22. A man bought 27 yds. of cloth, and had 192 How much was left? yds. of it made into clothes. 97 A. 84 yds. gals. of brandy in a cask, How much had he left? 23. A merchant had 56,9 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. 4937 gals. acres of ground, How many acres A. 81 Acres. DIVISION. MENTAL EXERCISES. LI. 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 7? 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 40? is contained in 3 how many times? 3. 6. are in how many times? In? In ? 7. are in how many times? In 12? In 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 If they were not the same, they might be made so by § XLIV. Hence, to divide a Fraction by a Fraction, same. I. REDUCE THE FRACTIONS TO A COMMON DENOMINATOR, AND DIVIDE THE NUMERATOR of THE DIVIDEND BY THAT OF THE DIVISOR. 8. Divide by . The Fractions, then, 9. Divide by . = 4. Ans. The least com. den. (§xLv.) is 8. are g and 3. Then, 6÷3=2 Ans. Frac. reduced and . 40÷49 ió. Divide by 1.7 by by 4. 3 by 4. It will be observed that no use, whatever, is made of the Common Denominator, and that this denominator, is, in fact, lost, in the answer. It is very manifest, then, that, it is of no use to find this denominator 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 33. 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 70-27-19-21 Ans. X 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. 24x35=10=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 12 by 1. Ans. 398-104. 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 Ans. 11. contained in 31. 1. 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 divid end to its lowest terms before dividing. 20. Divide by 24. 24. 18 Ans. 4=1}. 21. Divide 1 by. 15. Ans. =2. 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 ? 7. How often is in 6? 8. How often is 3 in 11? in 6 ? in 5? 7 in 3? 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 I divide a 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. many spools of thread 9. At of a dollar a piece, how can be bought with 2 dollars? First find the ninths in 2. 9 times. Ans. Then say are in 1, 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, 11,=. Ans. 23=28. 12. Divide 84 by 10. 13. Divide 22 by 23. Ans. 8. NOTE. The divisor, if a mixed number, must always be reduced to an improper Fraction. 21. Divide 37 by 43. A. 82 127 by 9. 948 by 8. 53 by 85. A. 6 44. 1,847 by 15 1. 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 1, what is? Ans. 225. 15. 178 is Ans. 623. of what number? 16. 256 is 20. 7,587,648 is of what number? Ans. 23,444,046,633. of what number? 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? 6. Divide by 6. § by 3. by 5. 4 by 4. 4 by 2. § by 8. & by 4. by 2. 11 by 11. by 2. by 4. 3 by 3. by 4. 1 by 6. 12 by 4. 12 by 3. 12 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. 7. Divide 13 by 3. Dividing the numerator divides the value. (§ XLI.) Ans. 251. 785 8. Divide by 27. A. 3T 4 by 9. A. 7. T. 38 9. Divide 227 by 2. 2 will not divide the numerator, but (§ XLI.) multiplying the denominator divides the value. A. 7. 297 1290 10. Divide 1 by 4. A. . 8 by 5. A. 5. 212 7887 389 A.65. Hence the rules, I. DIVIDE THE NUMERATOR OF THE FRACTION BY THE WHOLE NUMBER. OR, II. MULTIPLY THE DENOMINATOR OF THE FRACTION BY THE WHOLE NUMBER. 11. Divide 737 by 7,896,437. A. 77878777 7898437 57678777 12. Divide by 8. 8=4×2. We may divide by these factors successively. ( xxix.) 18÷2=2. $+ 48 Ans. This method is often convenient. 976 13. Divide 37 by 9. A. t. 7 by 32. A. tr. 1 by 81. A.-16. 432 14. Divide 142 by 7. Divide the whole number and Fraction separately. Ans. 2. 15. Divide 575 12 by 25. Ans. 23 16. Divide 65 by 7. If you divide the whole number by 7, you obtain a remainder of 2; for 7 is contained in 65, 9 times, and 2 over. Now this remainder 2 is §, (§ XXVII.) and § added ៖ makes.÷7=1. The answer then is 9 1. to Hence, when a remainder is left from dividing the whole number, it must, with the given Fraction, be reduced to an improper Fraction, and then divided. LIV. 1. What part of a hogshead of wine is 1 gallon is 57 gallons? 2. What part of a gallon is 1 quart? 3 quarts? 3. What part of an hour is 1 minute? 5 minutes ? 4. What part of a week is 1 day? 2 days? 3 days? 5. What part of a yard of cloth is 1 quarter? 6. What part of a minute is 1 second? 10 seconds? The following are to be written. qr.? Compound 7. What part of a pound is 3d. 1 numbers cannot be used in a Fraction. fore reduce 3d. 1 qr. to qrs.=13 qrs. qrs. Therefore, (§ xxvIII.) 3d. 1 qr. is 8. What part of 1 cwt. is 3 qrs. 15 lb.? 9. What part of 1 hogshead of wine, We must there- 455 262 This process is called bringing whole numbers of lower denominations, to Fractions of a higher. The rule seems to be, REDUCE THE GIVEN NUMBERS TO THE LOWEST DENOMINATION MENTIONED FOR A NUMERATOR, AND AN INTEGER OF THE NEXT HIGHER DENOMINATION, TO THE SAME DENOMINATION, FOR A DENOMINATOR. NOTE. Reduce the Fraction, thus obtained to its lowest terms. 10. What part of a pound is 13s. 4d.? Ans. 100=4. |