Question

If $$\$ 105,000$$ is spent on land and building a house, how much did the land cost if the cost of building the house was \(25 \%\) less than the cost of the land? A. $$\$ 45,000$$ B. $$\$ 50,000$$ C. $$\$ 55,000$$ D. $$\$ 60,000$$ E. $$\$ 65,000$$

Short answer

The land cost \$\ 60,000.

Step-by-step solution

Define Variables

Let the cost of the land be denoted as \( L \). According to the problem, the cost of building the house is \( 25\% \) less than the cost of the land. This means the cost of building the house is \( 0.75L \).

Write the Cost Equation

The total cost given is \$\$ 105,000, which is the sum of the cost of the land and the cost of building the house. Therefore, we can write the equation: \[ L + 0.75L = 105,000 \]

Combine Like Terms

Add the terms on the left side of the equation: \[ 1.75L = 105,000 \]

Solve for \( L \)

To find the cost of the land, divide both sides by 1.75: \[ L = \frac{105,000}{1.75} \] Calculating the right-hand side, we get: \[ L = 60,000 \]

Key concepts

Cost Calculation

In this problem, we need to determine how much the land costs given a total expense and a condition regarding the relative costs of land and building the house. Calculating costs involves understanding the relationship between different components of the total expenditure. Here, we break down the total cost and relate the parts using simple algebra.
The problem specifies that the total cost, including both land and building expenses, is $105,000. We represent the cost of the land with a variable and use the given percentage relationship to express the cost of the house in terms of this variable.

Percentages

Percentages help us understand how one quantity relates to another in terms of parts per hundred. In this exercise, the building's cost is 25% less than the cost of the land. Here’s how we use percentages:
Understanding that 25% less means the cost of building the house is 75% of the land cost. This can be expressed mathematically as:

• If the cost of the land is L, then the cost of building is 0.75L.

Using percentages helps us convert verbal relationships into a mathematical form that we can then solve systematically for unknown quantities.

Problem-Solving Steps

To solve GMAT math problems effectively, follow a series of structured steps:
1. Understand and define the variables. This clarifies what you need to find and sets up your equation. In this case, we start by letting L be the cost of the land.
2. Translate the problem into an equation. Write down the relationship using mathematical expressions. Here we noted that the total cost equals the sum of the land and the house cost, expressed as L + 0.75L = 105,000.
3. Simplify and solve the equation. Combine like terms to form an easier equation, then solve for the variable.
4. Check your solution by substituting it back into the context of the problem to ensure it makes sense.

Variable Definition

Defining variables is crucial in algebra to solve equations. In this exercise, we define L as the cost of the land. It’s a placeholder that helps us write out other relationships directly:
By setting up L, determining the cost of building the house as 0.75L becomes straightforward. This step is pivotal since it translates the verbal problem into a form where we can apply algebraic operations.
After defining our variables, we create an equation reflecting the total cost (land + house). This allows us to solve for L by combining terms and performing basic arithmetic operations, leading to the final value for the land cost.