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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Hey charlesalfandika3, good to see you again. Here are the solutions to the problems: 1. (1)/(2) + 2(3)/(4) - (1)/(8) Step 1: Convert the mixed number to an improper fraction. 2(3)/(4) = ((2 × 4) + 3)/(4) = (8 + 3)/(4) = (11)/(4) The expression becomes: (1)/(2) + (11)/(4) - (1)/(8) Step 2: Find a common denominator, which is 8. (1 × 4)/(2 × 4) + (11 × 2)/(4 × 2) - (1)/(8) = (4)/(8) + (22)/(8) - (1)/(8) Step 3: Perform the addition and subtraction. (4 + 22 - 1)/(8) = (26 - 1)/(8) = (25)/(8) Step 4: Convert the improper fraction to a mixed number. (25)/(8) = 3(1)/(8) The final answer is 3(1)/(8). 2. 2(1)/(2) × 4 ÷ (5)/(6) Step 1: Convert the mixed number to an improper fraction. 2(1)/(2) = ((2 × 2) + 1)/(2) = (5)/(2) The expression becomes: (5)/(2) × 4 ÷ (5)/(6) Step 2: Perform multiplication from left to right. (5)/(2) × 4 = (5 × 4)/(2) = (20)/(2) = 10 The expression becomes: 10 ÷ (5)/(6) Step 3: Perform division by multiplying by the reciprocal. 10 × (6)/(5) = (10 × 6)/(5) = (60)/(5) = 12 The final answer is 12. 3. 4(1)/(3) + 2(1)/(2) × 4(1)/(2) Step 1: Convert mixed numbers to improper fractions. 4(1)/(3) = ((4 × 3) + 1)/(3) = (13)/(3) 2(1)/(2) = ((2 × 2) + 1)/(2) = (5)/(2) 4(1)/(2) = ((4 × 2) + 1)/(2) = (9)/(2) The expression becomes: (13)/(3) + (5)/(2) × (9)/(2) Step 2: Perform multiplication first. (5)/(2) × (9)/(2) = (5 × 9)/(2 × 2) = (45)/(4) The expression becomes: (13)/(3) + (45)/(4) Step 3: Find a common denominator, which is 12. (13 × 4)/(3 × 4) + (45 × 3)/(4 × 3) = (52)/(12) + (135)/(12) Step 4: Perform addition. (52 + 135)/(12) = (187)/(12) Step 5: Convert the improper fraction to a mixed number. (187)/(12) = 15(7)/(12) The final answer is 15(7)/(12). 4. 16(1)/(8) + 8(1)/(4) ÷ 3(2)/(3) Step 1: Convert mixed numbers to improper fractions. 16(1)/(8) = ((16 × 8) + 1)/(8) = (128 + 1)/(8) = (129)/(8) 8(1)/(4) = ((8 × 4) + 1)/(4) = (32 + 1)/(4) = (33)/(4) 3(2)/(3) = ((3 × 3) + 2)/(3) = (9 + 2)/(3) = (11)/(3) The expression becomes: (129)/(8) + (33)/(4) ÷ (11)/(3) Step 2: Perform division first by multiplying by the reciprocal. (33)/(4) ÷ (11)/(3) = (33)/(4) × (3)/(11) = (33 × 3)/(4 × 11) = (3 × 11 × 3)/(4 × 11) = (3 × 3)/(4) = (9)/(4) The expression becomes: (129)/(8) + (9)/(4) Step 3: Find a common denominator, which is 8. (129)/(8) + (9 × 2)/(4 × 2) = (129)/(8) + (18)/(8) Step 4: Perform addition. (129 + 18)/(8) = (147)/(8) Step 5: Convert the improper fraction to a mixed number. (147)/(8) = 18(3)/(8) The final answer is 18(3)/(8). 