Solving a Real-World Problem with Inequalities: Tanya's Apple and Orange Purchase

Imagine a scenario where Tanya has $17.50 to spend on apples and oranges. Let's break down the problem into manageable parts and use mathematical inequalities to solve it. This example not only helps us understand inequalities but also how to apply them in real-life contexts.

Understanding the Problem

Tanya has $17.50 to buy apples and oranges. Apples cost $1.50 per pound, and oranges cost $0.50 per pound. There's a restriction on the quantity available: there are only 10 pounds of apples in stock and 15 pounds of oranges in stock. We need to determine the possible combinations of apples and oranges Tanya can purchase given these constraints.

Formulating Inequalities

We will use variables to represent the number of pounds of apples and oranges Tanya can buy. Let a be the number of pounds of apples and o be the number of pounds of oranges.

Step-by-Step Inequality Formation

Given the quantity constraints:

a ≤ 10, since Tanya can buy at most 10 pounds of apples.

o ≤ 15, since Tanya can buy at most 15 pounds of oranges.

The total cost of apples and oranges should not exceed $17.50. Therefore, the cost constraint becomes: 1.5a 0.5o ≤ 17.5.

Combining these, we have the following system of inequalities:

a ≤ 10

o ≤ 15

1.5a 0.5o ≤ 17.5

Interpreting the Inequalities

The inequalities represent a feasible region on a coordinate plane. The points on the line and within the region defined by these inequalities are the possible combinations of a and o.

A Practical Solution

One possible solution is to buy 10 pounds of apples and 5 pounds of oranges. Let’s verify:

Cost of 10 pounds of apples: 10 * $1.50 $15.00

Cost of 5 pounds of oranges: 5 * $0.50 $2.50

Total cost: $15.00 $2.50 $17.50, which fits within her budget.

Apple and Orange Comparisons

One must be careful with comparisons involving different quantities. Initially, Tanya could afford 20 pounds of apples or 15 pounds of oranges, but due to the different costs per pound (apples $1.50, oranges $0.50), a direct comparison isn't straightforward.

Converting Quantities

If we convert the quantities to a common basis, we can make a more intuitive comparison:

20 pounds of apples at $1.50 per pound is equivalent to 10 pounds of apples at $1.50 per pound.

15 pounds of oranges at $0.50 per pound is the same as 15 pounds of oranges, which would cost the same as 7.5 pounds of apples at $1.50 per pound.

This shows that a direct comparison in pounds is misleading. The correct way to compare is by cost.

Inequalities in Context

The inequalities help us understand the constraints and visualize the possible solutions. By solving the system of inequalities, Tanya can determine how many pounds of each fruit she can purchase without exceeding her budget.

Conclusion

This problem demonstrates the power and practicality of using inequalities to solve real-world problems. Understanding and applying inequalities can help students, educators, and anyone looking to make informed decisions, especially when budgeting and comparing quantities of different items.

Key Points

Inequality Formulation: Inequalities help define feasible regions for real-world scenarios.

Cost Consideration: Comparing quantities should consider the cost per unit, not just the raw quantity.

Feasible Solutions: The region defined by inequalities represents all possible solutions.