An Interactive Algebraic Puzzle Involving Pens, Notebooks, and Pencils

In this article, we delve into an intriguing algebraic puzzle that deals with the expenditure of two individuals, S and C, at a bookshop. Using algebraic equations and step-by-step reasoning, we aim to determine the amount spent by S on pens. This not only showcases the practical application of algebra in solving real-world problems but also enhances understanding of basic algebraic principles.

Introduction to the Puzzle

S and C went to a bookshop where S purchased 5 pens, 3 notebooks, and 9 pencils, and used up all her money. Meanwhile, C purchased 6 pens, 6 notebooks, and 18 pencils and paid 50 more than what S paid. The puzzle is to find the fraction of the amount S spent on pens. This article aims to break down the problem, solve it using algebraic methods, and provide a detailed explanation.

Setting Up the Equations

Let's denote the price of a pen as x, the price of a notebook as y, and the price of a pencil as z. We also define the total amount spent by S as s. The given information can be translated into the following equations:

S's Expenditure

Equation 1: [5x 3y 9z s] (Equation for S's expenditure)

C's Expenditure

Equation 2: [6x 6y 18z s 50] (Equation for C's expenditure, which is 50 more than S's expenditure)

Simplifying the Equations

To simplify these equations, we can use substitution and elimination techniques. First, let's subtract Equation 2 from Equation 1:

Subtracting Equations

[5x 3y 9z - (6x 6y 18z) s - (s 50)]

[5x 3y 9z - 6x - 6y - 18z -50]

[-x - 3y - 9z -50]

This simplifies to:

[x 3y 9z 50] (Equation 3)

Further Simplification

Next, we can substitute Equation 1 into Equation 3 to eliminate variables:

p>5x 3y 9z 50]

From Equation 1, we have:

[5x 3y 9z s]

Substituting s into Equation 3:

[x 3y 9z frac{s}{5}]

Using Equation 3, we get a simplified equation:

[x 3y 9z frac{s}{5}] (Equation 4)

Notice that the left side of Equation 4 is the same as the left side of Equation 3. This means:

[x 3y 9z frac{s}{5}]

Multiplying both sides by 5:

[5x 15y 45z s]

Now, we can see that:

[5x 15y 45z s] (Equation 5)

Solving for the Amount Spent on Pens

From Equation 5, it is clear that the amount spent by S on pens is:

[5x]

Dividing Equation 5 by s gives:

[5x/s frac{5}{6}]

Multiplying both sides by 100 to convert this into a percentage:

[5x/s times 100 frac{5}{6} times 100 83.33%]

Therefore, the amount S spent on pens is approximately 83.33% of her total expenditure.

Conclusion

This article has demonstrated how to solve the given algebraic puzzle involving pounds, notebooks, and pencils. By setting up equations and simplifying them step-by-step, we were able to determine that S spent approximately 83.33% of her money on pens. This exercise not only provides a practical example of algebra in real-life scenarios but also reinforces the importance of systematic problem-solving in mathematics.