Question

The numbers \(m, n,\) and \(T\) are all positive, and \(m>n .\) A weekend farm stand sells only peaches. On Saturday, the farm stand has \(T\) peaches to sell, at a profit of \(m\) cents each. Any peaches remaining for sale on Sunday will be marked down and sold at a profit of \((m-n)\) cents each. If all peaches available for sale on Saturday morning are sold by Sunday evening, how many peaches, in terms of \(T, m,\) and \(n,\) does the stand need to sell on Saturday in order to make the same profit on each day? a. \(\frac{T m}{m-n}\) b. \(\frac{T m}{n-m}\) c. \(\frac{m(m-n)}{T}\) d. \(\frac{T}{m-n}\) e. \(\frac{T(m-n)}{2 m-n}\)

Short answer

The stand needs to sell \(\frac{T(m-n)}{2m-n}\) peaches on Saturday (Option e).

Step-by-step solution

- Understanding the Profit Terms

Identify the profit earned per peach on each day. On Saturday, the farm stand earns a profit of \(m\) cents per peach. On Sunday, the profit per peach decreases to \(m-n\) cents each.

- Total Profit Calculation for Each Day

To make the same profit on both days, calculate the total profit for each day in terms of the number of peaches sold on Saturday (let this be \(x\)) and Sunday (\(T-x\)).

- Profit from Saturday

The total profit from Saturday is given by the product of the number of peaches sold on Saturday \(x\) and profit per peach \(m\). Hence, the profit is \(xm\) cents.

- Profit from Sunday

The total profit from Sunday is given by the product of the number of peaches sold on Sunday \(T-x\) and profit per peach on Sunday \(m-n\). Hence, the profit is \((T-x)(m-n)\) cents.

- Setting Profits Equal

To make the same profit on each day, set the two profit expressions equal: \(xm = (T-x)(m-n)\)

- Solving the Equation

Solve for \(x\): Distribute the right-hand side: \(xm = T(m-n) - x(m-n)\) Combine like terms to isolate \(x\): \(xm + x(m-n) = T(m-n)\) Factor out \(x\) from the left-hand side: \(x[m+(m-n)] = T(m-n)\) Simplify the expression inside the brackets: \(x[2m-n] = T(m-n)\) Finally, solve for \(x\): \(x = \frac{T(m-n)}{2m-n}\)

- Choosing the Correct Option

Compare the derived expression with the given options. The derived expression \(\frac{T(m-n)}{2m-n}\) matches option e.

Key concepts

Profit Calculation

Profit calculation is an essential concept in business and GMAT math problems.
When we talk about profit, we are often concerned with how much money is earned after considering all costs. In the given exercise, the farm stand has to calculate profit across two days. On Saturday, the profit per peach is higher, and on Sunday, it is reduced by a specific amount, denoted as \(n\).

To calculate the total profit properly:

• First, determine how much profit is made per item on each day.
• Then, analyze how many items (peaches) are sold on each day.
• Finally, equate the total profits if the goal is to have them be the same on both days.

Doing these calculations correctly ensures that decisions on pricing and sales strategy maximize earnings.

Understanding how to equate profits and correctly isolate variables, as shown in the steps of the solution, are valuable skills in solving such problems efficiently.

Algebraic Equations

Algebraic equations form the backbone of understanding and solving many GMAT math problems.
In our given problem, it's crucial to set up and solve an algebraic equation that represents the equality of profits on both days. Here’s a simplified breakdown:

1. **Expression for profit on Saturday:** \(xm\) (where \ x\ represents the number of peaches sold on Saturday).
2. **Expression for profit on Sunday:** \((T - x)(m - n)\) (where \ T\ is the total number of peaches and \ x\ is the number sold on Saturday).

• These expressions represent how the total profit can be formulated based on the number of items sold and the respective profit per item on different days.

Setting these equal to each other built our core algebraic equation:
\(xm = (T - x)(m - n)\)
This equation can then be solved by distributing, combining like terms, and isolating the variable \ x\ to find the required number of peaches to sell on Saturday.
This problem also reinforces the importance of methodically solving algebraic equations by following each step carefully, ensuring accuracy in the result.

GMAT Preparation

Effective GMAT preparation involves practicing various types of math and verbal problems to ensure a strong understanding.
The exercise focuses on profit and algebra, which are common in GMAT problems. To prepare efficiently:

**Focus on Key Math Concepts:**

• Review algebraic equations and practice setting up equations from word problems.
• Understand profit calculations and how to balance costs and revenues.

**Develop Problem-Solving Strategies:**

• Always read through the problem carefully and identify the key variables.
• Practice breaking down complex problems into simpler parts.
• Follow step-by-step solutions to understand the logical progression of solving the problem.

**Time Management:**

• Simulate GMAT test conditions to improve time management.
• Practice regularly under timed conditions to become comfortable with the time constraints.

By focusing on essential concepts and practicing regularly, you can significantly improve your problem-solving ability and overall GMAT performance.