1 PERFORMED. 7x5x11 385, the common denominator. 6×5×11=330, the numerator for, which therefore equals 330 385 8 4x7x11=308, the numerator for , therefore, 38 8x5x7=280, the numerator for, therefore, fr The required fractions, therefore, are 339, 388, 388. 60 280 280 2. Reduce,,,, to a common denominator. 120 3. Reduce 62 168 72 84 63 252 252 252 252 4. Reduce 4 II 120 ° and, to a common denominator. 3 and 1 5 312, 2304 10560 240 1620 2880 2880 2880 2880 5. Reduce 32 810 1620 6 3 91 1080 1440 Ans. Ans. 16, to a common denominator. Ans. , and, to a common denominator. Ans. 486 6. Reduce,,, and to a common denominator. Ans. At the commencement of this rule, the scholar was instructed relative to the peculiar form and nature of fractions, and made acquainted with certain principles of universal application. In the course of the preceding eight cases, he has been shown the various changes of which fractions are susceptible, while their value remains unaffected. His attention will now be directed to those operations by which their value is affected. ADDITION OF FRACTIONS. If the scholar will turn back to Simple Addition, he will there find it stated, that numbers, or quantities of the same kind only, can be reduced to a single number or quantity by adding. The same is true of fractions. It is obvious that of a shilling, and of a penny, make neither of a shilling nor of a penny. But of a shilling makes 12 of a penny; and to this, we can add of a penny, and the amount will be 13 of a penny. Therefore, before we can add fractions, they must be reduced to the same denomination. (See Case 5th.) It is equally impossible to add fractions whose denominators are unlike. of a shilling added to of a shilling, makes neither of a shilling nor of a shilling. But of a shilling =2 of a shilling; and 4+ of a shilling. Fractions must therefore be reduced to a common denominator, before they can be united. (See Case 8th.) Hence we have the following rule. RULE.-Reduce all the fractions to the same denomination, and also to a common denominator; then add their numerators and place their sum over the common denominator. If the fractions produced be improper, reduce them to a whole number or mixed quantity. Note.-If any of the fractions are compound, they must be reduced to simple ones, before they can be reduced to a common denominator. (See Case 4th.) and 11, and Ex. 1. What is the sum of and? These fractions, reduced to a common denominator by Case 8th, become 21+1=1 or 12, Ans. 63 48 2. What is the sum of of, and of? By Case 4th, of =2, and of 5, and 12+3=10+103, (see Case 8th,) and +10=101 or 1788. 3. What is the sum of and? Ans. or 1. 4. What is the amount of of and of? Ans. 28. 5. What is the amount of 18 and 16, and of ? Ans. 34. Note 2d. When whole numbers are combined in the same operation with fractions, add each separately and unite them, as in the above sum. 6. What is the amount of 21, 7, 3, and of? Ans. 2941. 7. What is the amount of of a penny added to of a £.? Explanation. of a £. of a penny; and of a penny + of a penny-329 of a penny, and this equals 2 s. 3 d. 13 qr. Ans. = 8. What is the amount of of a yard and of a nail? Ans. 3 qr. Of na. of a pound added to of a shilling? 9. What is the sum of Ans. 103 s. 17 s. 2 d. 10. What is the sum of 11. What is the sum of Ans. 12. What is the sum of 5 of a ton added to of a cwt.? Ans. 113 cwt. As we can subtract a quantity only from another of the same kind, it is obvious that the same preparations are necessary to perform operations in this rule as in the preceding; therefore, RULE.-Prepare the fraction as in addition, then subtract the less numerator from the greater, and place the remainder over the common denominator. It will be obvious that the difference of the numerators is the difference sought. 8 Ex. 1. From take . Operation, }-1=31—2=4, Ans. 68 72 96 2. From take÷¬λ=-13=189, Ans. & 3. From take. Ans. 4. In this last example it is evident, that, as the denominators are the same, the operation consists in subtracting the numerators only. The same is true of all similar examples, provided only that the fractions are of the same denomination. Ans. 16 8. From of a pound take of a shilling. of a pound 60 =20 of a shilling: and 20-60 of a shilling. Therefore, - of a shilling, and 55 of a shilling 9 s. 2 d. 9. From of a league take of a mile. 1 league 3 miles; therefore, of a league of a mile; and of a mile =2 of the same; hence, 2-2 or 13 of a mile, which is the distance required. 10. From of a shilling take of a penny. Ans. 2 or 54 pence. 11. From of a day take of an hour. Ans. 144, or 204 hours. 9 10 12. From of a pound Avoirdupois take of an ounce. Ans. 273 or 1313 ounces. 20 13. From take of of. Ans. . of 3 of 4=4, and -4=2 or 1. 14. From or 2 yards. of an ell English take of a yard. Ans. Ans. 131. 15. From 9 take 6. Ans. 3. 16. From 19 yards take 57 yards. 17. From 7 Ells English take 4 yards, or 3 Ells English, yards. Ans. =44 18. From of a pound sterling take of a penny. Ans. 31 of a penny, or 2 s. 111 d. Ans. or 10 feet. 19. From of a rod take of a foot. 20. From of an ounce take of a pwt. 1611 pwt. MULTIPLICATION OF FRACTIONS. A fraction may be multiplied by a whole number, either by multiplying the numerator or dividing the denominator by that number. This has been fully illustrated in Section 4th, of Fractions. A fraction is multiplied into a whole number equal to its denominator, by rejecting that denominator. 9. Multiply by 21. By dividing part of 15; if, then, this by 21, I take part be repeated 21 times, it is evident that the value of all the parts will equal 15. RULE.-Multiply the numerators of the given fractions together for the required numerator, and the denominators, for the required denominator; then by Case 1st reduce the terms as far as practicable. Note.-Mixed numbers are to be reduced to improper fractions before multiplying; or we may first multiply the integris and then the fractions, and add their products. Ex. 1. Multiply by 2. Multiply by . 3 3. Multiply 7 by 3. =15025, or 25, Ans. . 3x=2, and ÷3=1, Ans. ==4, Ans. 7=15, and 3=10, and 15 × 10 RULE FOR CANCELING.-Place the numerators of the given fractions above a horizontal line, and their denominators below the same. Cantel, multiply, &c. as before. One important advantage of the above rule will be found in the fact, that it always gives the answer in its lowest possible terms. 5. Multiply of of by of of }}. therefore, 11=numerator, and 2×6×9×12=1296, denomi |