Problem 46 - Algebra A Combined Function

Question

The price of a biology book recently increased by \(10 \%\). If this book originally cost \(\$ 89.90,\) find the mark-up and the new price.

Step-by-Step Solution

Answer

The mark-up is $8.99 and the new price is $98.89.

Step 1: Calculate the Mark-up

The mark-up of an item when its price increases by a percentage is found by multiplying the original price by the percentage increase. Here, the original price of the book is \(89.90 and it increased by 10%. Calculate the mark-up by using the formula: \( \text{Mark-up} = \text{Original Price} \times \text{Percentage Increase} \). Substitute the values: \( \text{Mark-up} = 89.90 \times 0.10 = 8.99 \). Thus, the mark-up is \)8.99.

Step 2: Determine the New Price

To find the new price of the book, add the mark-up to the original price. The original price is \(89.90 and the mark-up calculated in Step 1 is \)8.99. Use the formula: \( \text{New Price} = \text{Original Price} + \text{Mark-up} \). Substitute the values: \( \text{New Price} = 89.90 + 8.99 = 98.89 \). Therefore, the new price of the book is $98.89.

Key Concepts

Mark-up Calculation

When the price of an item increases, we often refer to this increase as a "mark-up." Essentially, the mark-up is the additional amount added to the original price. To calculate the mark-up, you need to know both the original price and the percentage increase. You perform the calculation by multiplying the original price by the percentage increase in decimal form.

For example, if a book originally costs $89.90 and the price goes up by 10%, the calculation for the mark-up would be:

• First, convert the percentage to a decimal: 10% becomes 0.10.
• Multiply the original price by this decimal: $89.90 multiplied by 0.10 gives an $8.99 mark-up.

This $8.99 is the additional cost due to the increase. Understanding mark-up calculations helps you see exactly how much more you are paying after a price increase.

Original and New Price

The "original price" refers to the initial cost of an item before any increases or mark-ups are applied. When the price of an item increases, you can find the "new price" by adding the calculated mark-up to the original price.

Let's consider the example from before: the book's original price is $89.90.

• To find the new price, calculate the mark-up (as we did, it's $8.99).
• Add this mark-up to the original price: $89.90 + $8.99.

The outcome is a new price of $98.89. This simple addition shows how the mark-up affects what you will pay after the price increase. Understanding these terms clarifies how much more an item costs compared to its original pricing.

Algebraic Expressions

In the world of finances and shopping, we often deal with numbers using algebraic expressions—simple mathematical statements that express relationships between variables and constants. When calculating price changes, algebra becomes very useful.

Let's explore this with the recent price change of the book:

• We start with the formula for mark-up: \( \text{Mark-up} = \text{Original Price} \times \text{Percentage Increase} \)
• To find the new price, use the formula for adding the mark-up: \( \text{New Price} = \text{Original Price} + \text{Mark-up} \)

These basic algebraic formulas allow you to plug in numbers to find both the mark-up and the new price efficiently. Algebra helps simplify the complexity of these calculations, making it easier to understand and predict financial changes. Using algebraic expressions ensures that results are precise and easy to verify.