Question

When calculating the cost of a good or service, you often have to consider price markups and discounts. Consider these two statements regarding a

discounted item:
• The list price of the item is 80 percent of the original price.
• The price of the item has been reduced by 80 percent.
Write a pair of linear equations using variables of your choice to prove that these two statements are not equivalent. Explain how a calculation for
change in percentage (increase or decrease) is different from a calculation that involves multiplying by percentages. Why is the wording of percentage
problems so important? Give examples to illustrate your point.
20 points:D

Answer 1

Step-by-step explanation:

Let’s assume the original price of the item is “x.” Then, using the first statement, the list price of the item is 80% of the original price, or 0.8x. If there is a discount applied, let’s say “d,” then the discounted price would be (1-d)(0.8x).

Using the second statement, if the price of the item has been reduced by 80%, then the discounted price would be 0.2x. This can be expressed as (1 - 0.8)(x).

So, we have the following two equations:

(1 - d)(0.8x) = 0.2x

0.64x - 0.8dx = 0.2x

Simplifying this equation, we get:

0.44x = 0.8dx

d = 0.55

This means that the discount applied in the first statement is 55%, not 80%.

The calculation for change in percentage (increase or decrease) involves finding the difference between two values and expressing it as a percentage of the original value. This is different from multiplying by percentages, which involves finding a percentage of the original value and subtracting or adding it to the original value.

The wording of percentage problems is important because it can affect the way the problem is interpreted and the calculation that is used to solve it. For example, the phrase “increased by 50%” could be interpreted as multiplying the original value by 1.5, while the phrase “increased to 50%” could be interpreted as finding 50% of the original value and adding it to the original value.

Examples:

A shirt is on sale for 30% off its original price of $50. The discounted price is (1-0.3)($50) = $35.

A company’s revenue increased from $100,000 to $120,000. The percentage increase in revenue is ((120,000 - 100,000) / 100,000) x 100% = 20%.

Answer 2

To prove that the two statements are not equivalent, we can write a pair of linear equations using variables and solve for them.

Step-by-step explanation:

To prove that the two statements are not equivalent, we can write a pair of linear equations using variables:

Let x be the original price of the item and let y be the discounted price of the item.

Equation 1: y = 0.8x (The list price of the item is 80 percent of the original price)

Equation 2: y = x - 0.8x (The price of the item has been reduced by 80 percent)

By solving these equations, we can find that the two statements indeed yield different results. The wording of percentage problems is important because it can change the interpretation of the calculation. For example, in the first statement, the original price is multiplied by 0.8 to get the list price, while in the second statement, the original price is multiplied by 0.2 to get the discount amount.