Tan Bought X Apples For $10. One-fifth Of The Apples Were Spoilt. He Sold The Remaining Apples At $0.50 Each And Earned A Profit. (a) Find The Number Of Remaining Apples In Terms Of X. (b) Find An Expression, In Terms Of X, For The Profit Earned.
Tan's Apple Purchase and Profit Calculation
In this mathematical problem, we delve into the scenario of Tan's apple purchase, where he buys a certain quantity of apples for a fixed price, encounters spoilage, sells the remaining apples, and ultimately aims to make a profit. This problem involves algebraic expressions, fractions, and profit calculation. To effectively solve this, we must break it down into manageable parts, focusing on finding the number of remaining apples and then determining the expression for the profit earned.
Initially, Tan purchased 'x' apples for $10. This sets the stage for the problem. The core challenge lies in understanding how the spoilage affects the quantity of apples available for sale and, consequently, the profit. One-fifth of the apples being spoilt is a crucial piece of information, as it directly impacts the number of apples Tan can sell. To find the number of remaining apples, we need to subtract the number of spoilt apples from the total number of apples purchased. This involves basic fraction manipulation and algebraic thinking.
Following this, the problem introduces the selling price of the remaining apples, which is $0.50 each. This piece of information is vital for calculating the revenue generated from the sale. The revenue is the product of the number of remaining apples and the selling price per apple. However, earning a profit is not just about generating revenue; it's about the revenue exceeding the initial cost. Therefore, to find the profit, we must subtract the initial cost of the apples from the revenue generated.
(a) Determining the Remaining Apples in Terms of x
The first part of the problem requires us to express the number of remaining apples in terms of 'x.' Given that one-fifth of the apples were spoilt, we need to calculate how many apples this represents. Mathematically, this can be expressed as (1/5) * x. These spoilt apples are not available for sale, so we need to subtract this quantity from the total number of apples purchased, which is 'x.' Therefore, the number of remaining apples can be calculated as x - (1/5) * x.
To simplify this expression, we need to combine the 'x' terms. We can rewrite 'x' as (5/5) * x, which allows us to subtract (1/5) * x more easily. The expression then becomes (5/5) * x - (1/5) * x. When we subtract the fractions, we get (4/5) * x. This simplified expression tells us that four-fifths of the original number of apples remain after the spoilage.
Therefore, the remaining apples in terms of x are (4/5)x. This expression is crucial for the next part of the problem, where we need to calculate the profit earned. It represents the quantity of apples that Tan can sell to generate revenue. The clarity and accuracy of this expression are paramount for the subsequent profit calculation. A mistake here would propagate through the rest of the problem, leading to an incorrect final answer. Thus, it is essential to double-check the calculations and ensure that the expression accurately represents the number of remaining apples.
(b) Finding the Expression for Profit Earned in Terms of x
The second part of the problem is focused on determining an expression for the profit earned, using 'x' as the variable. This requires a clear understanding of the concept of profit, which is the difference between the revenue generated from sales and the total cost incurred. In this scenario, the revenue is the total income from selling the remaining apples, and the cost is the initial amount Tan spent on purchasing the apples.
We already know from part (a) that the number of remaining apples is (4/5)x. We also know that Tan sold each of these apples for $0.50. Therefore, the total revenue can be calculated by multiplying the number of remaining apples by the selling price per apple. This gives us a revenue of (4/5)x * $0.50. Simplifying this expression, we get $0.40x. This expression represents the total income Tan received from selling the apples.
However, to find the profit, we need to subtract the initial cost from the revenue. Tan initially bought the apples for $10. Therefore, the profit can be calculated as the revenue ($0.40x) minus the cost ($10). This gives us the final expression for the profit earned: $0.40x - $10. This algebraic expression represents the profit Tan made, and it is directly dependent on the initial number of apples purchased (x).
This expression highlights the relationship between the number of apples purchased and the profit earned. A higher value of 'x' (more apples purchased) will potentially lead to a higher profit, provided that the spoilage rate remains constant and the selling price stays the same. However, the negative $10 in the expression signifies the initial investment, indicating that Tan needs to sell enough apples to cover this cost before making a true profit. The expression $0.40x - $10 provides a clear and concise way to determine Tan's profit based on the number of apples he initially bought.
Printing Costs Problem
Transitioning from the apple problem, we now encounter a new mathematical scenario involving printing costs. This problem focuses on the financial aspects of printing, specifically examining the relationship between the cost of printing, the number of copies printed, and any fixed charges involved. Understanding these relationships is crucial in various real-world scenarios, such as managing business expenses, budgeting for personal projects, or analyzing the economics of publishing.
The core challenge in this type of problem lies in identifying the different components of the total cost. Typically, printing costs involve a fixed component, such as the cost of setting up the printing equipment or a base service fee, and a variable component, which depends on the number of copies printed. The variable cost is usually expressed as a cost per copy. The total cost is the sum of these two components. To solve printing cost problems effectively, it's essential to break down the information provided, identify the fixed and variable costs, and then use algebraic expressions to represent the relationships.
For instance, if a printing service charges a fixed setup fee plus a certain amount per copy, we can represent the total cost as a linear equation. The fixed fee would be the constant term in the equation, and the cost per copy would be the coefficient of the variable representing the number of copies. By understanding this basic structure, we can formulate equations, solve for unknowns, and make informed decisions based on the cost implications of printing different quantities.
Furthermore, printing cost problems may also involve comparisons between different printing services or options. In such cases, it's necessary to formulate equations for each option and then compare them to determine the most cost-effective choice. This often involves analyzing the break-even point, which is the number of copies at which the costs of two options are equal. Beyond the break-even point, one option may become more economical than the other. Therefore, a thorough understanding of algebraic principles and cost analysis is essential for tackling printing cost problems.
In conclusion, the problems presented here cover a range of mathematical concepts, from algebraic expressions and profit calculations to cost analysis in printing scenarios. By breaking down these problems into smaller, manageable parts, carefully analyzing the given information, and applying the appropriate mathematical techniques, we can arrive at accurate solutions and gain valuable insights into real-world situations.