SUBTRACTION. MENTAL EXERCISES. § L. 1. A boy had of a quart of chesnuts, and gave away. How many eighths had he left? 2. A man bought of a barrel of flour, and gave to some labourers. How much had he left? 3. Two boys together gathered of a peck of strawberries, and one of them took 3 of a peck, as his share. How many had the other? 4. from leaves how many ninths? 5. from leaves how much? 6? from 47? from? from 11? from 8? from 5 ? from But we shall often be required to make subtractions between Fractions whose denominators are different. In this case we may reduce them to a common denominator, and then we can subtract as above. 63 from 16. from 51. 92 189 NOTE. When mixed numbers occur, perform the subtraction of the whole, and fractional parts separately, if it can be done. 13. From 2013 take 63. 14. From 3711 take 202. A. 14. 15. From 30 take 151. A. 157. 16. From 21 take 63. A. 147. NOTE. When the Fraction in the subtrahend is greater than that in the minuend, it will be necessary to borrow a unit. Sometimes, likewise, there will be no Fraction in the minuend. In this case, it will also be necessary to borrow. 17. From 225, take 71, A. 153. 18. From 6213 take 4975. A. 12311. 19. From 525 take 326. A. 198149. 20. From 2,983217 take 1,84334. A. 1,1393333. 21. From 21488 take 13327049. A. 81898 8 8 7 7 7 7 7. Hence, to perform Subtraction of Fractions, 2 28 REDUCE THE FRACTIONS TO A COMMON DENOMINATOR, FIND THE DIF FERENCE OF THE NUMERATORS, AND WRITE IT OVER THE COMMON DE NOMINATOR. 22. A man bought 27 yds. of cloth, and had 192 yds. of it made into clothes. How much was left? A. 84 yds. 23. A merchant had 56,7 gals. of brandy in a cask, How much had he left? and 7 gals. leaked out. 24. A man had a lot containing 97 from which he fenced off 14 acres. were left in the lot? A. 49378 gals. acres of ground, How many acres A. 811 Acres. $ LI. DIVISION. MENTAL EXERCISES. 1. At of a dollar a yard, how many yards of calico can be bought with of a dollar? Hom many with ? With ? With? With? With 24? 2. At of a dollar a bushel, how many bushels of oats can be bought for 4 of a dollar? How many for ? For ? For 4o ? 3. is contained in 3 how many times? In g? 6. are in In 1? In how many times? In ? 7.3 are in how times? many The pupil will see, that he has been finding how often one Fraetion is contained in another; that is, he has been dividing a Fraction by a Fraction. In the above examples, the denominators are the same. If they were not the same, they might be made so by § XLIV. Hence, to divide a Fraction by a Fraction, I. REDUCE THE FRACTIONS TO A COMMON DENOMINATOR, AND DIVIDE THE NUMERATOR OF THe dividend BY THAT OF THE DIVISOR. 8. Divide by 3. The least com. den. (§xLv.) is 8. The Fractions, then, are and 3. Then, 6-3=2 Ans. 9. Divide by . Frac. reduced 18 and 18. 40:49 1. Ans. 49 io. Divide by .by. by 4. by 4. 3 swer. It will be observed that no use, whatever, is made of the Common Denominator, and that this denominator, is, in fact, lost, in the an. It is very manifest, then, that, it is of no use to find this de. aominator ITSELF, if we only find the numerators, AS THOUGH we were reducing the fractions to a common denominator. For these numer. ators are the only numbers concerned in the process, Thus, 11. Divide by 35. Find the numerators as though for a common denominator; which is done (§ XLIV.) by multiplying 35 into 2=70, and 27 into 1=27. Then 70÷27=19=214 Ans. The answer obtained, is evidently the same we should have found, if we had turned over, or inverted the Divisor, and then, multiplied the two upper numbers together for a numerator, and the two lower for a denominator. X 27x35-14=21 as before. But this process after inverting the Divisor, is exactly like the rule for Multiplication of Fractions. (§ XLIX.) Hence, to divide a Fraction by a Fraction, II. INVERT the divisor and proceed AS IN MULTIPLICATION. 12. Divide 13 by 15. Ans. 1818. Ans.. 2456 13. Divide 24 by 1912 14. Divide 9876 by 3233. 10859 555 Ans. 28665-26937 15. Divide 253611 by 3030301. 1002001 16. Divide 2 by . Ans =63. 10859 10859 A. 1567338861044 30 36 36 46 32 301 NOTE. When a mixed number occurs, it is to be reduced to an improper Fraction. ( XXVII.) 17. Divide 5 by 84. 18. How often is 2 19. How often is 6 contained in 131.6 may be made an improper Fraction by placing 1 under it, thus f. Ans ==23. NOTE. It will be best usually to reduce the divisor, and, sometimes the dividend to its lowest terms before dividing. 