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Make r the subject of the relation y = 5/r - 1 and find the value of r when y = -1 and x = 19. A woman bought 240 tubers of yams for GHc 60.00. She sold all of them at 3 for GHc20.00. Find her percentage profit.

Mathematics

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Part 1· 5 steps

Make rr the subject of the relation y=r−1r+1y = \frac{r-1}{r+1} and find the value of rr when y=1y = 1.

  1. Multiply both sides of the equation by (r+1)(r+1) to remove the denominator.

    y(r+1)=r−1y(r+1) = r-1 yr+y=r−1yr + y = r-1
  2. Gather all terms containing rr on one side of the equation and all other terms on the opposite side.

    yr−r=−1−yyr - r = -1 - y
  3. Factor out rr from the terms on the left side.

    r(y−1)=−(1+y)r(y-1) = -(1+y)
  4. Divide both sides by (y−1)(y-1) to make rr the subject.

    r=−(1+y)y−1r = \frac{-(1+y)}{y-1} r=1+y1−yr = \frac{1+y}{1-y}

    The expression for rr is r=1+y1−y\boxed{r = \frac{1+y}{1-y}}.

  5. Find the value of rr when y=1y=1.

    Substitute y=1y=1 into the expression for rr.

    r=1+11−1=20r = \frac{1+1}{1-1} = \frac{2}{0}

    Division by zero is undefined. Therefore, there is no value of r for which y=1y=1.
    The value of rr when y=1y=1 is undefined\boxed{\text{undefined}}.

Part 2· 4 steps

A woman bought 240 tubers of yam for GH¢1200.00. She later sold all of them at 5 for GH¢20.00. Find her percentage profit.

  1. Calculate the selling price of one tuber of yam.

    Selling Price per tuber=GH¢20.005 tubers=GH¢4.00/tuber\text{Selling Price per tuber} = \frac{\text{GH¢}20.00}{5 \text{ tubers}} = \text{GH¢}4.00/\text{tuber}
  2. Calculate the total selling price for all 240 tubers.

    Total Selling Price=240 tubers×GH¢4.00/tuber=GH¢960.00\text{Total Selling Price} = 240 \text{ tubers} \times \text{GH¢}4.00/\text{tuber} = \text{GH¢}960.00
  3. Calculate the profit or loss.

    Profit/Loss=Total Selling Price−Cost Price\text{Profit/Loss} = \text{Total Selling Price} - \text{Cost Price} Profit/Loss=GH¢960.00−GH¢1200.00=−GH¢240.00\text{Profit/Loss} = \text{GH¢}960.00 - \text{GH¢}1200.00 = -\text{GH¢}240.00

    Since the result is negative, the woman incurred a loss of GH¢240.00.

  4. Calculate the percentage profit (which will be a negative value, indicating a loss).

    Percentage Profit=Profit/LossCost Price×100%\text{Percentage Profit} = \frac{\text{Profit/Loss}}{\text{Cost Price}} \times 100\% Percentage Profit=−GH¢240.00GH¢1200.00×100%=−0.2×100%=−20%\text{Percentage Profit} = \frac{-\text{GH¢}240.00}{\text{GH¢}1200.00} \times 100\% = -0.2 \times 100\% = -20\%

    The woman made a \boxed{\text{20% loss}}.

Part 3

Transformations of triangle PQR with P(2, 3), Q(6, 5), R(2, 5).

Part iii

Draw the image P₁Q₁R₁ of triangle PQR under enlargement with scale factor 2 from the origin.

For an enlargement from the origin with scale factor kk, a point (x,y)(x, y) maps to (kx,ky)(kx, ky).

  • P₁: (2×2,3×2)=(4,6)(2 \times 2, 3 \times 2) = (4, 6)
  • Q₁: (6×2,5×2)=(12,10)(6 \times 2, 5 \times 2) = (12, 10)
  • R₁: $(2 \times 2, 5 \
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