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Sagot :
Sure, let's solve Jason's problem step-by-step:
4.1 Calculate the selling price of each type of hat with a 45% markup:
For Type A:
- Cost per hat: R75
- Markup: 45%
- Selling price = Cost + Markup
[tex]\[ \text{Selling price of Type A} = 75 + (75 \times 0.45) = 75 + 33.75 = R108.75 \][/tex]
For Type B:
- Cost per hat: R70
- Markup: 45%
- Selling price = Cost + Markup
[tex]\[ \text{Selling price of Type B} = 70 + (70 \times 0.45) = 70 + 31.50 = R101.50 \][/tex]
For Type C:
- Cost per hat: R86
- Markup: 45%
- Selling price = Cost + Markup
[tex]\[ \text{Selling price of Type C} = 86 + (86 \times 0.45) = 86 + 38.70 = R124.70 \][/tex]
4.2 Calculate the profit if all hats are sold at the marked-up price:
For Type A:
- Profit per hat = Selling price - Cost
[tex]\[ \text{Profit per Type A hat} = 108.75 - 75 = R33.75 \][/tex]
- Total profit for Type A:
[tex]\[ \text{Total profit for Type A} = 33.75 \times 150 = R5062.50 \][/tex]
For Type B:
- Profit per hat = Selling price - Cost
[tex]\[ \text{Profit per Type B hat} = 101.50 - 70 = R31.50 \][/tex]
- Total profit for Type B:
[tex]\[ \text{Total profit for Type B} = 31.50 \times 300 = R9450.00 \][/tex]
For Type C:
- Profit per hat = Selling price - Cost
[tex]\[ \text{Profit per Type C hat} = 124.70 - 86 = R38.70 \][/tex]
- Total profit for Type C:
[tex]\[ \text{Total profit for Type C} = 38.70 \times 200 = R7740.00 \][/tex]
Adding these up:
[tex]\[ \text{Total profit if all are sold at full price} = 5062.50 + 9450.00 + 7740.00 = R22252.50 \][/tex]
4.3 Calculate total income if Type C hats are sold at a 30% discount after selling 20 at full price:
- Number of Type C hats sold at full price: 20
- Number of Type C hats sold at a discount: 200 - 20 = 180
- Discounted selling price of Type C hat:
[tex]\[ \text{Discounted selling price} = 124.70 \times (1 - 0.30) = 124.70 \times 0.70 = R87.29 \][/tex]
Calculate total income:
- Income from 20 hats sold at full price:
[tex]\[ \text{Income from 20 hats} = 20 \times 124.70 = R2494.00 \][/tex]
- Income from 180 hats sold at a discount:
[tex]\[ \text{Income from 180 hats} = 180 \times 87.29 = R15712.20 \][/tex]
Total income from all Type C hats:
[tex]\[ \text{Total income from Type C hats} = 2494.00 + 15712.20 = R18206.20 \][/tex]
4.4 Verify if Jason makes no profit due to the reduced price:
Calculate the profit for all hats considering the discount:
For Type C:
- Profit per hat sold at full price:
[tex]\[ \text{Profit for 20 hats} = (124.70 - 86) \times 20 = 38.70 \times 20 = R774.00 \][/tex]
- Profit per hat sold at a discount:
[tex]\[ \text{Profit for discounted hats} = (87.29 - 86) \times 180 = 1.29 \times 180 = R232.20 \][/tex]
- Total profit for Type C:
[tex]\[ \text{Total profit for Type C} = 774.00 + 232.20 = R1006.20 \][/tex]
Adding this with the previously calculated profits for Type A and Type B:
[tex]\[ \text{Total profit} = 5062.50 + 9450.00 + 1006.20 = R15518.70 \][/tex]
So, Matthew's statement that Jason would make no profit due to reducing the price is incorrect. Jason would still make a profit, but it would be less than originally projected:
- Initial profit if all Type C hats sold at full price:
[tex]\[ R7740.00 \][/tex]
- Reduced profit including discount on Type C hats:
[tex]\[ R1006.20 \][/tex]
- Net profit comparison shows Jason still earns a profit:
[tex]\[ R15518.70 \][/tex]
Thus, these calculations confirm that Jason will still make a profit even after selling the Type C hats at a discounted price.
