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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a) 589 + 759 Step 1: Add the numbers. 589 \\ + 759 \\ 1348 The sum is 1348. b) 8589 - 4289 Step 1: Subtract the numbers. 8589 \\ 4289 \\ 4300 The difference is 4300. c) 325 × 569 Step 1: Multiply 325 by 9. 325 × 9 = 2925 Step 2: Multiply 325 by 60. 325 × 60 = 19500 Step 3: Multiply 325 by 500. 325 × 500 = 162500 Step 4: Add the partial products. 325 \\ × 569 \\ 2925 & (325 × 9) \\ 19500 & (325 × 60) \\ 162500 & (325 × 500) \\ 184925 The product is 184925. d) 0.27 ÷ 0.3 Step 1: Convert the division to an equivalent form with a whole number divisor by multiplying both numbers by 10. 0.27 ÷ 0.3 = (0.27 × 10) ÷ (0.3 × 10) = 2.7 ÷ 3 Step 2: Perform the division. 2.7 ÷ 3 = 0.9 The quotient is 0.9. e) 2(3)/(8) ÷ 7(1)/(2) Step 1: Convert the mixed numbers to improper fractions. 2(3)/(8) = ((2 × 8) + 3)/(8) = (16 + 3)/(8) = (19)/(8) 7(1)/(2) = ((7 × 2) + 1)/(2) = (14 + 1)/(2) = (15)/(2) Step 2: Rewrite the division as multiplication by the reciprocal of the second fraction. (19)/(8) ÷ (15)/(2) = (19)/(8) × (2)/(15) Step 3: Multiply the fractions. (19)/(8) × (2)/(15) = (19 × 2)/(8 × 15) = (38)/(120) Step 4: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2. (38)/(120) = (38 ÷ 2)/(120 ÷ 2) = (19)/(60) The result is (19)/(60). f) 2(1)/(3) + 4(2)/(5) Step 1: Convert the mixed numbers to improper fractions. 2(1)/(3) = ((2 × 3) + 1)/(3) = (6 + 1)/(3) = (7)/(3) 4(2)/(5) = ((4 × 5) + 2)/(5) = (20 + 2)/(5) = (22)/(5) Step 2: Find a common denominator for the fractions. The least common multiple of 3 and 5 is 15. (7)/(3) = (7 × 5)/(3 × 5) = (35)/(15) (22)/(5) = (22 × 3)/(5 × 3) = (66)/(15) Step 3: Add the fractions. (35)/(15) + (66)/(15) = (35 + 66)/(15) = (101)/(15) Step 4: Convert the improper fraction back to a mixed number. 101 ÷ 15 = 6 with a remainder of 11 So, (101)/(15) = 6(11)/(15). The sum is 6(11)/(15). g) 4002 ÷ 23 Step 1: Perform long division. Divide 40 by 23. 23 × 1 = 23. r 1 \\ 23)4002 \\ -23 \\ 17 Step 2: Bring down the next digit (0). Divide 170 by 23. 23 × 7 = 161. r 17 \\ 23)4002 \\ -23 \\ 170 \\ -161 \\ 9 Step 3: Bring down the next digit (2). Divide 92 by 23. 23 × 4 = 92. r 174 \\ 23)4002 \\ -23 \\ 170 \\ -161 \\ 92 \\ -92 \\ 0 The quotient is 174.