4 20. Divide by 24. 24=1. Ans. 4=1}. 18 21. Divide 1 by.. Ans. 5-21. 18 MENTAL EXERCISES. LII. 1. A man bought cotton cloth, to the amount of 5 dollars, at of a dollar a yard. How many yards did he buy? 2. A boy bought marbles at a cent apiece, to the amount of 12 cents. How many marbles did he buy? 3. How many fifths are there in 4? In 5? 4. How many sevenths are there in 3? In 4? 5. A man put 6 barrels of beer into kegs that held of a barrel each. How many kegs did it take? 6. How many times in 10? in 9? in 8? in 4? in 6 ? in 5? 7 in 3? σ' in 6? in 4? 7. How often is in 6? 8. How often is 3 in 11? In the above examples, it is required to find how often a Fraction is contained in a whole number, that is, to divide a whole number by a Fraction. The quotient will be seen to be larger than the dividend, and it ought so to be whenever a proper Fraction is the divisor; for (§ XLI.) the smaller the divisor, the greater the quotient. There are twice as many half-pints in a pail of water, as there are pints. If I divide & whole number by 1, the quotient will be the same whole number. Thus 1 is in 8, 8 times; 1 is in 12, 12 times, &c. If I divide by a number less than 1, then, I ought to obtain a greater quotient. Thus is in 8, 16 times; is in 12, 24 times, &c.; in which cases, I have a quotient, twice as great as my dividend. 9. At of a dollar a piece, how many spools of thread can be bought with 2 dollars? First find the ninths in 2=. 9 times. Ans. Then say are in 3, Hence, to divide a whole number by a Fraction, MULTIPLY THE WHOLE NUMBER BY THE DENOMINATOR OF THE FRAC TION, AND DIVIDE THE PRODUCT BY THE NUMERAtor. 11. Divide 24 by 1. It is best to reduce the Fraction to its lowest terms, 1,4. Ans. 28=28. 12. Divide 84 by 9. Ans. 901. 13. Divide 22 by 2 2-. Ans. 8. NOTE.-The divisor, if a mixed number, must always be reduced to an improper Fraction. 21. Divide 37 by 43. 127 by 9. 948 by 84. A. 8. 53 by 8. A. 64. 1,847 by 15 12. To this case belong such questions as these; 4 is of what number; 6 is of what number, &c.; that is, all questions where a certain part is given, to find a whole. 14. 125 is of what number? Ans. 225. Perhaps it may be well for beginners, to accustom themselves to solve these examples by analysis, thus. If 125 is 5 ninths, what is one ninth? Ans. 25. If 25 is what is? 15. 178 is Ans. 225. of what number? Ans. 623. 16. 256 is of what number? Ans. 352. 17. 1,874 is Ans. 14,055. 18. 29,864 is of what number? 19. 425,763,891 is of what number? Ans. 23,444,046,633, 20. 7,587,648 is 72 of what number? 227 21, 1,343,826 is 18 of what number? 968 MENTAL EXERCISES. LIII. 1. A lady, having of a yard of ribbon, cut it into 5 equal parts. How much was there in each part? 2. Thomas had of a pint of chesnuts, and distributed them equally among 4 of his companions. How many had each? 3. A man divided of an acre of ground into 7 equal parts. How much was there in each part? 4. A boy divided of an orange among 5 of his companions. How much did he give each? 5. A grocer put of a hogshead of brandy into 6 casks, putting an equal quantity into each. How much did he put in each? 8. 6. Divide by 6. by 4. by 2. 1 by 3. by 5. 4 by 4. 4 by 2. by by 11. by 2. 4 by 4. 3 by 3. by 4. 12 by 6. 13 by 4. 12 by 3. 1 by 2. In the above examples, a whole number is the divisor, and a fraction, the dividend; that is, it is required to divide a fraction by a whole number. Let the following be written. 762 7. Divide by 3. Dividing the numerator divides the value. (§ XLI.) Ans. 25. 8. Divide 2 by 27. A. 4 by 9. A. 385. 9. Divide by 2. 2 will not divide the numerator, but (§ XLI.) multiplying the denominator divides the value. A. 297 1290 6 8 10. Divide by 4. A. 7. 28 by 5. A. 73. Hence the rules, I. DIVIDE THE NUMERATOR OF THE FRACTION BY THE WHOLE NUMOR, BER. II. MULTIPLY THE DENOMINATOR OF THE FRACTION BY THE WHOLE NUMBER. 11. Divide 7898437, by 7,896,437. 5767877 25 A. 37678777° 12. Divide by 8. 8=4×2. We may divide by these factors successively. (§ xxix.) 13÷2=25• 25+ 4 Ans. This method is often convenient. 976 1143 13. Divide 373 by 9. A. . by 32. A. 7917. 432 by 81. A. 17. 15. Divide 575 by 25. Ans. 23 16. Divide 65 by 7. If you divide the whole number by 7, you obtain |