Solving For Books Purchased Hugh's Shopping Spree Problem

Let's dive into a mathematical problem involving Hugh's recent shopping trip. Hugh went to the store and purchased a combination of magazines and books. Each magazine was priced at $3.95, while each book had a price tag of $8.95. After his shopping spree, Hugh's total expenditure amounted to $47.65. The challenge we face is to determine the number of books Hugh bought, given that he purchased 3 magazines.

Understanding the Problem and the Equation

The core of this problem lies in understanding the relationship between the quantities and prices of the items Hugh purchased. We are provided with an equation that models this relationship:

$$3.95m + 8.95b = 47.65$$

Where:

• 'm' represents the number of magazines Hugh bought.
• 'b' represents the number of books Hugh bought.

This equation essentially states that the total cost of magazines (3.95 multiplied by the number of magazines) plus the total cost of books (8.95 multiplied by the number of books) equals the total amount Hugh spent ($47.65). We are also given that Hugh bought 3 magazines, so we know that m = 3. Our goal is to find the value of 'b', which represents the number of books Hugh purchased.

Step-by-Step Solution

To solve this problem, we will follow a step-by-step approach:

1. Substitute the known value: We know that Hugh bought 3 magazines, so we substitute m = 3 into the equation:

   $$3. 95(3) + 8.95b = 47.65$$

2. Simplify the equation: Next, we multiply 3.95 by 3:

   $$11. 85 + 8.95b = 47.65$$

3. Isolate the variable term: To isolate the term with 'b', we subtract 11.85 from both sides of the equation:

   $$12. 95b = 47.65 - 11.85$$

   $$13. 95b = 35.80$$

4. Solve for 'b': Finally, to find the value of 'b', we divide both sides of the equation by 8.95:

   $$b = \frac{35.80}{8.95}$$

   $$b = 4$$

Therefore, Hugh bought 4 books.

Detailed Explanation of Each Step

Substitution

The first crucial step in solving this problem involves substitution. We are given that Hugh purchased 3 magazines. In mathematical terms, this translates to m = 3. The original equation, which models the total cost of Hugh's purchases, is:

$$3. 95m + 8.95b = 47.65$$

To incorporate the information about the number of magazines, we replace 'm' with its known value, which is 3. This substitution is a fundamental technique in algebra, allowing us to simplify equations by replacing variables with their numerical values. By substituting m = 3, we transform the equation into:

$$3. 95(3) + 8.95b = 47.65$$

This new equation now contains only one unknown variable, 'b', representing the number of books. This simplification brings us closer to our goal of determining the number of books Hugh purchased.

Simplification

Following the substitution step, our equation looks like this:

$$3. 95(3) + 8.95b = 47.65$$

To further simplify the equation, we perform the multiplication operation: 3.95 multiplied by 3. This calculation represents the total cost of the magazines Hugh bought. Performing this multiplication, we get:

$$14. 85 + 8.95b = 47.65$$

This step is crucial because it combines the constant terms, making the equation more manageable. The equation now clearly shows the sum of the cost of magazines ($11.85) and the total cost of books (8.95b) equaling the total expenditure ($47.65). By simplifying the equation, we are one step closer to isolating the variable 'b' and finding its value.

Isolating the Variable Term

After simplifying the equation, we have:

$$15. 85 + 8.95b = 47.65$$

The next crucial step is to isolate the term that contains our variable, 'b'. This means we need to get the term 8.95b by itself on one side of the equation. To achieve this, we employ the principle of inverse operations. Since 11.85 is being added to 8.95b, we perform the inverse operation, which is subtraction. We subtract 11.85 from both sides of the equation. This maintains the balance of the equation while moving us closer to isolating 'b'.

Subtracting 11.85 from both sides gives us:

$$16. 95b = 47.65 - 11.85$$

Performing the subtraction on the right side, we get:

$$17. 95b = 35.80$$

Now, the term with 'b' is isolated on the left side, making it easier to solve for the number of books.

Solving for 'b'

Having isolated the variable term, our equation now stands as:

$$18. 95b = 35.80$$

To finally determine the value of 'b', which represents the number of books Hugh purchased, we need to isolate 'b' completely. Currently, 'b' is being multiplied by 8.95. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 8.95. This ensures that the equation remains balanced while isolating 'b'.

Dividing both sides by 8.95, we get:

$$b = \frac{35.80}{8.95}$$

Performing the division, we find:

$$b = 4$$

This result tells us that Hugh bought 4 books. This is the final step in solving the problem, providing a clear and concise answer to the question posed.

Verification

To ensure the accuracy of our solution, we can verify it by substituting the values we found back into the original equation. We determined that Hugh bought 3 magazines and 4 books. Plugging these values into the original equation:

$$3. 95m + 8.95b = 47.65$$

We substitute m = 3 and b = 4:

$$3. 95(3) + 8.95(4) = 47.65$$

Now, we perform the calculations:

$$19. 85 + 35.80 = 47.65$$

Adding the two values, we get:

$$40. 65 = 47.65$$

This confirms that our solution is correct. The total cost of 3 magazines and 4 books at the given prices does indeed equal $47.65, which is the total amount Hugh spent. This verification step adds a layer of confidence to our answer, ensuring that we have accurately solved the problem.

Conclusion

In this mathematical exploration, we successfully determined the number of books Hugh bought by carefully analyzing the given information and applying algebraic principles. The problem presented us with an equation that modeled Hugh's spending on magazines and books:

$$3. 95m + 8.95b = 47.65$$

We were also given that Hugh bought 3 magazines, which meant m = 3. Our goal was to find the value of 'b', representing the number of books.

By following a step-by-step approach, we:

1. Substituted the known value of 'm' into the equation.
2. Simplified the equation by performing multiplication.
3. Isolated the variable term by subtracting from both sides.
4. Solved for 'b' by dividing both sides.

Our calculations revealed that Hugh bought 4 books. To ensure the accuracy of our solution, we verified it by plugging the values back into the original equation and confirming that the total cost matched the given amount.

This problem demonstrates the power of algebra in solving real-world scenarios. By understanding the relationships between variables and applying mathematical operations, we can effectively tackle complex situations and arrive at accurate solutions. The ability to translate word problems into equations and solve them is a valuable skill that extends far beyond the classroom, enabling us to make informed decisions and solve problems in various aspects of life.