Problem 65 - Algebra 1

Question

Use the following information. You are in charge of the music for a school dance. The school's budget allows only \(\$ 300\) for music, which is enough to hire a disc jockey for 4 hours. You would rather hire a live band, but the band charges \(\$ 135\) per hour. Your school does not allow students to be charged an admission fee. To raise the money for a live band, you obtain permission for a voluntary contribution of \(\$ 1.25\) per person. How much extra money do you need to raise?

Step-by-Step Solution

Answer

The additional amount needed is $240

Step 1: Understand the problem

First, understand that you have to calculate the difference between the total cost of hiring a live band for 4 hours and the available fund of $300.

Step 2: Translate the problem into mathematical equation

Let's denote the cost per hour for the live band as \(b\) which equals to $135. We know the disc jockey can be hired for 4 hours with the available fund of $300. So, if we still hire the live band for 4 hours, the total cost will be 4 * \(b\) = 4 * 135 = $540.

Step 3: Calculate the required extra amount

Now, subtract the available fund $300 from the total cost of hiring the live band for 4 hours $540. This gives the extra amount required which equals to $540 - $300 = $240.

Key Concepts

Budget Management in Real-Life Situations

Budget management is a crucial skill, particularly when confronted with limitations and fixed constraints, like the $300 budget in this scenario. Effective budget management involves evaluating what you can afford within the confines of available capital. In this exercise, we explore how to prioritize spending by comparing the cost of a disc jockey with the higher expense of a live band.
This requires a strategic approach to managing funds by determining essential expenses versus additional desires. To make informed financial decisions, consider these steps:

• Assess and identify available funds – Know exactly how much money is in your budget upfront.
• Evaluate costs – Understand how much each option (disc jockey vs. live band) will cost.
• Identify fundraising opportunities – Look for creative solutions like the voluntary $1.25 contribution per attendee.

Solving this problem teaches valuable lessons in financial literacy, such as efficient allocation of resources and strategizing to meet goals, even with limited funds.

Using Mathematical Equations to Solve Problems

Mathematical equations are powerful tools for solving real-world problems, such as deciding how to allocate funds for a school dance. In this scenario, equations help us calculate the money needed to hire a live band instead of a disc jockey.
By translating details into a mathematical formula, we gain clarity and precision in understanding the financial shortfall. Begin by defining variables: let the cost of the band per hour be represented as \(b\) and equal to \(135. The question asks for the total cost of employing the band for 4 hours, which can be expressed as:\[ 4 \times b = 4 \times 135 = 540 \]Then, compare this to the existing budget of \)300 to find the extra amount needed:\[ 540 - 300 = 240 \]Equations simplify complex scenarios and make problem-solving more manageable. They provide a structured approach to challenge analysis, fostering a better understanding and clearer insights into potential solutions.

Formulating a Problem-Solving Strategy

Developing a problem-solving strategy is key when tackling any complex task, such as budgeting for entertainment within constraints. The strategy involves breaking down the problem into manageable steps, as was done in this exercise. Start by understanding what you're trying to solve. For this problem, it is the deficit between the allotted budget and the desired spending. Break it down accurately by first calculating total costs, then identifying any discrepancies between those and the funds available.

• Define the problem clearly – Recognize that this involves exceeding the $300 budget.
• Use mathematical calculations to find precise answers.
• Seek alternative solutions – like encouraging donations for the shortfall.

Engaging in systematic problem-solving enhances critical thinking skills and equips you with the tools needed for effective decision-making in similar real-life challenges. Planning, analysis, and creativity combine to create effective and actionable strategies.