This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. 2. Find the product. (a) (2)/(7) × (7)/(9) (2)/(7) × (7)/(9) = (2 × 7)/(7 × 9) = (2)/(9) (2)/(9) (b) (5)/(8) × (3)/(7) (5)/(8) × (3)/(7) = (5 × 3)/(8 × 7) = (15)/(56) (15)/(56) (c) (4)/(15) × (9)/(16) (4)/(15) × (9)/(16) = (4 × 9)/(15 × 16) = (1 × 3)/(5 × 4) = (3)/(20) (3)/(20) (d) 4 × (3)/(5) 4 × (3)/(5) = (4 × 3)/(5) = (12)/(5) (12)/(5) (e) 7 × (3)/(5) 7 × (3)/(5) = (7 × 3)/(5) = (21)/(5) (21)/(5) (f) (17)/(12) × 6 (17)/(12) × 6 = (17 × 6)/(12) = (17)/(2) (17)/(2) 3. Find the product. (a) (8)/(15) × 20 (8)/(15) × 20 = (8 × 20)/(15) = (8 × 4)/(3) = (32)/(3) (32)/(3) (b) (5)/(9) × 15 (5)/(9) × 15 = (5 × 15)/(9) = (5 × 5)/(3) = (25)/(3) (25)/(3) (c) 3 (1)/(8) × 16 Step 1: Convert the mixed number to an improper fraction. 3(1)/(8) = (3 × 8 + 1)/(8) = (24+1)/(8) = (25)/(8) Step 2: Multiply. (25)/(8) × 16 = (25 × 16)/(8) = 25 × 2 = 50 50 (d) 3 (6)/(7) × 4 (2)/(3) Step 1: Convert mixed numbers to improper fractions. 3(6)/(7) = (3 × 7 + 6)/(7) = (21+6)/(7) = (27)/(7) 4(2)/(3) = (4 × 3 + 2)/(3) = (12+2)/(3) = (14)/(3) Step 2: Multiply. (27)/(7) × (14)/(3) = (27 × 14)/(7 × 3) = (9 × 2)/(1 × 1) = 18 18 (e) 5 (5)/(6) × 1 (5)/(7) Step 1: Convert mixed numbers to improper fractions. 5(5)/(6) = (5 × 6 + 5)/(6) = (30+5)/(6) = (35)/(6) 1(5)/(7) = (1 × 7 + 5)/(7) = (7+5)/(7) = (12)/(7) Step 2: Multiply. (35)/(6) × (12)/(7) = (35 × 12)/(6 × 7) = (5 × 2)/(1 × 1) = 10 10 (f) 4 (1)/(8) × 2 (10)/(11) Step 1: Convert mixed numbers to improper fractions. 4(1)/(8) = (4 × 8 + 1)/(8) = (32+1)/(8) = (33)/(8) 2(10)/(11) = (2 × 11 + 10)/(11) = (22+10)/(11) = (32)/(11) Step 2: Multiply. (33)/(8) × (32)/(11) = (33 × 32)/(8 × 11) = (3 × 4)/(1 × 1) = 12 12 4. Find the product. (a) 1 (3)/(5) × 2 (2)/(3) Step 1: Convert mixed numbers to improper fractions. 1(3)/(5) = (1 × 5 + 3)/(5) = (8)/(5) 2(2)/(3) = (2 × 3 + 2)/(3) = (8)/(3) Step 2: Multiply. (8)/(5) × (8)/(3) = (64)/(15) (64)/(15) (b) 12 (1)/(2) × 6 (1)/(4) Step 1: Convert mixed numbers to improper fractions. 12(1)/(2) = (12 × 2 + 1)/(2) = (25)/(2) 6(1)/(4) = (6 × 4 + 1)/(4) = (25)/(4) Step 2: Multiply. (25)/(2) × (25)/(4) = (625)/(8) (625)/(8) (c) (5)/(8) × 1000 (5)/(8) × 1000 = (5 × 1000)/(8) = 5 × 125 = 625 625 (d) 2 (4)/(15) × 1 (12)/(17) Step 1: Convert mixed numbers to improper fractions. 2(4)/(15) = (2 × 15 + 4)/(15) = (30+4)/(15) = (34)/(15) 1(12)/(17) = (1 × 17 + 12)/(17) = (17+12)/(17) = (29)/(17) Step 2: Multiply. (34)/(15) × (29)/(17) = (2 × 17)/(15) × (29)/(17) = (2 × 29)/(15) = (58)/(15) (58)/(15) (e) (12)/(25) × (15)/(28) × (35)/(36) (12)/(25) × (15)/(28) × (35)/(36) = (12 × 15 × 35)/(25 × 28 × 36) = ((12 × 3) × (5 × 7))/((5 × 5) × (4 × 7) × (3 × 12)) = (1)/(5 × 4) = (1)/(20) (1)/(20) (f) (10)/(27) × (28)/(65) × (39)/(56) (10)/(27) × (28)/(65) × (39)/(56) = (10 × 28 × 39)/(27 × 65 × 56) = ((2 × 5) × (28) × (3 × 13))/((3 × 9) × (5 × 13) × (2 × 28)) = (1)/(9) (1)/(9) 5. Find the product. (a) 1 (1)/(7) × 1 (1)/(22) × 1 (1)/(15) Step 1: Convert mixed numbers to improper fractions. 1(1)/(7) = (1 × 7 + 1)/(7) = (8)/(7) 1(1)/(22) = (1 × 22 + 1)/(22) = (23)/(22) 1(1)/(15) = (1 × 15 + 1)/(15) = (16)/(15) Step 2: Multiply. (8)/(7) × (23)/(22) × (16)/(15) = (8 × 23 × 16)/(7 × 22 × 15) = (8 × 23 × 16)/(7 × (2 × 11) × 15) = (4 × 23 × 16)/(7 × 11 × 15) = (1472)/(1155) (1472)/(1155) (b) 3 (1)/(16) × 7 (3)/(7) × (25)/(39) Step 1: Convert mixed numbers to improper fractions. 3(1)/(16) = (3 × 16 + 1)/(16) = (48+1)/(16) = (49)/(16) 7(3)/(7) = (7 × 7 + 3)/(7) = (49+3)/(7) = 5