I. MULTIPLY THE NUMERATOR OF THE FRACTION BY THE WHOLE NUMBER. This was also shown in § XLI. In the same §, likewise, it was shown, that dividing the denominator multiplies the value. Hence, another rule. II. DIVIDE THE DENOMINATOR OF THE FRACTION BY THE WHOLE NUMBER. NOTE. This is altogether the best rule, and should always be used when the denominator admits of Division. In performing the following examples for practice, let the pupil divide the denominator, in all cases, where it is practicable. The results should be reduced, if possible, to whole or mixed numbers as in Addition. 11. Multiply 25 by 4. A. 25=1& 7853 12. Multiply 32 by 7,953. A. 328-328. Hence, A Fraction is multiplied into a number equal to its denominatar, by removing the denominator. 120 A. =233. 14. How much is 40 times 13. How much is 27 times 27 is a composite number, having the factors 9 and 3. We may multiply by these factors, successively, instead of multiplying by 27 at once. (§ XI.) Now 9 will divide the denominator 45, though 27 will not. Therefore, X9, and x3=183 A. This mode is often convenient. 15. A man bought 12 yards of cloth at 5 yard. How much did the whole come to. 31 dollars a A. $62. To multiply a mixed by a whole number, multiply the whole, and fractional parts separately, and unite the products. 16. Multiply 35 by 42. A. 129. 17. Multiply 7 by 40. A. 2811. 18. Multiply 15 by 471. A. 294. 19. Multiply 967 1000 by 138. A. 133388. 20. What will 1333 hogsheads of wine come to at $1.23 per gallon? A. $115.34. 21. What will 43 tons of iron come to at $4.00 pr. cwt.? A. $350. 22. What will Sg hhds. of molasses come to at $0.48 pr. gal.? A. $360.063. 23. What cost 74 cwt. of sugar at $0.08 pr. lb. ? A. $63.66. 24. What cost 2313 pipes of brandy at $1.43 pr. gal.? A. $4.231.653 § XLVIII. In the last section you were taught to multiply a Fraction by a whole number. Multiplying a whole number by a Fraction, is exactly the same thing, for, (§ xI.) it matters not which factor is made the multiplicand nor which the multiplier. This case was noticed, § xxxIII. To this case belong such questions as these: what is of 24; what is of 40, &c.: that is all questions where a whole is given to find a certain part. EXAMPLES FOR PRACTICE. 1. Three men, A. B. and C. drew a prize in a lottery, of $25, 000.00, A. owned the ticket; B.; and C. 1. What amount ought each to have? Ans. A. of $25,000-$12,500. B. j of. $25,000-$8,333.33, and C. of $25,000 $4,166.66}. 2. For the same ticket was paid $16.00. What did each pay? Ans. A. $8.00. B. $5.33. C. $2.66. 3. A man, worth $32,750.25, wished to invest of it in bank capital. How much would he have left? A. $14,328.237. 4. At of a dollar a yard, what cost 164 yards of linen? 16 A. $123.00. 5. At 75 cts. a yard, what cost 164 yards of linen ?75 cts. is of a dollar; therefore the question is the same as the last. 6. At 50 cts a yard, what cost 230 yards of cambric ? 7. At 4d. a pr. pt. what cost 135 pts. of molasses? 8. At 33 33 cts. is 9. At 16 16 cts. is cts. pr. lb. what cost 15,618 lbs. of coffee? A. $5,206. of a dollar, and § of 15,618=$5,206, A. cts. pr. lb. what cost 5,766 lbs. of sugar? A. $991. of a dollar, and of 5,766 $991.00 A. 10. At 25 cts. pr. lb. what cost 9,496 lbs. of raisins? A. $2,374. 25 cts. is of a dollar, and 4 of 9,495 $2,374 A, 11. At 5s. a yard what cost 3,656 yds. of cloth? A. £914. 12. At 6d. pr. lb. what cost 2,864 lbs. of sugar? A. £71; 12In the above examples, the price is an even part of a higher de. nomination, as a dollar, a shilling, &c. An even part is called an ALIQUOT part. Hence, when the price of a single thing, (as a gallon, pound, &c.) is an aliquot part of a higher denomination of money, MULTIPLY THE QUANTITY GIVEN, BY THE FRACTION EXPRESSING THAT PART. NOTE, This rule may be found in other books, under the head of PRAC TICE; which name is given it on account of its practical utility. |