CHALLENGE An apple costs the same as 2 oranges. Together, an...

CHALLENGE An apple costs the same as 2 oranges. Together, an orange and a banana cost $10 \notin$ more than an apple. Two oranges cost $15 \notin$ more than a banana. What is the cost for one of each fruit?

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Key Concepts

System of Linear Equations and Inequalities

Many algebraic word problems involve establishing a system of equations or inequalities that represent the constraints provided. This system allows for a structured approach to finding the values of unknowns by ensuring that all given relationships are satisfied simultaneously.

Ratio and Proportion

When a problem involves statements like one item costing as much as a certain number of another, the concept of ratio and proportion is essential. It allows one to express the relationship between the costs of different items as a fraction or ratio, which can then be used to set up accurate equations.

Word Problem Translation

This concept entails deciphering a verbal description to create mathematical representations. In algebra, problems are often presented in a narrative format and must be translated into equations and inequalities that capture the relationships between different quantities.

Algebraic Manipulation and Substitution

After setting up the equations, algebraic manipulation—such as substitution and elimination methods—is used to solve for the unknown variables. This process entails rearranging the equations to isolate variables and substituting expressions from one equation into another to find a solution that meets all constraints.

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