Problem 76 - Algebra and Trigonometry

Question

Buying a Cottage A group of friends decides to buy a vacation home for \(\$ 120,000,\) sharing the cost equally. If they can find one more person to join them, each person's contribution will drop by \(\$ 6000\) . How many people are in the group?

Step-by-Step Solution

Answer

4 people are in the group initially.

Step 1: Define Variables

Let the original number of people in the group be \( n \). Each person contributes \( \frac{120,000}{n} \).

Step 2: Set Up Equation for New Contribution

If one more person joins, the number of people becomes \( n + 1 \). The new contribution per person is \( \frac{120,000}{n+1} \).

Step 3: Establish the Contribution Difference Equation

According to the problem, each person's contribution decreases by \( 6000 \) when one more person joins. Hence, we can write the equation: \( \frac{120,000}{n} - \frac{120,000}{n+1} = 6000 \).

Step 4: Solve the Equation

Rearrange and simplify the equation \( \frac{120,000}{n} - \frac{120,000}{n+1} = 6000 \) to find \( n \): Multiply both sides by \( n(n+1) \): \[ 120,000(n+1) - 120,000n = 6000n(n+1) \].Simplify: \[ 120,000 = 6000n^2 + 6000n \].Divide everything by 6000: \[ 20 = n^2 + n \].Rearrange: \[ n^2 + n - 20 = 0 \].Solve the quadratic equation using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1 \), \( b=1 \), and \( c=-20 \):\[ n = \frac{-1 \pm \sqrt{1 + 80}}{2} \].\[ n = \frac{-1 \pm 9}{2} \].

Step 5: Calculate and Check Solutions

Calculate the possible solutions: \( n = \frac{-1 + 9}{2} = 4 \) or \( n = \frac{-1 - 9}{2} = -5 \). Since the number of people cannot be negative, the solution is \( n = 4 \).

Key Concepts

Equation Solving

Solving equations can seem like a puzzle at first, but it is a vital skill in mathematics. When approaching an equation, the goal is to find the value of the unknown variable that makes the equation true. In our cottage problem, the unknown variable is the number of friends initially planning to purchase the vacation home. We denote this number as \( n \). Setting up an equation helps us organize the given information. We know each person’s contribution changes based on new information (e.g., an additional person joining). The first step is to translate the word problem into a mathematical equation. This involves understanding what each part of the problem means numerically and how they relate:

• Translate relationship to contribution: \( \frac{120,000}{n} \)-original and \( \frac{120,000}{n+1} \)-new.
• Differentiate the values via subtraction.
• Equate this difference to the given decrease amount ($6000). This forms our equation.

Thus, breaking complex situations into simpler components is crucial for equation-solving.

Mathematics

Mathematics is not just about numbers; it's about understanding relationships and changes. In many problems like our cottage buying scenario, math helps in visualizing how individual contributions adjust when circumstances change, such as adding one more friend.The equation helps us see the impact of adding another group member mathematically. This is explored through division and subtraction operations. This equation was simplified to determine the likely scenarios:

• Multiply through by expressions like \( n(n+1) \) to clear fractions.
• Simplify the equivalent to find a more direct representation.

This practice allows us to see how concepts build upon each other to help solve real-life arithmetic problems effectively.

Algebra Concepts

Algebra is concerned with finding unknowns and establishing variable relationships. In the process of buying a cottage, setting \( n \) helps us calculate contributions based on different member numbers.Key steps in solving a quadratic equation involve rearranging it into a standard form. For this problem, a quadratic-like structure \( n^2 + n - 20 = 0 \) emerges after simplifying initial equations:

• Adjust terms to get a zero on one side, creating a basic quadratic format.
• Apply the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find potential values for \( n \).
• Analyze results to determine valid (realistic) solutions.

The formula captures the essence of finding roots in an equation, showing which values satisfy initial conditions. Real-world application of these concepts uncovers practical answers to situational challenges, like friend contributions.