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 for questions 4a, 5a, and 5b. Question 4a: Express (2+b)/(a) + (a+b)/(2a) as a single fraction. Step 1: Find the Least Common Multiple (LCM) of the denominators. The denominators are a and 2a. The LCM of a and 2a is 2a. Step 2: Rewrite each fraction with the common denominator. For the first term, multiply the numerator and denominator by 2: (2+b)/(a) = (2(2+b))/(2a) = (4+2b)/(2a) The second term already has the denominator 2a: (a+b)/(2a) Step 3: Add the fractions. (4+2b)/(2a) + (a+b)/(2a) = (4+2b+a+b)/(2a) Step 4: Simplify the numerator by combining like terms. (a+3b+4)/(2a) The expression as a single fraction is (a+3b+4)/(2a). Question 5a: Express (1)/(ab) + (2)/(bc) + (3)/(ac) as a single fraction. Step 1: Find the LCM of the denominators. The denominators are ab, bc, and ac. The LCM of ab, bc, and ac is abc. Step 2: Rewrite each fraction with the common denominator. For the first term, multiply the numerator and denominator by c: (1)/(ab) = (1 × c)/(ab × c) = (c)/(abc) For the second term, multiply the numerator and denominator by a: (2)/(bc) = (2 × a)/(bc × a) = (2a)/(abc) For the third term, multiply the numerator and denominator by b: (3)/(ac) = (3 × b)/(ac × b) = (3b)/(abc) Step 3: Add the fractions. (c)/(abc) + (2a)/(abc) + (3b)/(abc) = (c+2a+3b)/(abc) Step 4: Rearrange the terms in the numerator in alphabetical order. (2a+3b+c)/(abc) The expression as a single fraction is (2a+3b+c)/(abc). Question 5b: Express (a)/(2) - (2)/(ef) + (4)/(3f) as a single fraction. Step 1: Find the LCM of the denominators. The denominators are 2, ef, and 3f. The LCM of the numerical coefficients (2 and 3) is 6. The LCM of the variable parts (ef and f) is ef. So, the LCM of 2, ef, and 3f is 6ef. Step 2: Rewrite each fraction with the common denominator. For the first term, multiply the numerator and denominator by 3ef: (a)/(2) = (a × 3ef)/(2 × 3ef) = (3aef)/(6ef) For the second term, multiply the numerator and denominator by 6: -(2)/(ef) = -(2 × 6)/(ef × 6) = -(12)/(6ef) For the third term, multiply the numerator and denominator by 2e: (4)/(3f) = (4 × 2e)/(3f × 2e) = (8e)/(6ef) Step 3: Combine the fractions. (3aef)/(6ef) - (12)/(6ef) + (8e)/(6ef) = (3aef - 12 + 8e)/(6ef) Step 4: Rearrange the terms in the numerator for clarity. (3aef + 8e - 12)/(6ef) The expression as a single fraction is (3aef + 8e - 12)/(6ef). That's 2 down. 3 left today — send the next one.