Question

Each writer for the local newspaper is paid as follows: $a$ dollars for each of the first $n$ stories each month, and $a+b$ dollars for each story thereafter, where $a>b$. How many more dollars will a writer who submits $n+a$ stories in a month earn than a writer who submits $n+b$ stories? A. $$(a-b)(a+b+n)$$ B. $$a-b$$ C. $$a^2-b^2$$ D. $$n(a-b)$$ E. $$an+bn-an$$

Short answer

The writer who submits $n+a$ stories earns $a^2-b^2$ dollars more than the writer who submits $n+b$ stories. The answer is C.

Step-by-step solution

Understanding the payment system

Each writer is paid $a$ dollars for the first $n$ stories. For each additional story beyond $n$, they are paid $a+b$ dollars.

Calculate total earnings for submitting $n+a$ stories

For the first $n$ stories, the writer is paid $na$ dollars. The remaining $a$ stories are paid at $a+b$ dollars per story. Therefore, the earnings for these stories are $a(a+b)$. Summing these, the total earnings are: $$na+a(a+b)=na+a^2+ab$$.

Calculate total earnings for submitting $n+b$ stories

For the first $n$ stories, the writer is paid $na$ dollars. The remaining $b$ stories are paid at $a+b$ dollars per story. Therefore, the earnings for these stories are $b(a+b)$. Summing these, the total earnings are: $$na+b(a+b)=na+ab+b^2$$.

Compare the two earnings

Subtract the earnings for $n+b$ stories from the earnings for $n+a$ stories: $$na+a^2+ab-(na+ab+b^2)=na+a^2+ab-na-ab-b^2=a^2-b^2$$.

Conclusion

The writer who submits $n+a$ stories earns $a^2-b^2$ dollars more than the writer who submits $n+b$ stories.

Key concepts

Linear Equations

Linear equations are equations where the highest power of the variable is one. In the GMAT problem we are exploring, we see a direct relationship between the number of stories written and the payment. The formula for payment after a certain number of stories involves linear components like $a$, $b$, and $n$. For example, the equation for total earnings for $n+a$ stories was derived as: $na+a^2+ab$. This kind of equation is linear because each term combines coefficients and variables linearly, without any variable being squared or having a higher power.

Understanding linear equations is crucial since they form the foundation for more complex algebraic manipulations and problem-solving techniques. They are easy to solve and interpret, providing clear answers in many practical scenarios.

To solve linear equations, remember to:

• Isolate the variable by combining like terms on each side of the equation.
• Use basic operations like addition, subtraction, multiplication, and division to simplify the equation.
• Always keep the equation balanced by performing the same operation on both sides.
• Check your solutions by substituting them back into the original equation.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. In this GMAT problem, we use algebraic expressions to represent the total earnings for different numbers of stories submitted. For example, the expression $na+a(a+b)$ can be expanded and simplified to $na+a^2+ab$.

Becoming comfortable with algebraic expressions involves understanding:

• How to expand and simplify expressions: Combining like terms and using distributive properties helps simplify complex expressions.
• How to factor expressions: For instance, $a^2-b^2$ can be factored into $(a+b)(a-b)$. This technique is useful for solving and simplifying equations.
• How to substitute values: This is often necessary to solve word problems where values for variables are known or can be derived from context.

Working through these principles helps in manipulating algebraic expressions to make them easier to analyze and solve. This can be especially helpful in standard tests like the GMAT, where quick and accurate problem-solving is vital.

Problem-Solving Strategies

Effective problem-solving strategies are key to tackling GMAT math problems. They involve breaking down the problem, applying systematic methods, and checking the results. Here’s how we can apply these strategies to our newspaper writer problem:

• Understand the problem: Begin by clearly understanding the writer's payment system. Identifying the relationships and dependencies helps in setting up the correct equations.
• Set up equations: Translate the word problem into mathematical equations. For the given task, we used linear equations to represent total earnings for different numbers of stories submitted.
• Break down the problem: Simplify and solve the equations step by step. We calculated the earnings for $n+a$ and $n+b$ stories separately before comparing them.
• Check your work: After solving, always go back and check your calculations and ensure your logical steps follow through. We verified that the additional earnings for more stories were represented correctly as $a^2-b^2$.
• Practice different types of problems: Regular practice with a variety of problems helps build familiarity with common patterns and techniques. This increases speed and accuracy.

Remember, the more you practice these strategies, the more instinctive they'll become. Applying a structured approach ensures that even the toughest problems can be tackled effectively.