Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, consider this scenario: Fred earns $$\$ 40$$ mowing lawns. He spends $$\$ 10$$ on \(\mathrm{mp} 3 \mathrm{~s}\), puts half of what is left in a savings account, and gets another $$\$ 5$$ for washing his neighbor's car. How much money does Fred keep?

Step-by-Step Solution

Answer

Fred keeps $20.

Step 1: Calculate the Remaining Money After Spending on MP3s

Fred initially earns \\(40 by mowing lawns. He spends \\)10 on MP3s. We need to subtract \\(10 from \\)40 to find out how much money he has left.Remaining money = \\(40 - \\)10 = \$30.

Step 2: Determine Amount Put into Savings

Fred puts half of what is left after purchasing MP3s into a savings account. From Step 1, we know he has \\(30 remaining.Money put in savings = \( \frac{1}{2} \times \\)30 = \$15 \).

Step 3: Calculate Money Kept After Savings

We know from Step 1 that Fred has \\(30 left after the MP3 purchase. After placing \\)15 in savings, we subtract this amount from the remaining balance.Money kept after savings = \\(30 - \\)15 = \$15.

Step 4: Add Money Earned from Washing Neighbor's Car

Fred earns an additional \\(5 for washing his neighbor's car. We need to add this to the money he kept after savings to find the total amount he keeps.Total money kept = \\)15 + \\(5 = \\)20.

Key Concepts

Arithmetic Operations

Understanding arithmetic operations is key to solving many algebra word problems. In Fred's scenario, we use basic operations like addition and subtraction. These operations help us track Fred's earnings and expenses.
Fred initially earns \\(40, and we subtract \\)10 for the mp3s he purchases. The operation used here is subtraction:

• Remaining money = \(\\(40 - \\)10 = \\(30\)

After calculating the remaining amount, Fred's savings are determined by taking half of what's left. This requires the use of multiplication with a fraction:

• Money put in savings = \(\frac{1}{2} \times \\)30 = \$15\)

Using these basic operations helps make each step clearer and manageable, demonstrating how arithmetic is foundational to solving such problems.

Savings Calculation

Calculating savings involves determining what portion of available money should be set aside for future use. Fred's exercise illustrates a simple way to calculate savings. After spending on his mp3s, Fred decides to save half of his remaining money.

The calculation looks like this: for the \\(30 left, Fred puts into his savings:

• Amount saved = \(\frac{1}{2} \times \\)30 = \$15\)

This teaches us that savings can be calculated by identifying a certain percentage or fraction of one's remaining funds. This principle can be applied to personal finance—deciding how much to set aside from our own earnings.

Money Management

Money management is about making informed and balanced decisions regarding one's finances. In Fred's case, he demonstrates sound money management practices through planned saving and spending. Initially, he starts by using a portion of his income to purchase something (mp3s) he desires and subsequently allocates a portion of his funds to savings.

This is a simple yet powerful strategy for managing money:

• Allocate a portion for immediate expenses
• Save part of what's left for future needs
• Continue to earn additional income through other tasks

By adhering to such a routine, Fred ensures that he keeps part of his earnings available for any unforeseen expenses or future plans, illustrating practical money management.

Step-by-Step Problem Solving

Breaking down problems into smaller, manageable steps is essential in math and finance. The step-by-step solution approach not only simplifies complex problems but also makes them easier to understand.

Fred's problem is dissected into logical steps:

• Transfer of his expenditures into deductions
• Calculation of his savings
• Consideration of additional income from other jobs

This strategy allows us to tackle the problem one piece at a time. By understanding and solving each part, Fred—and similarly, anyone gaining these skills—can confidently come to a well-reasoned solution. This methodology exemplifies a systematic approach to problem-solving that can be applied far beyond this singular example.