Solving Real-World Price Problems: A Comprehensive Guide

Real-world mathematics often involves solving the pumpkin problem, where prices and quantities of different items are given, and the task is to find the cost per unit. This article will delve into a specific example involving two different types of fruits: strawberries and apples. We will explore two methods to solve the problem: a straightforward approach without equations and an algebraic method using linear equations.

Method 1: Direct Calculation

In the first method, we solve the problem by direct calculation. Let's consider the scenario where Diana initially purchases 6 pounds of strawberries and 4 pounds of apples for $18.90. Later, she realizes she needs more and purchases an additional 3 pounds of each for $10.74.

To find the cost per pound for each type of fruit, we start by subtracting the cost of the additional purchase from the initial cost:

18.90 - 10.74  8.16

This $8.16 difference represents the cost of 3 pounds of strawberries and 1 pound of apples in the additional purchase.

Now, let's distribute this cost to the respective fruits:

3 pounds of strawberries: $8.16 / 3 $2.72 per pound 1 pound of apples: $8.16 - $8.16 / 3 $5.44 / 1 $5.44

However, this direct method has an issue as it does not accurately represent the true cost per pound. Instead, we need to distribute the $8.16 proportionally to all 10 pounds.

The true distribution would be:

9 pounds of strawberries: $2.72 * 9 $24.48 5 pounds of apples: $5.44 * 5 $27.20

Adding these together, we get $24.48 for strawberries and $27.20 for apples, which doesn't match the $39.64 total cost. This indicates the need for a more accurate approach.

Method 2: Algebraic Solution Using Linear Equations

The second method involves using linear equations to solve the problem accurately. Let's define:

- S as the cost per pound of strawberries. - A as the cost per pound of apples.

From the problem, we can set up the following linear system of equations:

6S 4A 18.90 (from the initial purchase) 3S 3A 10.74 (from the additional purchase)

We can simplify the second equation by dividing both sides by 3:

S A 3.58

Now we solve this system of equations. First, we can isolate A in terms of S from the simplified second equation:

A 3.58 - S

Substitute this into the first equation:

6S 4(3.58 - S) 18.90 6S 14.32 - 4S 18.90 2S 14.32 18.90 2S 4.58 S 2.29

Now, substitute S 2.29 back into the equation A 3.58 - S:

A 3.58 - 2.29 A 1.29

Thus, the cost per pound for strawberries is $2.29, and for apples is $1.29.

Conclusion

Solving real-world problems using mathematics requires a thorough approach, often involving linear equations for greater accuracy. The algebraic method demonstrated in this article provides a precise and reliable way to determine unit prices, which is essential for both academic and practical applications.

Key Takeaways

Understanding the pumpkin problem and its real-world applications. Using linear equations to solve price problems accurately. Mastering the algebraic approach for precise unit pricing.

Keywords

keywords: pumpkin problem, algebraic solution, linear equations