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1. Sarah wants to start selling school scarves for the winter. She can buy them in bulk for R55 and decides she wants to add a 45% profit markup on these scarves.

1.1 Determine the amount of profit she will get for each scarf.
1.2 Determine the selling price of one scarf.

2. Jason buys 3 different types of bucket hats and sells them at markets.

[tex]\[
\begin{tabular}{|l|l|l|}
\hline
& \text{Cost per Hat} & \text{Number bought} \\
\hline
\text{Type A} & R75 & 150 \\
\hline
\text{Type B} & R70 & 300 \\
\hline
\text{Type C} & R86 & 200 \\
\hline
\end{tabular}
\][/tex]

2.1 His markup is 45% on all his hats. Calculate the selling price of each type of hat.
2.2 Calculate the profit he makes if he sells all of the hats.
2.3 After 4 months, he has only sold 20 of the Type C hats. He decides to offer a 30% discount on the selling price of these hats. Calculate his total income if he sells all the Type C hats at the reduced price.
2.4 Matthew states that Jason would make no profit due to reducing the price. Verify, showing all calculations, if this statement is correct.

Sagot :

GinnyAnsweridn
Let's address each question step-by-step:

### 1. Sarah's Winter Scarves

1.1 Determine the amount of profit per scarf:
- The cost of one scarf is R55.
- The profit markup is 45%, which is [tex]\(0.45\)[/tex] in decimal form.

To find the profit per scarf:
[tex]\[ \text{Profit per scarf} = \text{Cost} \times \text{Markup}\][/tex]
[tex]\[ \text{Profit per scarf} = 55 \times 0.45 = 24.75 \][/tex]

So, Sarah will get R24.75 profit for each scarf.

1.2 Determine the selling price of one scarf:
- The selling price is calculated as the cost plus the profit.

[tex]\[ \text{Selling price} = \text{Cost} + \text{Profit} \][/tex]
[tex]\[ \text{Selling price} = 55 + 24.75 = 79.75 \][/tex]

So, the selling price of one scarf is R79.75.

### 2. Jason's Bucket Hats

Jason buys three types of hats and applies a 45% markup on each.

2.1 Calculate the selling price of each type of hat:
- For hat Type A:
- Cost: R75
- Markup: 45%

[tex]\[ \text{Selling price} = 75 + (75 \times 0.45) = 75 + 33.75 = 108.75 \][/tex]

- For hat Type B:
- Cost: R70
- Markup: 45%

[tex]\[ \text{Selling price} = 70 + (70 \times 0.45) = 70 + 31.5 = 101.5 \][/tex]

- For hat Type C:
- Cost: R86
- Markup: 45%

[tex]\[ \text{Selling price} = 86 + (86 \times 0.45) = 86 + 38.7 = 124.7 \][/tex]

So, the selling prices are:
- Type A: R108.75
- Type B: R101.5
- Type C: R124.7

2.2 Calculate the total profit if he sells all of the hats:
- The number of hats and their costs are respectively:
- Type A: 150 hats at R75 each
- Type B: 300 hats at R70 each
- Type C: 200 hats at R86 each

Profit per hat type:
[tex]\[ \begin{aligned} &\text{Type A Profit} = (108.75 - 75) \times 150 = 33.75 \times 150 = 5062.5 \\ &\text{Type B Profit} = (101.5 - 70) \times 300 = 31.5 \times 300 = 9450 \\ &\text{Type C Profit} = (124.7 - 86) \times 200 = 38.7 \times 200 = 7740 \\ \end{aligned} \][/tex]

Total profit:
[tex]\[ \text{Total Profit} = 5062.5 + 9450 + 7740 = 22252.5 \][/tex]

So, the total profit is R22,252.5.

2.3 Calculate total income if he sells all Type C hats at the reduced price after 4 months:
- Full price sales:
- 20 hats sold at R124.7 each.

[tex]\[ \text{Income from 20 full price hats} = 20 \times 124.7 = 2494 \][/tex]

- Reduced price (30% discount) for remaining 180 hats:
- Discounted price: [tex]\( 124.7 \times (1 - 0.30) = 124.7 \times 0.70 = 87.29 \)[/tex]

[tex]\[ \text{Income from 180 discounted hats} = 180 \times 87.29 = 15712.2 \][/tex]

Total income from all Type C hats:
[tex]\[ \text{Total Income Type C} = 2494 + 15712.2 = 18206.2 \][/tex]

So, the total income from selling all Type C hats, considering the discount, is R18,206.2.

2.4 Verify if Jason makes no profit due to reducing the price:
- The total number of Type C hats purchased is 200.

Total cost of Type C hats:
[tex]\[ \text{Total cost Type C} = 86 \times 200 = 17200 \][/tex]

Income from selling all Type C hats at the calculated prices:
[tex]\[ \text{Total Income from Type C} = 18206.2 \][/tex]

Profit calculation for Type C:
[tex]\[ \text{Profit from Type C} = 18206.2 - 17200 = 1006.2 \][/tex]

Since the profit is still R1006.2, Matthew's statement that Jason would make no profit due to reducing the price is incorrect. Jason still makes a profit, albeit a smaller one, of R1006.2.
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