If the price of a bicycle is dropped from $800 to $600, find the discount percentage. In a holiday, the manager of a shop declares a discount of 15% on a coat. a) Find the linear r... If the price of a bicycle is dropped from $800 to $600, find the discount percentage. In a holiday, the manager of a shop declares a discount of 15% on a coat. a) Find the linear relation between the new price y and the original price x. b) This coat is not sold during the holiday; he decided to raise his price by 15%. What will be the total percentage of variation? c) If the original price of the coat was $120, what will its price be after the two variations? Read more

Steps to Solve

1. Finding the Discount Percentage for the Bicycle

To find the discount percentage, use the formula:

$$ \text{Discount Percentage} = \left( \frac{\text{Original Price} - \text{New Price}}{\text{Original Price}} \right) \times 100 $$

Here, the original price is $800 and the new price is $600.

2. Calculating the Discount Amount

Calculate the discount amount first:

$$ \text{Discount Amount} = \text{Original Price} - \text{New Price} = 800 - 600 = 200 $$

3. Calculating the Discount Percentage

Now plug the values into the formula:

$$ \text{Discount Percentage} = \left( \frac{200}{800} \right) \times 100 = 25% $$

4. Finding the Linear Relationship for the Coat

For the coat, the manager offers a discount of 15%. The linear relationship between the new price (y) and the original price (x) can be described as:

$$ y = x - 0.15x = 0.85x $$

This means the new price is 85% of the original price.

5. Price Increase After Discount

If the coat's price is increased by 15% after the discount, we first calculate the raised price:

$$ \text{Raised Price} = y + 0.15y = 1.15y $$

Substituting the previous linear relation into this:

$$ \text{Raised Price} = 1.15(0.85x) = 0.9775x $$

6. Calculating Total Percentage of Variation

To find the total percentage of variation from the original price (x) to the new price (0.9775x):

$$ \text{Total Percentage of Variation} = \left( \frac{\text{Original Price} - \text{New Price}}{\text{Original Price}} \right) \times 100 $$

Substituting the values:

$$ \text{Total Percentage of Variation} = \left( \frac{x - 0.9775x}{x} \right) \times 100 = 2.25% $$

7. Calculating Final Price after Two Variations

If the original price of the coat was $120, we can calculate the price after each variation:

After the discount: $$ \text{Price after discount} = 0.85 \times 120 = 102 $$

After the price increase: $$ \text{Price after increase} = 1.15 \times 102 = 117.30 $$