Question

A new homeowner drew a floor plan of her new house, in which 1 inch on the floor plan is equivalent to 18 inches on the actual floor. If the actual longest side of the floor in one of the bedrooms is 153 inches, what is the length of the longest side of the same bedroom in the floor plan?

Short answer

The length of the longest side of the bedroom in the floor plan is 8.5 inches.

Step-by-step solution

Set up the proportion

We know that the scale is 1 inch on the floor plan equals 18 inches on the actual floor. Let x be the length of the longest side of the bedroom in the floor plan. We can set up this proportion: \[\frac{1\text{ inch (floor plan)}}{18\text{ inches (actual)}} = \frac{x\text{ inches (floor plan)}}{153\text{ inches (actual)}}\]

Solve for x

To solve for x, we can cross-multiply and then solve for x: \[1 * 153 = 18 * x\]

Simplify the equation

Now, we can simplify the equation: \[153 = 18x\]

Divide both sides by 18

In order to isolate x, we need to divide both sides of the equation by 18: \[\frac{153}{18} = x\]

Calculate x

Now we can calculate x: \[x = 8.5\]

Interpret the result

The length of the longest side of the bedroom in the floor plan is 8.5 inches.

Key concepts

Proportion Word Problems

Understanding proportion word problems is essential in many fields including mathematics, science, and even everyday situations. Proportions are statements that two ratios are equal. They can be used to solve real-world problems such as converting scales on a map or floor plan, as seen in our original exercise.

To tackle a proportion problem, you generally follow these steps:

• Identify the two pairs of quantities that have a proportional relationship.
• Set up a proportion equation by equating the two ratios.
• Solve for the unknown variable using cross-multiplication.

In our example, the homeowner needed to convert the scale of a floor plan to actual size. By setting up a proportion between the floor plan scale (1 inch on paper to 18 inches in reality) and the actual length side of the floor (153 inches), the problem simplifies to finding the corresponding length on the floor plan.

Geometry

In geometry, we often deal with a variety of scales and measurements which are crucial for accurate representation of shapes and spaces. When drawing models or constructing plans, such as those for a house, we use scale drawings. These are two-dimensional diagrams that represent a three-dimensional space on a smaller, more manageable scale.

To make a scale drawing, specific rules are followed to ensure that every dimension and angle is proportional to those of the actual object or space. This involves:

• Understanding and applying the scale factor correctly.
• Solving for unknown dimensions using the scale factor.
• Analyzing drawings to extract measurements accurately.

Our exercise is a prime example of applying these principles in geometry to create a scale factor that accurately reflects the actual dimensions of the bedroom.

SAT Math Practice

Preparing for standardised tests like the SAT can be daunting. One key component of SAT Math is the ability to solve problems involving ratios, proportions, and geometry. It’s not just about knowing the concepts but also about applying them effectively.

For successful SAT Math practice, consider these tips:

• Understand and review the fundamental concepts regularly.
• Solve a wide variety of practice problems to build familiarity with different question types.
• Work on enhancing your problem-solving speed and accuracy.

The scale drawing problem from our exercise could very well appear on such an exam, making it a suitable practice problem. Not only does it require knowledge of basic geometric measurements and proportions, but it also tests your ability to apply that knowledge in a practical context.