Question

A chef is making cookies from scratch. He requires a set period of time to gather the ingredients and to get everything set up to make the cookies. Then the chef needs a set period of time to make each individual cookie. If c represents the total number of cookies he is making and if t represents the total amount of time it takes to make $c$ cookies, what is the meaning of the 20 in this equation: $\mathrm{t}=20+10\mathrm{c}$ ? (A) How much time it takes to make each individual cookie (B) The fixed cost of the cookie ingredients (C) The maximum number of cookies he can make in 10 minutes (D) The amount of time it takes him to set things up prior to making a cookie

Short answer

(D) The amount of time it takes him to set things up prior to making a cookie.

Step-by-step solution

Read and Understand the Equation

The equation given is $t=20+10c$. In this equation, $c$ represents the total number of cookies, and $t$ is the total time needed.

Identify the Structure of the Equation

The equation $t=20+10c$ consists of two parts: a constant (20) and a variable term (10c). This type of equation suggests an initial or fixed time (represented by the constant) and a time that varies with the number of cookies (represented by the variable term).

Interpret the Constant in the Equation

In the equation $t=20+10c$, the constant is 20. It represents the initial fixed amount of time needed by the chef to set things up before making any cookies, as the time to gather ingredients and set up doesn't depend on the number of cookies.

Analyze the Meaning of the Constant

The 20 in the equation reflects a constant time period that is not influenced by the number of cookies ($c$) being made. Therefore, options involving variable aspects like the time per cookie or number of cookies in a fixed time (Options A and C) do not apply here.

Final Answer Selection

Since the 20 is a fixed time period that occurs before any cookies are made, it fits with Option D, which states it's the time taken for setup before making cookies.

Key concepts

Linear Equations

Linear equations are a type of mathematical expression used to model real-world situations. They express a relationship between two variables with a consistent rate of change. This means as one variable changes, the other varies in a predictable manner. In a typical linear equation like $t=20+10c$, the variable $t$ represents the total time, while $c$ is the number of cookies. The structure of the equation reveals two components: a constant and a variable term.

The constant is a fixed value, indicating a starting point or base amount. The variable part, multiplied by the rate (in this case, 10), shows how the outcome (time, $t$) changes with each unit change in the variable $c$. By understanding these components, students can predict how variations like an increase in cookies affect total time efficiently.

Interpretation of Constants

Interpreting constants in equations is crucial in math word problems. The constant provides an anchor point, helping us understand the scenario without changeable components. In our problem, the equation $t=20+10c$ has a constant of 20. This represents a fixed duration which, according to the problem, is the setup time before the chef starts baking.

Constants simplify initial assumptions and provide clarity in equations that model repetitive processes. They signify aspects of a scenario that remain unchanged regardless of other variables' variations. Recognizing the role of 20 as the setup time helps in understanding that no matter how many cookies are made, the setup requires a consistent amount of time.

Problem-Solving Techniques

Mathematical problem-solving often requires a structured approach. For word problems, the following techniques can be effective:

• **Understand the Problem**: Fully comprehend the situation described. Identify what each part of the equation represents.
• **Identify Variables and Constants**: Break the equation down into its constant and variable terms. Understand what each term means in the context of the problem.
• **Formulate a Strategy**: Decide how to solve for unknowns, verify consistencies, or simply explain each part like the constant time in the setup.
• **Check the Solution**: Once interpreted, cross-check with possible answers to ensure completeness.

By integrating these techniques, students not only solve the specific problem but develop a framework for tackling similar linear equations in PSAT math and beyond.

Time Management in Problems

Efficient time management in solving math problems can impact performance positively. Here are ways students can manage their time effectively when approaching problems like this one:

• **Prioritize Understanding**: Spend the initial minutes reading and understanding the full scope of the problem. This reduces mistakes later on.
• **Avoid Overthinking**: Trust your understanding of constants and variables without getting trapped in excessive calculations.
• **Practice Regularly**: Familiarity with word problems, like those involving linear equations, builds speed and accuracy.
• **Skip and Return**: If a problem stalls progress, move to the next one and return if time permits. This ensures fluid and dynamic exam performance.

Managing time isn't just about speed; it's about making strategic decisions that keep you moving and thinking critically through the test.