4.1 Calculate the selling price of each type of hat with a 45% markup:
For Type A:
- Cost per hat: R75
- Markup: 45%
- Selling price = Cost + Markup
[tex]\[ \text{Selling price of Type A} = 75 + (75 \times 0.45) = 75 + 33.75 = R108.75 \][/tex]
For Type B:
- Cost per hat: R70
- Markup: 45%
- Selling price = Cost + Markup
[tex]\[ \text{Selling price of Type B} = 70 + (70 \times 0.45) = 70 + 31.50 = R101.50 \][/tex]
For Type C:
- Cost per hat: R86
- Markup: 45%
- Selling price = Cost + Markup
[tex]\[ \text{Selling price of Type C} = 86 + (86 \times 0.45) = 86 + 38.70 = R124.70 \][/tex]
4.2 Calculate the profit if all hats are sold at the marked-up price:
For Type A:
- Profit per hat = Selling price - Cost
[tex]\[ \text{Profit per Type A hat} = 108.75 - 75 = R33.75 \][/tex]
- Total profit for Type A:
[tex]\[ \text{Total profit for Type A} = 33.75 \times 150 = R5062.50 \][/tex]
For Type B:
- Profit per hat = Selling price - Cost
[tex]\[ \text{Profit per Type B hat} = 101.50 - 70 = R31.50 \][/tex]
- Total profit for Type B:
[tex]\[ \text{Total profit for Type B} = 31.50 \times 300 = R9450.00 \][/tex]
For Type C:
- Profit per hat = Selling price - Cost
[tex]\[ \text{Profit per Type C hat} = 124.70 - 86 = R38.70 \][/tex]
- Total profit for Type C:
[tex]\[ \text{Total profit for Type C} = 38.70 \times 200 = R7740.00 \][/tex]
Adding these up:
[tex]\[ \text{Total profit if all are sold at full price} = 5062.50 + 9450.00 + 7740.00 = R22252.50 \][/tex]
4.3 Calculate total income if Type C hats are sold at a 30% discount after selling 20 at full price:
- Number of Type C hats sold at full price: 20
- Number of Type C hats sold at a discount: 200 - 20 = 180
- Discounted selling price of Type C hat:
[tex]\[ \text{Discounted selling price} = 124.70 \times (1 - 0.30) = 124.70 \times 0.70 = R87.29 \][/tex]
Calculate total income:
- Income from 20 hats sold at full price:
[tex]\[ \text{Income from 20 hats} = 20 \times 124.70 = R2494.00 \][/tex]
- Income from 180 hats sold at a discount:
[tex]\[ \text{Income from 180 hats} = 180 \times 87.29 = R15712.20 \][/tex]
Total income from all Type C hats:
[tex]\[ \text{Total income from Type C hats} = 2494.00 + 15712.20 = R18206.20 \][/tex]
4.4 Verify if Jason makes no profit due to the reduced price:
Calculate the profit for all hats considering the discount:
For Type C:
- Profit per hat sold at full price:
[tex]\[ \text{Profit for 20 hats} = (124.70 - 86) \times 20 = 38.70 \times 20 = R774.00 \][/tex]
- Profit per hat sold at a discount:
[tex]\[ \text{Profit for discounted hats} = (87.29 - 86) \times 180 = 1.29 \times 180 = R232.20 \][/tex]
- Total profit for Type C:
[tex]\[ \text{Total profit for Type C} = 774.00 + 232.20 = R1006.20 \][/tex]
Adding this with the previously calculated profits for Type A and Type B:
[tex]\[ \text{Total profit} = 5062.50 + 9450.00 + 1006.20 = R15518.70 \][/tex]
So, Matthew's statement that Jason would make no profit due to reducing the price is incorrect. Jason would still make a profit, but it would be less than originally projected:
- Initial profit if all Type C hats sold at full price:
[tex]\[ R7740.00 \][/tex]
- Reduced profit including discount on Type C hats:
[tex]\[ R1006.20 \][/tex]
- Net profit comparison shows Jason still earns a profit:
[tex]\[ R15518.70 \][/tex]
Thus, these calculations confirm that Jason will still make a profit even after selling the Type C hats at a discounted price.
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