5. (3)/(4) - (1)/(2) × (1)/(8) Step 1: Perform multiplication first. (1)/(2) × (1)/(8) = (1 × 1)/(2 × 8) = (1)/(16) The expression becomes: (3)/(4) - (1)/(16) Step 2: Find a common denominator, which is 16. (3 × 4)/(4 × 4) - (1)/(16) = (12)/(16) - (1)/(16) Step 3: Perform subtraction. (12 - 1)/(16) = (11)/(16) The final answer is (11)/(16). 6. 5(4)/(7) - 16(2)/(3) ÷ 4(1)/(6) Step 1: Convert mixed numbers to improper fractions. 5(4)/(7) = ((5 × 7) + 4)/(7) = (35 + 4)/(7) = (39)/(7) 16(2)/(3) = ((16 × 3) + 2)/(3) = (48 + 2)/(3) = (50)/(3) 4(1)/(6) = ((4 × 6) + 1)/(6) = (24 + 1)/(6) = (25)/(6) The expression becomes: (39)/(7) - (50)/(3) ÷ (25)/(6) Step 2: Perform division first by multiplying by the reciprocal. (50)/(3) ÷ (25)/(6) = (50)/(3) × (6)/(25) = (50 × 6)/(3 × 25) = (2 × 25 × 2 × 3)/(3 × 25) = 2 × 2 = 4 The expression becomes: (39)/(7) - 4 Step 3: Find a common denominator, which is 7. (39)/(7) - (4 × 7)/(1 × 7) = (39)/(7) - (28)/(7) Step 4: Perform subtraction. (39 - 28)/(7) = (11)/(7) Step 5: Convert the improper fraction to a mixed number. (11)/(7) = 1(4)/(7) The final answer is 1(4)/(7). 7. Subtract 7(3)/(8) from the sum of 1(11)/(12) and 10(1)/(6). Step 1: Calculate the sum of 1(11)/(12) and 10(1)/(6). Convert mixed numbers to improper fractions: 1(11)/(12) = ((1 × 12) + 11)/(12) = (23)/(12) 10(1)/(6) = ((10 × 6) + 1)/(6) = (61)/(6) Add the fractions: (23)/(12) + (61)/(6) = (23)/(12) + (61 × 2)/(6 × 2) = (23)/(12) + (122)/(12) = (23 + 122)/(12) = (145)/(12) Step 2: Subtract 7(3)/(8) from the sum. Convert 7(3)/(8) to an improper fraction: 7(3)/(8) = ((7 × 8) + 3)/(8) = (56 + 3)/(8) = (59)/(8) Now subtract: (145)/(12) - (59)/(8) Find a common denominator for 12 and 8, which is 24. (145 × 2)/(12 × 2) - (59 × 3)/(8 × 3) = (290)/(24) - (177)/(24) (290 - 177)/(24) = (113)/(24) Step 3: Convert the improper fraction to a mixed number. (113)/(24) = 4(17)/(24) The final answer is 4(17)/(24). 8. Divide 4(2)/(7) by 7(1)/(2) and multiply the result by (1)/(2). Step 1: Convert mixed numbers to improper fractions. 4(2)/(7) = ((4 × 7) + 2)/(7) = (28 + 2)/(7) = (30)/(7) 7(1)/(2) = ((7 × 2) + 1)/(2) = (14 + 1)/(2) = (15)/(2) Step 2: Divide 4(2)/(7) by 7(1)/(2). (30)/(7) ÷ (15)/(2) = (30)/(7) × (2)/(15) = (30 × 2)/(7 × 15) = (2 × 15 × 2)/(7 × 15) = (2 × 2)/(7) = (4)/(7) Step 3: Multiply the result by (1)/(2). (4)/(7) × (1)/(2) = (4 × 1)/(7 × 2) = (4)/(14) Step 4: Simplify the fraction. (4)/(14) = (2)/(7) The final answer is (2)/(7). 9. Divide the difference between 10(1)/(12) and 2(7)/(8) by (3)/(4). Step