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Express (2+b)/(a) + (a+b)/(2a) as a single fraction.

Mathematics

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Here are the solutions for questions 4a, 5a, and 5b.

Question 4a: Express 2+ba+a+b2a\frac{2+b}{a} + \frac{a+b}{2a} as a single fraction.

  1. Find the Least Common Multiple (LCM) of the denominators.

    The denominators are aa and 2a2a.
    The LCM of aa and 2a2a is 2a2a.

  2. Rewrite each fraction with the common denominator.

    For the first term, multiply the numerator and denominator by 2:

    2+ba=2(2+b)2a=4+2b2a\frac{2+b}{a} = \frac{2(2+b)}{2a} = \frac{4+2b}{2a}

    The second term already has the denominator 2a2a:

    a+b2a\frac{a+b}{2a}
  3. Add the fractions.

    4+2b2a+a+b2a=4+2b+a+b2a\frac{4+2b}{2a} + \frac{a+b}{2a} = \frac{4+2b+a+b}{2a}
  4. Simplify the numerator by combining like terms.

    a+3b+42a\frac{a+3b+4}{2a}

    The expression as a single fraction is a+3b+42a\boxed{\frac{a+3b+4}{2a}}.

    Question 5a: Express 1ab+2bc+3ac\frac{1}{ab} + \frac{2}{bc} + \frac{3}{ac} as a single fraction.

  5. Find the LCM of the denominators.

    The denominators are ab,bc,ab, bc, and acac.
    The LCM of ab,bc,ab, bc, and acac is abcabc.

  6. Rewrite each fraction with the common denominator.

    For the first term, multiply the numerator and denominator by cc:

    1ab=1×cab×c=cabc\frac{1}{ab} = \frac{1 \times c}{ab \times c} = \frac{c}{abc}

    For the second term, multiply the numerator and denominator by aa:

    2bc=2×abc×a=2aabc\frac{2}{bc} = \frac{2 \times a}{bc \times a} = \frac{2a}{abc}

    For the third term, multiply the numerator and denominator by bb:

    3ac=3×bac×b=3babc\frac{3}{ac} = \frac{3 \times b}{ac \times b} = \frac{3b}{abc}
  7. Add the fractions.

    cabc+2aabc+3babc=c+2a+3babc\frac{c}{abc} + \frac{2a}{abc} + \frac{3b}{abc} = \frac{c+2a+3b}{abc}
  8. Rearrange the terms in the numerator in alphabetical order.

    2a+3b+cabc\frac{2a+3b+c}{abc}

    The expression as a single fraction is 2a+3b+cabc\boxed{\frac{2a+3b+c}{abc}}.

    Question 5b: Express a2−2ef+43f\frac{a}{2} - \frac{2}{ef} + \frac{4}{3f} as a single fraction.

  9. Find the LCM of the denominators.

    The denominators are 2,ef,2, ef, and 3f3f.
    The LCM of the numerical coefficients (2 and 3) is 6.
    The LCM of the variable parts (efef and ff) is efef.
    So, the LCM of 2,ef,2, ef, and 3f3f is 6ef6ef.

  10. Rewrite each fraction with the common denominator.

    For the first term, multiply the numerator and denominator by 3ef3ef:

    a2=a×3ef2×3ef=3aef6ef\frac{a}{2} = \frac{a \times 3ef}{2 \times 3ef} = \frac{3aef}{6ef}

    For the second term, multiply the numerator and denominator by 6:

    −2ef=−2×6ef×6=−126ef-\frac{2}{ef} = -\frac{2 \times 6}{ef \times 6} = -\frac{12}{6ef}

    For the third term, multiply the numerator and denominator by 2e2e:

    43f=4×2e3f×2e=8e6ef\frac{4}{3f} = \frac{4 \times 2e}{3f \times 2e} = \frac{8e}{6ef}
  11. Combine the fractions.

    3aef6ef−126ef+8e6ef=3aef−12+8e6ef\frac{3aef}{6ef} - \frac{12}{6ef} + \frac{8e}{6ef} = \frac{3aef - 12 + 8e}{6ef}
  12. Rearrange the terms in the numerator for clarity.

    3aef+8e−126ef\frac{3aef + 8e - 12}{6ef}

    The expression as a single fraction is 3aef+8e−126ef\boxed{\frac{3aef + 8e - 12}{6ef}}.

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