Question

Rosa has already eaten 10 pretzels from a bag that originally contained \(p\) pretzels. If Rosa is able to eat each remaining pretzel in 18 seconds, which of the following represents the amount of additional time, in seconds, needed for Rosa to eat all the pretzels in the bag? A) \(10(18-p)\) B) \(10(p-18)\) C) \(18(10-p)\) D) \(18(p-10)\)

Short answer

The correct answer is D) \(18(p - 10)\).

Step-by-step solution

Determine the number of remaining pretzels in the bag

To find out how many pretzels are left in the bag after Rosa ate 10 of them, we need to subtract the number of eaten pretzels (10) from the total number of pretzels in the bag (p): \(p - 10\).

Find the expression for the total additional time to eat the remaining pretzels

Now that we know there are \(p - 10\) pretzels remaining in the bag, we can calculate the total time needed to eat all of them by multiplying the number of remaining pretzels by the time it takes to eat one pretzel (18 seconds): \[18(p - 10)\]. This expression represents the total amount of additional time, in seconds, needed for Rosa to eat all the pretzels in the bag.

Match the expression with the answer choices

Comparing the obtained expression from Step 2 with the given answer choices, we find that it matches with option D. Therefore, the correct answer is: D) \(18(p - 10)\).

Key concepts

Problem Solving

Problem solving in mathematics often involves breaking down a complex situation into manageable steps. This begins with understanding what the problem is asking. In our exercise, the problem asks how much more time Rosa needs to eat the remaining pretzels. The first step is to identify what we know: the total initial number of pretzels and how many she has eaten. From there, we define what we need to find, which is the time taken to eat the rest. By converting words into numbers and symbols, we create a clear mathematical expression. Problem solving, therefore, can be seen as a step-by-step journey from identifying the issue to finding a solution. This also involves a careful check at the end to make sure our results are reasonable and match one of the answer choices.

Algebra

Algebra is the branch of mathematics in which symbols, usually letters, represent numbers or quantities in formulas and equations. Here, the variable "\(p\)" stands for the number of pretzels. Algebra allows us to model real-life situations and find unknown values.

• The known value in the problem is that 10 pretzels are already eaten.
• The unknown is the remaining pretzels, represented as \(p - 10\).
• We use the operator "minus" because we subtract the pretzels already eaten from the total \(p\).

Once expressions are formed, algebra can be used to manipulate them to find solutions. In Rosa's pretzel problem, we used algebra to calculate the time it would take. The goal is to simplify and solve the equation correctly to represent a real-world action.

Linear Equations

Linear equations are mathematical statements of equality between two expressions. These equations involve variables raised only to the power of one (i.e., they form straight lines when graphed). In the context of this problem, the equation we use is based on the relationship between pretzels and time.

• The equation \(18(p - 10)\) follows the form \(ax + b = c\).
• Here, the solution to the equation represents the total extra time needed, a direct result of the relationship between the number of pretzels remaining and the time per pretzel.

The simplicity of linear equations makes them useful in describing a wide range of real-life scenarios, like Rosa's. Solving linear equations involves isolating the variable of interest (here, the number of pretzels) and evaluating the entire expression to describe the scenario effectively. This offers us both clarity and precision, key elements in clear mathematical presentation.