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Calculate the fraction of the farm given to the wife and the first child.

Mathematics

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Let TT be the total acres of the cocoa farm.

  1. Calculate the fraction of the farm given to the wife and the first child.

    The wife received 25\frac{2}{5} of the acres.
    The first child received 13\frac{1}{3} of the acres.
    The combined fraction received by the wife and the first child is:

    25+13=2×35×3+1×53×5=615+515=1115\frac{2}{5} + \frac{1}{3} = \frac{2 \times 3}{5 \times 3} + \frac{1 \times 5}{3 \times 5} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}
  2. Calculate the fraction of the farm remaining for the other three children.

    The remaining fraction of the farm is 1−11151 - \frac{11}{15}:

    1−1115=1515−1115=4151 - \frac{11}{15} = \frac{15}{15} - \frac{11}{15} = \frac{4}{15}

    This 415\frac{4}{15} of the total farm was shared among the three children in the ratio 3:5:23:5:2.

  3. Determine the total parts in the ratio and the value of one part.

    The sum of the ratio parts is 3+5+2=103 + 5 + 2 = 10 parts.
    The child who had the least share corresponds to the smallest part in the ratio, which is 2 parts.
    We are given that the child with the least share received 8 acres.
    So, 2 parts = 8 acres.
    Therefore, 1 part = 8 acres2=4 acres\frac{8 \text{ acres}}{2} = 4 \text{ acres}.

  4. Calculate the total acres shared among the three children.

    The total acres shared among the three children is 10 parts:
    Total acres for three children = 10×4 acres=40 acres10 \times 4 \text{ acres} = 40 \text{ acres}.

  5. Calculate the total acres of the farm (TT).

    We know that the 40 acres represents 415\frac{4}{15} of the total farm TT.

    415T=40 acres\frac{4}{15} T = 40 \text{ acres}

    To find TT, multiply both sides by 154\frac{15}{4}:

    T=40×154T = 40 \times \frac{15}{4} T=10×15T = 10 \times 15 T=150 acresT = 150 \text{ acres}

    i) The total acres of the man left (meaning the total acres of the farm) is 150 acres\boxed{\text{150 acres}}.

  6. Calculate the number of acres the wife received.

    The wife received 25\frac{2}{5} of the total acres TT.

    Wife’s acres=25×150 acres\text{Wife's acres} = \frac{2}{5} \times 150 \text{ acres} Wife’s acres=2×1505 acres\text{Wife's acres} = 2 \times \frac{150}{5} \text{ acres} Wife’s acres=2×30 acres\text{Wife's acres} = 2 \times 30 \text{ acres} Wife’s acres=60 acres\text{Wife's acres} = 60 \text{ acres}

    ii) The number of acres the wife received is 60 acres\boxed{\text{60 acres}}.

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