REM.—Multiplying 1 by 1 is the same as dividing it by 2, and multiplying it by j is the same as dividing it by 3 ; hence, to multiply a unit by a fraction whose numerator is unity, the product will have unity for its numerator, and the denominator of the fraction for its denominator ; if both factors are fractional, the product of their denominators is the denominator of the product; if the numerator of either factor is increased, the product is similarly increased. COR.—The product of two or more fractions will have for its numerator the product of all the numerators, and for its denominator the product of all the denominators. DIVISION OF FRACTIONS. PROBLEMS. 1. Divide 4 by. ;; Ans. 4. Ans. Ans. , 4. Ans. $,$, 4 Ans. , , 3 3 PRIN. 1.- When the dividend and divisor have a common denominator, divide the numerator of the dividend by the numerator of the divisor. Prin. 2.-Any fraction having unity for its numerator, is contained in unity as many times as there are units in its denominator ; if the fraction has 2 for its numerator, it is contained one-half as many times; if it has 3 for its numerator, one-third as many times; etc. Prin. 3.-Two fractions, each having unity for its numerator, are divided by making the denominator of the divisor the dividend, and that of the dividend the divisor, regarding these numbers as integers and performing the division. For, 1:1= 2; 1:$=3; 1:1= 4; 1:= 5. 1=4; 1:}=0; 1:1=8; = 10. 1:1=1; }:f== 11; 1:1==2; *:}= = 21. PRIN. 4.-If the dividend is increased, the quotient is increased just as many times; if the dividend is diminished, the quotient is diminished in like manner. COR.-To divide any number, either integral or frac . tional, by a fraction, invert the divisor and then multiply the dividend by it. ADDITION AND SUBTRACTION OF FRACTIONS. 79 ADDITION AND SUBTRACTION OF FRACTIONS. Add 1. , 4, 6, . Ans. 28 2. }, h, jo, and it Ans. 1 $. 3. }, }, kes Ans. 36 Ans. 2316 4. $, t, f, and . 5. 13, and 37. Ans. 48 6. 27, 34, and 5%. Ans. 1046 7. 11, 23, 34, and 41. Ans. 1117 Ans. 47 8. f of $, f of }, and 4 of 4. 9. , , and for Ans. 13. Ans. 1347 10. 21, 43, and 77. 11. From | take f. 용 Ans. 12. From take Pg. Ans. Ans. 16 13. From I take Ans. 14. From is take to Ans. 15. From I take Ans. 57. 16. What is 121 – 74 ? 17. What is 10% - 47 ? Ans. 57. Ans. 10%. 18. What is 214 – 101? 19. What is 254 - 1211 ? Ans. 1241 20. What is 364 – 214 ? Ans. 1521 Ans. 4841 21. What is 7524 – 264? 22. What is 1 of 11 - off? } 1 Ans. 3 23. What is } of} - of t? Ans. 27. Ans. 11. 24. What is 4 of 5% – of 34 ? 1 25. What is off - 1 of 1 ? Ans. 17 26. What is the difference between of of and of of 1? Ans. 124 . 23 . 840 27. What is the difference between of 23 and 1 Ans. l. of 34? What is the difference between Ans. to 28. f of 12 and 1 of 164 ? Ans. 113 29. of 20and f of 36 41 ? 30. What is the difference between two fractions whose numerators are each 1 and whose denominators are 5 and 7 ? Ans. REM.—The numerator will be the difference and the denominator the product of the denominators. MULTIPLICATION AND DIVISION OF FRACTIONS. 18 1. Multiply by 3. Ans. 2. Multiply 1 by 4. Ans. 1. 3. Multiply is by 5. Ans. 1. 4. Multiply if by 6. Ans. Ans. 4. 6. Multiply of 61 by of 57. Ans. 6000 7. Multiply of of 54 by of 44. Ans. 6478 8. Multiply 61, 51, 3f and of 51. Ans. 26918 9. Multiply 4, 4, 11, of 24, and of 51. Ans. tab. 10. Multiply 3, 5, 4, , and ft. Ans. 11. Multiply 31, 51, 63, $ of A1, and 74. Ans. 355. 12. Divide 7 by 4. Ans. 241. 13. Divide 9 by š. Ans. 24. 14. Divide 30 by 24. Ans. 12. 15. Divide by . Ans. 11. 16. Divide to by Ans. 11. 17. Divide by . Ans. 2. 18. Divide by is Ans. 3, 3.21 6. 19. Divide the by Tíz: Ans. 6. 20. Divide the by 128 Ans. 91. 21. Divide 17 by :}. Ans. 185 22. Divide $13 by to Ans. 3181 23. Divide of f of f of by of t1 of %. Ans. . Reduce to a simple fraction, 3 / 3 24. Ans. S. 26. Ans. 4. * 옳 25. Ans. 27. Ans. 1. 1 x 8 iz xe 28. Ans. 10. * * 29. · Ans.. 좋을 30 30. $+$. 1-4. 1 Ans. 1994. 1 1-1 1 + 3 š xt. $. 31. Ans. TÅT 1:1 1 x } Ans. 32. 32. 1x š x*x*x*: ****$*$. 33. 37 = 25$. Ans. 03 34. 2014 • 307}. Ans. 1346 35. 1000 - 1000. Ans. Toodoo 2048• NUMERATION, ADDITION AND SUBTRACTION OF DECIMALS. Write in figures, |