Solving Word Problems Involving Shopping: A Comprehensive Guide

Introduction

Solving word problems is an essential skill in mathematics, especially when it comes to real-world applications like shopping. In this article, we’ll explore a practical example involving the purchase of a handbag and blouses, and how to solve it step-by-step using algebraic equations.

The Problem

Diana paid a total of $205.50 for a handbag and 5 similar blouses. The handbag cost $100.50 more than each blouse. How much did she pay for the 5 blouses?

Step-by-Step Solution Using Variables

Let's represent the cost of one blouse as n.

Then, the cost of the handbag is n 100.50.

The total cost for the handbag and 5 blouses is given as:

[ (n 100.50) 5n 205.50 ]

This simplifies to:

[ 6n 100.50 205.50 ]

Now, isolate n to find the cost of one blouse:

[ 6n 205.50 - 100.50 ]

[ 6n 105.00 ]

[ n frac{105.00}{6} ]

[ n 17.50 ]

Since n represents the cost of one blouse, we now calculate the total cost of the 5 blouses:

[ 5n 5 times 17.50 87.50 ]

Alternative Approach Using Simultaneous Equations

Let's represent the cost of the handbag as x and the cost of one blouse as y.

From the problem, we have two equations:

[ x 5y 205.50 quad text{(Equation 1)} ]

[ x - 100.50 - y 0 quad text{(Equation 2)} ]

Rearrange Equation 2 to solve for x in terms of y:

[ x 100.50 y ]

Substitute this into Equation 1:

[ (100.50 y) 5y 205.50 ]

[ 100.50 6y 205.50 ]

[ 6y 105.00 ]

[ y frac{105.00}{6} ]

[ y 17.50 ]

Now, substitute y back into the expression for x:

[ x 100.50 17.50 118.00 ]

So, the total cost of the 5 blouses is:

[ 5y 5 times 17.50 87.50 ]

Conclusion

In summary, Diana paid $17.50 for each blouse, totaling $87.50 for the 5 blouses. This problem showcases how algebraic equations can be used to solve word problems efficiently, making it easier to understand and manage real-life scenarios involving shopping and budgeting.