Indien 'n wins van $$ 7,5 \% $$ gehef word op ' $$ n $$ produk, wat teen R510,60 verkoop word, wat was die kosprys van hierdie produk gewees? Rond jou antwoord af tot die tweede desimaal. R 472.31 ( Nadat die prys van in produk verhoog is met $$ \frac{-1}{6} $$ kos.dit R4.13. Wat was die oorspronklike prys van hierdie produk gewees? Rond jou antwoord af tot die tweede desimaal. R 225,28 Nadat die prys van in produk verlaag is met $$ \frac{1}{8} $$ kos dit R81,00. Wat was die oorspronklike prys van hierdie produk gewees? Rond jou antwoord af tot die tweede desimaal. R 70.88

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Let's solve each part of the problem step by step.

### Part 1: Finding the Cost Price of the Product

We know that a profit of $$ 7.5\% $$ is added to the cost price $$ C $$ to arrive at the selling price $$ S $$. The selling price is given as $$ R510.60 $$.

The relationship can be expressed as:
$$
S = C + 0.075C = 1.075C
$$

Substituting the selling price into the equation:
$$
510.60 = 1.075C
$$

To find the cost price $$ C $$, we can rearrange the equation:
$$
C = \frac{510.60}{1.075}
$$

Now, let's calculate $$ C $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{510.6}{1.075}$$
- step1: Convert the expressions:
  $$\frac{\frac{2553}{5}}{\frac{43}{40}}$$
- step2: Multiply by the reciprocal:
  $$\frac{2553}{5}\times \frac{40}{43}$$
- step3: Reduce the numbers:
  $$2553\times \frac{8}{43}$$
- step4: Multiply:
  $$\frac{2553\times 8}{43}$$
- step5: Multiply:
  $$\frac{20424}{43}$$
Calculating the cost price $$ C $$:
$$
C \approx 474.98
$$

Rounding to two decimal places, the cost price of the product is approximately **R474.98**.

### Part 2: Finding the Original Price After a Decrease

The problem states that after the price of the product was decreased by $$ \frac{1}{6} $$, it costs $$ R4.13 $$. Let the original price be $$ P $$.

The decrease can be expressed as:
$$
\text{New Price} = P - \frac{1}{6}P = \frac{5}{6}P
$$

Setting this equal to the new price:
$$
\frac{5}{6}P = 4.13
$$

To find the original price $$ P $$, we rearrange the equation:
$$
P = \frac{4.13 \times 6}{5}
$$

Now, let's calculate $$ P $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(4.13\times 6\right)}{5}$$
- step1: Remove the parentheses:
  $$\frac{4.13\times 6}{5}$$
- step2: Multiply the numbers:
  $$\frac{24.78}{5}$$
- step3: Convert the expressions:
  $$\frac{\frac{1239}{50}}{5}$$
- step4: Multiply by the reciprocal:
  $$\frac{1239}{50}\times \frac{1}{5}$$
- step5: Multiply the fractions:
  $$\frac{1239}{50\times 5}$$
- step6: Multiply:
  $$\frac{1239}{250}$$
Calculating the original price $$ P $$:
$$
P \approx 4.96
$$

Rounding to two decimal places, the original price of the product is approximately **R4.96**.

### Part 3: Finding the Original Price After a Decrease

The problem states that after the price of the product was decreased by $$ \frac{1}{8} $$, it costs $$ R81.00 $$. Let the original price be $$ Q $$.

The decrease can be expressed as:
$$
\text{New Price} = Q - \frac{1}{8}Q = \frac{7}{8}Q
$$

Setting this equal to the new price:
$$
\frac{7}{8}Q = 81.00
$$

To find the original price $$ Q $$, we rearrange the equation:
$$
Q = \frac{81.00 \times 8}{7}
$$

Now, let's calculate $$ Q $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(81\times 8\right)}{7}$$
- step1: Remove the parentheses:
  $$\frac{81\times 8}{7}$$
- step2: Multiply the numbers:
  $$\frac{648}{7}$$
Calculating the original price $$ Q $$:
$$
Q \approx 92.57
$$

Rounding to two decimal places, the original price of the product is approximately **R92.57**.

### Summary of Results
1. The cost price of the product is **R474.98**.
2. The original price after a decrease of $$ \frac{1}{6} $$ is **R4.96**.
3. The original price after a decrease of $$ \frac{1}{8} $$ is **R92.57**.
- **Cost Price:** R474.98 - **Original Price after 1/6 Decrease:** R4.96 - **Original Price after 1/8 Decrease:** R92.57

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