Problem 16 - Algebra 1

Question

Pover DESIGN In Exercises \(14-16\), you want the cover of a sports media guide to show two photos across its width. The cover is \(6 \frac{1}{2}\) inches wide, and the left and right margins are each \(\frac{3}{4}\) inch. The space between the photos is \(\frac{1}{2}\) inch. How wide should you make the photos? Solve the equation and answer the question.

Step-by-Step Solution

Answer

This is a mathematical absurdity, which implies that it is not possible to place two photos of the calculated width on the cover along with the margins and the space. Therefore, you should reconsider the dimensions of your cover, margins, or the space between the photos.

Step 1: Convert All Measurements to A Common Denominator

The first step involves converting the given measurements into fractions with a common denominator to simplify subtraction. The total width of the cover is \(6 \frac{1}{2}\) inches which is equivalent to \(\frac{13}{2}\) inches. The left and right margins are each \(\frac{3}{4}\) inch, equal to \(\frac{3}{4} \times \frac{2}{2} = \frac{6}{8}\) inches. The space between the photos is \(\frac{1}{2}\) inch, which is equivalent to \(\frac{4}{8}\) inch.

Step 2: Calculate Total Space Occupied by the Margins and the Space Between the Photos

In this step, add the measurements of the left and right margins to the space between the photos to find the total space not occupied by the photos. This is \(2 \times \frac{6}{8} + \frac{4}{8} = \frac{16}{8}\) inch.

Step 3: Determine the Width of the Photos

Now that we have the total space occupied by the covers and the space between the photos, we subtract this from the total width of the cover and divide by 2 (since two photos of the same size will be shown). So, the width of each photo will be \(\frac{\frac{13}{2} - \frac{16}{8}}{2} = \frac{\frac{26}{4} - \frac{16}{8}}{2} = \frac{\frac{26 - 32}{8}}{2} = \frac{-6}{8 \times 2} = \frac{-6}{16} = -\frac{3}{8}\) inches.

Key Concepts

Measurement Conversion

Measurement conversion is crucial when working with fractions, especially when dealing with mixed measurements in a problem. To solve our sports media guide cover problem, we needed to convert all given measurements, like the total width of the cover and the margins, into fractions with a common denominator. This helps in performing accurate arithmetic operations like addition or subtraction.

A mixed number, like the cover width of \(6 \frac{1}{2}\) inches, is first expressed as an improper fraction, \(\frac{13}{2}\), before adjusting to a common denominator with other fractions involved, such as \(\frac{3}{4}\). To do this, you multiply the numerator and denominator of a fraction by the same number to find equivalent fractions, resulting in \(\frac{6}{8}\) from \(\frac{3}{4}\) so it matches others like \(\frac{4}{8}\). These conversions are necessary to align all values for subsequent calculations.

Subtraction

Subtraction of fractions is an important skill in fractional arithmetic and is used to solve numerous real-world problems, such as determining the width of photos for the sports media guide. Once all measurements are converted to a common denominator, subtraction becomes a straightforward process.

In our given exercise, the need arises to subtract the total space occupied by the photos and margins from the entire width. Converting \(6 \frac{1}{2}\) inches into a proper fraction \(\frac{13}{2}\) aids smooth calculation. Simplifying further involves subtracting the fractional terms, like \(\frac{16}{8}\) from \(\frac{26}{4}\), ensuring accurate measurements for practical project planning.

Equation Solving

Equation solving identifies the critical steps involved in mathematically resolving a problem by determining unknown variables. In the context of our exercise, the challenge is to find an expression for the width of the photos. To do this, we equate the sum of the occupied margins and space in the formula, subtract from the overall width, then divide by 2 to account for the two photos.

The problem-solving sequence follows: \(\left(\frac{13}{2} - \frac{16}{8}\right) / 2\). This equation breaks down the solution into steps of subtraction and division, revealing the photo's intended width, and illustrates the precision involved in equation solving of fractional elements.

Problem Solving Steps

Breaking down a problem into manageable steps highlights the problem-solving process effectively, ensuring proper comprehension and application of mathematical principles in real situations. The structured approach in this exercise showcases how every step builds upon the previous one.

Initially, we convert all measurements. Then, we calculate the space taken up by non-photo elements, using addition and multiplication as necessary. Subsequently, the crucial subtraction step isolates the remaining space. Finally, the division determines each photo's width. By following these steps, we simplify complex-looking problems, thereby mastering strategic problem solving in a clear, replicable format.