Question

On the plans for a building, the drawing of a wall measures \(3 \frac{1}{3}\) inches tall by \(9 \frac{1}{2}\) inches wide. The scale says that 1 inch equals 6 feet. What will be the dimensions of the wall once it is built? A. 19 feet tall by 55 feet wide B. 19 feet tall by 57 feet wide C. 20 feet tall by 55 feet wide D. 20 feet tall by 57 feet wide

Short answer

D. 20 feet tall by 57 feet wide

Step-by-step solution

- Convert Mixed Numbers to Improper Fractions

The height of the wall on the plan is given as 3 \( \frac{1}{3} \) inches. Convert this to an improper fraction: \( 3 \frac{1}{3} = \frac{10}{3} \) inches. Similarly, the width is given as 9 \( \frac{1}{2} \) inches. Convert this to an improper fraction: \( 9 \frac{1}{2} = \frac{19}{2} \) inches.

- Use the Scale to Calculate the Actual Dimensions

According to the scale, 1 inch equals 6 feet. Multiply the height and width in inches by the scale factor to convert to feet. For the height: \[ \frac{10}{3} \times 6 = 20 \text{ feet} \] For the width: \[ \frac{19}{2} \times 6 = 57 \text{ feet} \]

- Identify the Correct Option

Compare the calculated dimensions (20 feet tall and 57 feet wide) with the given options. The correct option is: D. 20 feet tall by 57 feet wide

Key concepts

Converting Mixed Numbers

Mixed numbers are numbers made up of an integer and a fraction, like \(3 \frac{1}{3}\) or \(9 \frac{1}{2}\). To work with them in equations, it's often easier to convert them to improper fractions.

An improper fraction has a larger numerator (top number) than its denominator (bottom number).

Here's how to convert a mixed number:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Place that sum over the original denominator.

For example:

• Convert \(3 \frac{1}{3}\):
  \3 \times 3 + 1 = 10\, so \frac{10}{3}\.


• Convert \(9 \frac{1}{2}\):
  \9 \times 2 + 1 = 19\, so \frac{19}{2}\.

Now, these converted fractions are ready to be used in mathematical calculations, like finding actual dimensions using a scale.

Scale Factor

A scale factor is a number used to multiply dimensions to convert measurements from one scale to another.

In scale drawings, like building plans, the scale gives the relationship between the drawing and the actual size.

For our problem, the scale says that 1 inch on the drawing equals 6 feet in real life.

To find the real dimensions:

• Multiply the measurement on the drawing by the scale factor.

For example:

• The height on the plan is \frac{10}{3}\ inches. So, \[ \frac{10}{3} \times 6 = 20 \text{ feet} \].


• The width on the plan is \frac{19}{2}\ inches. So, \[ \frac{19}{2} \times 6 = 57 \text{ feet} \].

Knowing the scale factor helps convert the measurements accurately.

Improper Fractions

Improper fractions are used when the numerator is larger than the denominator.

For example, \frac{10}{3}\ is an improper fraction. These fractions are useful for calculations in algebra and geometry, as they eliminate the need for mixed numbers and make multiplication and division straightforward.

Converting mixed numbers to improper fractions involves multiplying and adding, which we discussed earlier, but here's a recap:

• Convert \(3 \frac{1}{3}\) to \frac{10}{3}\


• Convert \(9 \frac{1}{2}\) to \frac{19}{2}\

Using improper fractions simplifies working with scales and other proportional relationships, as they provide a clear, single fraction for calculations.

Dimension Calculation

Dimension calculation involves converting measures from a drawing to their real-world sizes. In our exercise:

• The given scale is 1 inch equals 6 feet.


• The drawing height is \3 \frac{1}{3}\ inches, converted to \frac{10}{3}\ inches as an improper fraction.


• The drawing width is \9 \frac{1}{2}\ inches, converted to \frac{19}{2}\ inches.

To find the actual height:
\[ \frac{10}{3} \times 6 = 20 \text{ feet} \]
To find the actual width:
\[ \frac{19}{2} \times 6 = 57 \text{ feet} \]
So, the wall's real dimensions will be 20 feet tall and 57 feet wide. Knowing how to convert and multiply properly ensures you get accurate results for any scaled drawing.