189 A shorter mode of finding numerators, may often be employed. Thus, - 10. Reduce la and 247 to the least com. den. A. iba and 160 3 | 12, 27, Process for the least com, mult. Then 3/4X9=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 pro. cess is similar for the other numerator. Thus, 3X9=27 the sec. ond den. Therefore, exclude 3 and 9, and multiply 4 the second numerator by 4, the remaining factor. Then 4X4=16 the second required numerator. 11. Reduce da, 37 and is to the least com. den. A. , 1344, 1344, 13410 12. Reduce an age of and to the least com. den. When seeking the least com. den. we must of course take care first to bring the given Fractions to their lowest terms. 13. Reduce 14, 5, 77, and 193 to the least com. den. A. , ii, 196, and . 14. Reduce , , an o't, f4 and 3 to the least com. den. Sometimes the denominator of one of the given Fractions may be a measure 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. Cominon 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 exer. cise his own ingenuity. 15. Take, for example the Fractions and 1. Mul-tiplying the former by 2 gives us is and i, having a com. den. Dividing the latter by 2 gives and y, having a least com. den. 16. Reduce į and it to a com. den. A. and is. 17. Reduce and into a com. den. A. 1 and To a least con. den. A. and h 18. · Reduce and is to a least com. den. A. and is and %. A. and f. its and f. and and 4 and is and 19. Reduce and, and to a least com. den. 20. Reduce and i and and it to a least com. den. of that of another. 8 T ADDITION. MENTAL EXERCISES. this way. § 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 g of a quart of chesnuts at one place, i at another, and at another. How many sixths did he buy in all ? 3. A man has of an acre of ground in his yard, in his garden, and in his corn lot. How many fourths has he in all ? 4. A man gave of a bushel of rye to one person, s to another, and to another. How many fifths did he give away? 5. How many eighths in +*+? In 1+*+*+? 6. How many twelfths in itýtis? In 't19 +5? In the above examples, the denominators are alike, and the pro. cess of Addition is very easy, since we only have to add the numerators. But when the denominators are different, we cannot add in But, by XLIV, we can make the denominators alike ; that is, we can reduce the Fractions to a common denominator, 7. Add and . and J. and and g. 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 bo reduced to lower terms, 9. Add 11 do and . A. i=. 10. Add 35. i and if. A. 2. 11. Add 20%, 11 and 1. A. 21918. 12. Add 1, 2, 3 and 3. A. 245. 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 54 13+5=18. ?t=14=11's: 18 +115=1915 A. 14. Add 177 and 164. A. '3311. A. 2839! 17. Add 38, 79, 259, 547398, 32928, 54 and 3zt. A. 1,1941%. 18. Add 13:45, 1776, 303, 99, 1011, 2018, 19, 20, and 15,44966A. 216. 19. Add 1373, 26,5 and 243 4. A. 4074. In this example, reducing is to its lowest terms, it becomes which added to makes ž, or 1. It is therefore unnecessary to reduce and is to the same denominator with . Such artifices may often be employed. 20. Add 31, 414, 334, 941, 16,'', and is. A. 38. 21. Add 1839, ji, 174, 11, 6,3 and 55. A. 59. 22. Add 13), 17, 231, 135, 14 and 53;} A. 102. 23. Add 431, 33, 10,9% 71, 6,877, 8, 10, 11 249, 491, 5,9 and 81: A. 609. MULTIPLICATION. MENTAL EXERCISES. XLVII. 1. If 1 dollar will buy of a bushel of wheat how many thirds will 2 dollars buy? 2. Ifl cent will buy 1 of an orange, how much will 5 cents buy? 3. If i 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 į are how many 6ths ? 7 times į? 7. 3 times are how many 5ths ? 6 times ? 8. 10 times are how many 4ths ? 5 times? 9. 5 times are how many 3ds ? 6 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, ($ xi.) 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, 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. (D 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 g XLI. In the same g, likewise, it was shown, that dividi the denominator multiplies the value. Hence, another rule. NUMBER. 31 Nore. 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 pupił 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 by 4. A. 41=14. 12. Multiply by 7,853. A. 328-328. Hence, A Fraction is multiplied into a number equal to its denominatar, by removing the denominator. 14. How much is 40 times M. A. Y=233. 13. How much is 27 times 31. 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. (1 x1.) Now 9 will divide the denominator 45, though 27 will not. Therefore, X9=3, and 4 X3="=183 A. This mode is often convenient. 15. A man bought 12 yards of cloth at 55 dollars a yard. How much did the whole come to. A. $623. 16. Multiply 3. by 42. A. 129). 20. What will 1971 hogsheads of wine come to at $1.23 per gallon? A. $115.34. 21. What will 44 tons of iron come to at $4.00 pr. cwt. ? A. $350. 22. What will 8 hhds, of molasses come to at $0.46 pr. gal. ? A. $360.063. 23. What cost 7jfo cwt. of sugar at $0.08 pr. Ib. ? A. $63.66,6 24. What cost 23,7 pipes of brandy at $1.43 pr. gal.? A. $4.231.653 ♡ XLVIII. In the last section you were taught to multiply a I'raction by a whole number. Multiplying a whole number by a Fraction, is exactly the same thing, for, iş xi.) it matters not which factor is made the multiplicand nor which the multiplier. This case was noticed, g xxxii. 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. f; and C. d. What amount ought each to have ? Ans. A. of $25,000=$12,500. B. fof. $25,000=$8,333.33}, and C. 7 of $25,000=$4,166.663. 2. For the same ticket was paid $16.00. What did each pay ? Ans. A. $8:00. B. $5.33}. C. $2.663. 3. A man, worth $32,750.25, wished to invest 16 of it in bank capital. How much would he have left ? A. $14,328.2376 4. At of a dollar a yard, what cost 164 yards of linen? 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 ? 50 cts. is a dollar, and 1 of 230=$115.00 A. 7. At 4d. a pr. pt. what cost 135 pts. of molasses ? 4d. is t of a shilling, and 1 of 135=45s.=£2; 5, A. 8. At 33} cts. pr. Ib. what cost 15,618 lbs. of coffee ? A. $5,206. 334 cts. is į of a dollar, and of 15,618=$5,206, A. 9. At 16 z cts. pr. lb. what cost 5,766 lbs. of sugar ? A. $991. 161 cts. is & of a dollar, and & of 5,766= $991.00 A. 10. At 25 cts. pr. Ib. what cost 9,496 lbs, of raisins ? A. $2,374. :25 cts. is of a dollar, and t of 9,495=$2,374 A, 11. At 5s. a yardówhat cost 3,656 yds, of cloth ? A. £914. 12. At 6d. pr. Ib. what cost 2,864 lbs. of sugar ? A. £71 ; 12. In 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 higher denomination of money, MULTIPLY THE. QUANTITY GIVEN, BY THE FRACTION EXPRESSING THAT PART. Note. This role may be found in other books, under the head of Prac TICE ; which name is given it on account of its practical utility. |