Problem 46 - Algebra 1

Question

You are designing a newspaper page with three photos. The page is \(13 \frac{1}{4}\) inches wide with 1 inch margins on both sides. You need to allow \(\frac{3}{4}\) inch between photographs. How wide should you make the photos if they are of equal size? Solve the equation and answer the question.

Step-by-Step Solution

Answer

Each photo should be \(3\frac{1}{4}\) inches wide.

Step 1: Determine the total width of the newspaper page minus margins

The total width of the newspaper is given as \(13\frac{1}{4}\) inches. Out of this, 1 inch on both sides are margins. Therefore, the width available for the photos is \(13\frac{1}{4} - 2 = 11\frac{1}{4}\) inches.

Step 2: Calculate the total gap space between photographs

Between the three photos, there are two gaps. Each gap must be \(\frac{3}{4}\) inch. Therefore, the total space taken up by gaps is \(2 \times \frac{3}{4} = 1\frac{1}{2}\) inches.

Step 3: Determine the total space available for the photos

Now, subtract the total gap space calculated in step 2 from the available photo width calculated in step 1: \(11\frac{1}{4} - 1\frac{1}{2} = 9\frac{3}{4}\) inches.

Step 4: Dividing the available space equally among three photos

The remaining available space from step 3 should be divided equally among the three photos. So each photo will be \( \frac{9\frac{3}{4}}{3} = 3\frac{1}{4} \) inches wide.

Key Concepts

Equations

In algebra, equations are mathematical statements that assert the equality of two expressions. They consist of variables, numbers, and arithmetic operators like addition, subtraction, multiplication, and division. Solving an equation means finding the value of the variable that makes the equation true. In the context of this exercise, equations help determine how to allocate space effectively for the photos on the newspaper page. An equation translates the word problem into a mathematical format which is easier to manipulate and solve. For example, if we need to find out how wide the photos should be, we would set up an equation that balances the total available space on one side and the combination of photo widths and space gaps on the other side. This turns the word problem into something more manageable by using variables to represent the widths of the photos, and using arithmetic to isolate and solve for these variables.

Fractions

Fractions are a way of expressing numbers that aren't whole numbers, often representing parts of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). In the exercise above, illustrating the width of the page and gaps between photos involves working with fractions. To correctly manage these widths, we must have a solid understanding of how to perform arithmetic with fractions.

• Adding and Subtracting Fractions: Ensure that the denominators are the same before performing the operation.
• Multiplying Fractions: Multiply the numerators together and the denominators together.
• Dividing Fractions: Multiply by the reciprocal of the divisor.

In this problem, such as calculating margins and gaps, requires subtracting and multiplying fractions, turning them into a routine aspect of calculations. An example includes subtracting the total margin width from the newspaper page width and calculating the total gap space.

Step-by-Step Solutions

Using step-by-step solutions is a powerful approach in problem-solving, especially in mathematics. It involves breaking down a complex problem into manageable steps, each involving a simple calculation or reasoning process. This methodology not only helps in reaching the correct solution but also enhances understanding. In the current problem, the following steps were employed:

• First, determining the available width by accounting for the margins. This was essential to move forward with the space left for photos.
• Then, identifying the total space occupied by the gaps between photos. Each step builds on the previous one, ensuring there are no gaps in understanding.
• Finally, distributing the remaining space equally among the photos to find their individual widths using division. By following these steps, you can ensure accuracy and clarity.

This kind of approach provides a clear path from the problem's initial phase to its solution, reducing errors and increasing comprehension.