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A number of people formed a cooperative society with the intention of buying a 250-hectare farm costing Sh 60,000 per hectare. After sometime, the management of the society realized that the mem

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3 parts
a
200
b

1.04 hectares1.04 \text{ hectares}

c

Sh 63020.83\text{Sh } 63020.83

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Here is the solution to question 48.

48. A number of people formed a cooperative society with the intention of buying a 250-hectare farm costing Sh 60,000 per hectare. After sometime, the management of the society realized that the members could not raise the amount on time. They decided to recruit 40 more members, with the result that each member would now pay Sh. 12,500 less.

First, calculate the total cost of the farm:

Total cost=250 hectares×Sh 60,000/hectare=Sh 15,000,000\text{Total cost} = 250 \text{ hectares} \times \text{Sh } 60,000/\text{hectare} = \text{Sh } 15,000,000

Part a· 8 steps

Find the original number of members who formed the society.

  1. Let xx be the original number of members.

    The original contribution per member was 15,000,000x\frac{15,000,000}{x}.

  2. After recruiting 40 more members, the new number of members is x+40x + 40.

    The new contribution per member is 15,000,000x+40\frac{15,000,000}{x + 40}.

  3. Set up the equation based on the information that each member pays Sh.

    12,500 less.

    15,000,000x−15,000,000x+40=12,500\frac{15,000,000}{x} - \frac{15,000,000}{x + 40} = 12,500

  4. Divide the entire equation by 12,500 to simplify.

    1200x−1200x+40=1\frac{1200}{x} - \frac{1200}{x + 40} = 1

  5. Multiply by x(x+40)x(x+40) to clear the denominators.

    1200(x+40)−1200x=x(x+40)1200(x + 40) - 1200x = x(x + 40) 1200x+48000−1200x=x2+40x1200x + 48000 - 1200x = x^2 + 40x 48000=x2+40x48000 = x^2 + 40x

  6. Rearrange into a quadratic equation.

    x2+40x−48000=0x^2 + 40x - 48000 = 0

  7. Solve the quadratic equation using the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

    x=−40±402−4(1)(−48000)2(1)x = \frac{-40 \pm \sqrt{40^2 - 4(1)(-48000)}}{2(1)} x=−40±1600+1920002x = \frac{-40 \pm \sqrt{1600 + 192000}}{2} x=−40±1936002x = \frac{-40 \pm \sqrt{193600}}{2} x=−40±4402x = \frac{-40 \pm 440}{2}

  8. Calculate the two possible values for xx.

    x1=−40+4402=4002=200x_1 = \frac{-40 + 440}{2} = \frac{400}{2} = 200 x2=−40−4402=−4802=−240x_2 = \frac{-40 - 440}{2} = \frac{-480}{2} = -240

    Since the number of members cannot be negative, we take the positive value.
    The original number of members is 200\boxed{200}.

Part b

1.04 hectares1.04 \text{ hectares}

After buying the farm, it was divided equally among all the members. Determine how much land each member got.

  1. Calculate the total number of members after recruitment.

    Total members=Original members+New members=200+40=240\text{Total members} = \text{Original members} + \text{New members} = 200 + 40 = 240

  2. Calculate the land each member got.

    Land per member=Total landTotal members=250 hectares240 members\text{Land per member} = \frac{\text{Total land}}{\text{Total members}} = \frac{250 \text{ hectares}}{240 \text{ members}} Land per member=2524 hectares≈1.04166... hectares\text{Land per member} = \frac{25}{24} \text{ hectares} \approx 1.04166... \text{ hectares}

    Rounding to two decimal places:
    Each member got 1.04 hectares\boxed{1.04 \text{ hectares}}.

Part c

Sh 63020.83\text{Sh } 63020.83

Each member had to pay surveyors fee at the rate of Sh 500 per hectare. Calculate the total amount each member paid for his piece of land.

  1. Calculate the cost of the land share per member.

    Cost of land share per member=Total cost of farmTotal members=15,000,000240=Sh 62,500\text{Cost of land share per member} = \frac{\text{Total cost of farm}}{\text{Total members}} = \frac{15,000,000}{240} = \text{Sh } 62,500

  2. Calculate the surveyor's fee per member.

    Surveyor’s fee per member=Land per member×Rate per hectare\text{Surveyor's fee per member} = \text{Land per member} \times \text{Rate per hectare} Surveyor’s fee per member=2524 hectares×Sh 500/hectare=1250024 Sh\text{Surveyor's fee per member} = \frac{25}{24} \text{ hectares} \times \text{Sh } 500/\text{hectare} = \frac{12500}{24} \text{ Sh} Surveyor’s fee per member≈Sh 520.83\text{Surveyor's fee per member} \approx \text{Sh } 520.83

  3. Calculate the total amount each member paid.

    Total amount per member=Cost of land share+Surveyor’s fee\text{Total amount per member} = \text{Cost of land share} + \text{Surveyor's fee} Total amount per member=Sh 62,500+Sh 1250024\text{Total amount per member} = \text{Sh } 62,500 + \text{Sh } \frac{12500}{24} Total amount per member=Sh 62500×24+1250024=Sh 1500000+1250024=Sh 151250024\text{Total amount per member} = \text{Sh } \frac{62500 \times 24 + 12500}{24} = \text{Sh } \frac{1500000 + 12500}{24} = \text{Sh } \frac{1512500}{24} Total amount per member≈Sh 63020.83\text{Total amount per member} \approx \text{Sh } 63020.83

    The total amount each member paid for his piece of land is Sh 63020.83\boxed{\text{Sh } 63020.83}.

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