2 thirds equal 8 twelfths (see 51); 5 twentieths equal 1 fourth; and 1 fourth equals 3 twelfths. (3) SUBTRACTION.-3 twelfths from 8 twelfths leave 5 twelfths. (4) RESULT.-Hence, 5 twentieths from 6 ninths leave RULE. Reduce the fractions to other fractions which shall have the same denominators; subtract the numerator of the subtrahend from that of the minuend; place the difference over the denominator. Prac REMARK 1.—In all operations in fractions, it is generally best to begin by reducing given fractions to their lowest terms. tice will soon teach the pupil when this preliminary step is without advantage. 5. Find the value of -;-; 1-3; 1-4; 1-1; 31 3 1 1. 3 Ans. ; ; ; ; 10; 15; 4. 6. Find the value of -;-;-3; —; 1-3; 17 3 # 1 1 Ans. 1 ; ↓ ; b ; 1. 24 § ; 4 ¦ ; 7. Find the value of 19-;-19; 18-12; 18-38; 8. Find the value of 31-14; 18-3;28-24; 31-24; 201 5 9 3 Ans. 0 24 36 40 20 0. 40 20i to. REMARK 2.-A proper fraction is subtracted from 1 by writing the difference between the numerator and denominator over the de nominator. Thus: 1-3; since—}=}. And 1—; since 1-4=1. 3 9. Find the value of 1-1; 1-3; 1-3; 1-3; 1—; 1-7; 1-8. Last Ans. = 10. Find the value of 2-1; 2-; 2-1; 2-1; 2-}; 2-7;2-5; 2-1. Ans. 1; 1; 1, etc. 13 11. Find the value of 3-3; 4-8; 5-7; 6-3; 7-1; 8-4; 9-; 27-3. Ans. 24; 3; 4, etc., to 26,8. 12. Copy and recite the following exercises: (1) (2) 9-93 33: (3) (4) RECITATION. (1) PROBLEM.-What is the difference between 83 and 33? (2) REDUCTION.-3 eighths equal 9 twenty-fourths; 2 thirds equal 16 twenty-fourths. (51.) (3) SUBTRACTION.-1 (=4) added to gives; 16 from leave 17. 1 added to 3 gives 4; 4 from 8 leaves 4. (4) RESULT.-Hence, 33 from 83 leaves 417. REMARK 3.—Hence, when the fraction in a mixed subtrahend is larger than the fraction in the minuend, the numerator of the upper fraction is added to the difference between the terms of the lower fraction for the numerator of the answer, and 1 is added to the whole number of the subtrahend to preserve the true differ(See 32, ex. 41. REMARK.) ence. 13. Copy and recite the following exercises: (4) (1) 2311=2311 (2) 140 =1418 63 615 (3) 133-1315 61=60 129 341 14. From 27 17. From 27 take 93 take 241. take 185. 18. From 125 take 274. 19. From 125 take 274. FRACTIONS. 20. A boy gave 3 quarters of an orange to his sister. How many quarters had he left? 21. If you give away the apple is left? Ans. 1 quarter. of an apple, what part of Ans.. 22. If of a week are past, what part of the week Ans. . is yet to come? 23. If I buy a barrel of flour for 147 dollars, and sell it for 15 dollars, what do I gain? Ans. § of a dollar. 24. If I buy 123 yards of calico, and sell 45% yards, how many yards have I left? Ans. 78 yards. 25. If you give of an orange to one boy, of it to another, and of it to another, what part of the orange remains? Ans. of the orange. MULTIPLICATION OF FRACTIONS. 56. To multiply a fraction by a whole number. (1) PROBLEM.-What is the product of multiplied by 6? (2) MULTIPLICATION.-6 times 3 eighths are 18 eighths; 18 equal 21. (3) RESULT.-Hence, 6 times are 21. RULE. (1.) Multiply the numerator by the whole number, and place the product over the denominator. REMARK 1.-All answers should be whole numbers, mixed num. bers, or proper fractions. (See 53. REMARK 1.) 2. Find the value of x3; ×4; x2; x7; x8; ×12;×6. Ans. 1; 1; 1; 21; 4; 5; 31. 3. Find the value of x7; x8; 2×8; x4; } x12; &×6; × 20. Ans. 2g; 43; 4; 13; 5; 31; 53. 4. Copy and recite the following exercises: (1) PROBLEM.-What is the product of § multiplied by 4? (2) MULTIPLICATION.-4 times 3 eighths are 3 halves. (3) RESULT.-Hence, 4 times are 12. RULE. (2) Divide the denominator by the whole number, if it is divisible without a remainder, and place the quotient under the numerator. REMARK 2.-Since the numerator of a fraction is a dividend and the denominator a divisor, it is evident that if the denominator be divided by a number, the quotient, or value of the fraction, will be multiplied by the same number. 5. Find the value of 12×4; ×3; 2×4; 2× 9. Ans. 6. Find the value of 105x27; 502x 19. Ans. 54 x2; 7×3; 13×6; 2 31; 81; 61; 51; 13. 6; 31; x 13; 8x 14; 8×16; 7×17; 63; 143; 273; 321; 52; 1271. 7. Multiply×2; 3; 4; 5; 6; 7; 8; 9; 10; 24; 36; Ans. 1; 1; ... 35. 8. Multiply by 2; 3; 4; 5; 6; 7; 8; 9; 10; 72; 108; 144; 360. 48; 60. Ans. 1; 1; 14; ... 130. 13 18 9. Multiply by 3; 18; 6; 9; 2; 27. Ans. 13, etc. FRACTIONS. REMARK 3.-Any fraction multiplied by its denominator gives the numerator for a quotient. 3 21 10. Find the value of x8; 3x5; 1×317; X 358; 512x 52; 1006x317. Ans. 7; 3; 4; 21; 519; 1006. 57. To multiply a mixed number by a whole number. (1) PROBLEM.—What is the product of 5g multiplied by 12? (2) MULTIPLICATION. -12 times 3 eighths are 36 eighths; 36-4 units. 12 times 5 units are 60 units; 60 units and 4 units are 641⁄2 units. (3) RESULT.-Hence, 12 times 5g are 64. Add the product RULE. the fraction multiplied by the whole number (found by 56) to the product of the whole numbers (found by ). 2. Multiply 14 by 2; 4; 6; 8; 22; 24. 3. Multiply 23 by 12; 21; 3; 15; 51. 10; 12; 14; 16; 18; 20; 4. Multiply 33 by 3; 5; 7; 9. 5. Multiply 61 by 4; 12 by 8; 61 by 16; 55 by 18. Ans. 3; 6; 9, etc. 18; 27; 36; 45; 6; 9; Ans. 32; 56; 8, etc. 163 6. Multiply 22, by 3; 1618 by 4; by 6; 81 by 8; 717 by 9. Ans. 111, etc. by 6; 331 by 3; Ans. 25; 100. 13, by 5; 111⁄2 Ans. 671. |