Problem 87 - Algebra 1

Question

SCHOOL BAKE SALE You have one hour to make cookies for your =school bake sale. You spend 20 minutes mixing the dough. It then takes 12 minutes to bake each tray of cookies. If you bake one tray at a time, which model can you use to find how many trays you can bake? (Review 3.3 and 3.6) A. \(x(20+12)=60\) B. \(12 x+20=60\)

Step-by-Step Solution

Answer

The correct model to use to find out how many trays of cookies can be baked is B) \(12x + 20 = 60\).

Step 1 - Understand the Problem

We have an hour to prepare and bake cookies. Mixing the dough takes 20 minutes and baking one tray of cookies takes 12 minutes.

Step 2 - Analyze the Models

We are given two mathematical models, A. \(x(20+12)=60\) and B. \(12x+20=60\). In option A, the time for mixing the dough and the baking time is multiplied by the number of trays which is incorrect since dough is mixed only once. In option B, the baking time (12 minutes) is multiplied by the number of trays (x) and then the dough mixing time (20 minutes) is added. This represents the correct division of time for the task.

Step 3 - Choose the Appropriate Model

Model B \(12x+20=60\) is the correct model because it properly represents the time spent on each task. The total time (60 minutes) is broken down into 12 minutes baking time per tray (12x) and 20 minutes for mixing dough (20).

Key Concepts

Time Management

Effectively managing your time is crucial, especially when you have fixed tasks to complete within a constrained time, like baking cookies for a school bake sale in one hour. In scenarios like this, it's important to understand the components of your task and allocate time accordingly.
When baking cookies:

• Mixing the dough consumes a set amount of time (20 minutes in this case). This is a fixed task that does not depend on the number of trays you're going to bake.
• Baking each tray takes an additional 12 minutes. This time is variable and depends on how many trays you decide to bake.

By breaking down the total task into these segments, you realize the importance of time management. You learn to recognize fixed tasks and adjust variable tasks, such as the number of trays, to fit into the available time. This is very similar to optimizing schedules in real life. Thinking this way helps ensure that every minute is used effectively, preventing wasted time and ensuring you fulfill all your goals within the allotted time.

Equation Solving

Solving equations often means choosing the correct mathematical model to describe a practical problem. In the bake sale example, the problem presents two models to choose from. Understanding each model’s representation is crucial.

• In model A, \(x(20+12)=60\), the assumption is that the total time for both tasks (mixing and baking) is multiplied by the number of trays \(x\). This leads to an illogical conclusion because dough mixing time wouldn't increase with each tray.
• Model B, \(12x+20=60\), breaks down the problem correctly. Here, \(12x\) represents the time for baking \(x\) trays, and \(20\) is a constant for mixing. This matches the reality of the task where you mix once and bake multiple times.

To solve an equation like Model B, isolating \(x\) involves subtracting the fixed time (20 minutes for dough) and then dividing by the baking time per tray. This method shows how understanding each part of an equation helps in finding accurate solutions to practical problems.

Mathematical Analysis

Mathematical analysis involves breaking down a problem into comprehensible components. In the bake sale exercise, we assessed the different parts of the task by using mathematical modeling to get a correct representation of time allocation.

• Identifying fixed versus variable components: The fixed component is the dough mixing time (20 minutes), while the variable component is the time for each tray of cookies (12 minutes per tray).
• Selecting the correct equation: By choosing model B, we accurately reflect our task's time distribution. Mathematical analysis helps us determine how to adjust one factor, the number of baked trays, without exceeding the total time (60 minutes).

With a clear understanding from mathematical analysis, you can adapt this approach to other scenarios, such as budgeting or scheduling, where you must distribute limited resources over different activities efficiently. This skill helps in planning and ensuring achievable outcomes under constraints.