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A businessman sells 60% of his wares at a profit of 15%. The remainder is sold at a loss of 25%. What is the percentage profit or loss on the sale of the whole wares?

Mathematics

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3 parts
a
(4, -4)

(5 - m, -n)

b
(1, 4)
c
5.1

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Part 1· 4 steps

A businessman sells 60% of his wares at a profit of 15%. The remainder is sold at a loss of 25%. What is the percentage profit or loss on the sale of the whole wares?

  1. Calculate the selling price for the first 60% of wares.

    Let the total cost price of the wares be CC.
    The cost price of 60% of the wares is 0.60C0.60C.
    The profit on this portion is 15%, so the selling price (SP1SP_1) is:

    SP1=0.60C×(1+0.15)=0.60C×1.15=0.69CSP_1 = 0.60C \times (1 + 0.15) = 0.60C \times 1.15 = 0.69C
  2. Calculate the selling price for the remaining 40% of wares.

    The remainder is 100%−60%=40%100\% - 60\% = 40\% of the wares.
    The cost price of this portion is 0.40C0.40C.
    The loss on this portion is 25%, so the selling price (SP2SP_2) is:

    SP2=0.40C×(1−0.25)=0.40C×0.75=0.30CSP_2 = 0.40C \times (1 - 0.25) = 0.40C \times 0.75 = 0.30C
  3. Calculate the total selling price and compare it to the total cost price.

    The total selling price (SPtotalSP_{total}) is the sum of SP1SP_1 and SP2SP_2:

    SPtotal=0.69C+0.30C=0.99CSP_{total} = 0.69C + 0.30C = 0.99C

    The total cost price is CC.
    Since SPtotal=0.99CSP_{total} = 0.99C is less than CC, there is an overall loss.

  4. Calculate the percentage loss.

    The total loss is C−0.99C=0.01CC - 0.99C = 0.01C.
    The percentage loss is:

    Percentage Loss=LossTotal Cost Price×100%\text{Percentage Loss} = \frac{\text{Loss}}{\text{Total Cost Price}} \times 100\% Percentage Loss=0.01CC×100%=0.01×100%=1%\text{Percentage Loss} = \frac{0.01C}{C} \times 100\% = 0.01 \times 100\% = 1\%

    The percentage profit or loss on the sale of the whole wares is a 1% loss.

Part 2

The coordinates of the vertices of a parallelogram are A(−2,3)A(-2, 3), B(2,−1)B(2, -1), C(5,0)C(5, 0) and D(m,n)D(m, n).

Part a(4, -4)

Find AB⃗\vec{AB} and DC⃗\vec{DC}.

  1. Find vector AB⃗\vec{AB}.

    To find AB⃗\vec{AB}, subtract the coordinates of A from B:

    AB⃗=B−A=(2−(−2),−1−3)=(2+2,−1−3)=(4,−4)\vec{AB} = B - A = (2 - (-2), -1 - 3) = (2 + 2, -1 - 3) = (4, -4)
  2. Find vector DC⃗\vec{DC}.

    To find DC⃗\vec{DC}, subtract the coordinates of D from C:

    DC⃗=C−D=(5−m,0−n)=(5−m,−n)\vec{DC} = C - D = (5 - m, 0 - n) = (5 - m, -n)

    So, AB⃗=(4,−4)\vec{AB} = \boxed{(4, -4)} and DC⃗=(5−m,−n)\vec{DC} = \boxed{(5 - m, -n)}.

Part b(1, 4)

Find the coordinates of D.

  1. Use the property of a parallelogram that opposite sides are equal and parallel.

    For a parallelogram ABCD, AB⃗=DC⃗\vec{AB} = \vec{DC}.
    Equating the components of the vectors from Step 1 and Step 2:

    (4,−4)=(5−m,−n)(4, -4) = (5 - m, -n)

    Equating the x-components:

    4=5−m  ⟹  m=5−4  ⟹  m=14 = 5 - m \implies m = 5 - 4 \implies m = 1

    Equating the y-components:

    −4=−n  ⟹  n=4-4 = -n \implies n = 4

    The coordinates of D are (1,4)\boxed{(1, 4)}.

Part c5.1

Calculate $|DB|$ correct to one decimal place.

  1. Calculate the distance between points D and B.

    The coordinates are D(1,4)D(1, 4) and B(2,−1)B(2, -1).
    The distance formula is (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

    ∣DB∣=(2−1)2+(−1−4)2|DB| = \sqrt{(2 - 1)^2 + (-1 - 4)^2} ∣DB∣=(1)2+(−5)2|DB| = \sqrt{(1)^2 + (-5)^2} ∣DB∣=1+25|DB| = \sqrt{1 + 25} ∣DB∣=26|DB| = \sqrt{26}

    Calculating the numerical value and rounding to one decimal place:

    ∣DB∣≈5.0990195...|DB| \approx 5.0990195... ∣DB∣≈5.1|DB| \approx \boxed{5.1}
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