Problem 12 - Algebra A Combined Function

Question

Solve. On an architect's blueprint, 1 inch corresponds to 4 feet. Find the length of a wall represented by a line that is \(3 \frac{7}{8}\) inches long on the blueprint.

Step-by-Step Solution

Answer

The wall is 15.5 feet long.

Step 1: Identify the Conversion Factor

In this problem, we need to convert a length from inches on a blueprint to feet in actual measurement. It is given that 1 inch on the blueprint corresponds to 4 feet in real life.

Step 2: Convert Mixed Fraction to a Decimal

The length on the blueprint is given as \(3 \frac{7}{8}\). First, convert the mixed fraction to a decimal. \(\frac{7}{8}\) can be converted to 0.875. Therefore, \(3 \frac{7}{8} = 3 + 0.875 = 3.875\) inches.

Step 3: Calculate Actual Length in Feet

Multiply the length in inches by the conversion factor (4 feet per inch) to find the actual length. So, the calculation is: \(3.875 \times 4 = 15.5\) feet.

Key Concepts

Understanding Blueprint Measurement

Blueprints are essential tools in architecture and engineering to represent real-world objects and structures on a manageable scale. They allow architects to communicate their designs effectively. Typically, a conversion factor is used to understand the relationship between measurements on the blueprint and actual measurements in real life. For example, a common practice might be that 1 inch on the blueprint equals a certain number of feet in real life. This conversion factor is crucial in translating blueprint measurements to actual sizes.

When working with a blueprint, you'll notice that its measurements are in smaller units due to its scaled nature. By knowing the conversion factor, you can easily convert these measurements into real-world dimensions. In our example, 1 inch on the blueprint is equivalent to 4 feet in actual size. This means to find the real length represented by a line on the blueprint, you simply multiply the number of inches by 4 to get feet.

Working with Mixed Fractions

Mixed fractions occur when whole numbers and fractions are combined. In architectural drawings, measurements might often be expressed in mixed fractions, such as in our problem with the given length of 3 \(\frac{7}{8}\) inches.

To work with these, it's often easier to convert them into decimal form. Let's break it down:

• The whole number is straightforward: 3.
• The fraction part, in this case, is \(\frac{7}{8}\). To convert it to a decimal, you divide 7 by 8, which is 0.875.

Thus, the mixed fraction \(3 \frac{7}{8}\) becomes 3.875 in decimal form. This conversion is helpful when further calculations or conversions are necessary, like applying the conversion factor from a blueprint.

Mastering Decimal Conversion

Decimal conversion simplifies mathematical operations by reducing mixed numbers into a single line of computation, often much easier to work with than fractions. Converting a mixed fraction to a decimal saves time and reduces the complexity of arithmetic operations.

Let's examine the conversion process:

• Start with the mixed fraction \(3 \frac{7}{8}\).
• The whole number is easy; it remains 3.
• The fractional part, \(\frac{7}{8}\), needs to be divided: 7 divided by 8 results in 0.875.
• Add the decimal to the whole number, resulting in 3.875.

Decimals streamline calculations, making multiplication tasks like converting blueprint measurements to real-world sizes quicker. In our example, multiplying the decimal inches (3.875) by 4 feet per inch quickly yields the real-world measurement of 15.5 feet.