Problem 40 - Prealgebra

Question

A newspaper is \(12 \frac{1}{4}\) inches wide and 22 inches long. This is \(1 \frac{1}{4}\) inches narrower and one-half inch longer than the old edition. What were the previous dimensions of the newspaper?

Step-by-Step Solution

Answer

The previous newspaper dimensions were 13 inches wide and 21.5 inches long.

Step 1: Understanding Current Dimensions

The current dimensions of the newspaper are given as width = \(12 \frac{1}{4}\) inches and length = 22 inches. We need to use these dimensions to find the previous dimensions.

Step 2: Calculate Previous Width

The current width is \(12 \frac{1}{4}\) inches, which is \(1 \frac{1}{4}\) inches narrower than the old edition. To find the old width, add \(1 \frac{1}{4}\) inches to \(12 \frac{1}{4}\) inches:\[\text{Old Width} = 12 \frac{1}{4} + 1 \frac{1}{4} = 12 + \frac{1}{4} + 1 + \frac{1}{4} = 12 + 1 + \frac{2}{4} = 13 \text{ inches}\]

Step 3: Calculate Previous Length

The current length is 22 inches, which is one-half inch longer than the old edition. To find the old length, subtract one-half inch from 22 inches:\[\text{Old Length} = 22 - \frac{1}{2} = 21 \frac{1}{2} \text{ inches}\]

Step 4: Conclusion on Previous Dimensions

Based on our calculations, the previous dimensions of the newspaper were 13 inches in width and \(21 \frac{1}{2}\) inches in length.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of a numerator and a denominator. The numerator is the top number, indicating how many parts we have or are considering, and the denominator is the bottom number, showing how many equal parts make up the whole. For example, the fraction \( \frac{1}{4} \) means that one part out of four equal parts is being considered.

When working with fractions in problems like the newspaper dimensions, it's important to understand how to add and subtract them. Take the width calculation exercise, we have fractions like \(12 \frac{1}{4}\). Here, \(\frac{1}{4}\) is the fractional part added to the whole number, 12. Converting mixed numbers into improper fractions can often make it easier to perform calculations, but in this case, working with the mixed number directly illustrates the step-by-step addition more clearly.

• Mixed numbers combine whole numbers and fractions (e.g., \(12 \frac{1}{4}\)).
• Add fractions by finding a common denominator if needed.
• Convert to improper fractions to simplify calculations if possible.

Length and Width Calculations

Calculating length and width involves understanding the dimensions of objects and how they change. In our exercise, the current newspaper's dimensions are known, and slight modifications tell us about its previous form.

By interpreting the text information:

• "\(1 \frac{1}{4}\) inches narrower" helps us understand that to find the old width, you add \(1 \frac{1}{4}\) to the current width.
• "One-half inch longer" means subtracting \(\frac{1}{2}\) inch from the current length to get the previous length.

By using straightforward addition and subtraction, and accurately accounting for both whole numbers and fractions, we solve for the dimensions without confusion. This approach is common in practical applications, such as adjusting fabric pieces in sewing or modifying design plans in construction.

Measurement Conversion

Measurement conversions often occur in practical scenarios where dimensions are compared or modified. While our exercise doesn't require conversion between units like inches to centimeters, understanding these principles is useful.

Consider: If the previous tutorial mentioned converting inches to feet, you might need to know that 12 inches make up one foot. However, in this problem, we're focusing on adjusting measurements slightly:

• "Adding" or "subtracting" straightforward amount: e.g., 1 inch, 5 centimeters.
• Look out for fractional parts - even in conversions, fractions retain their roles, like \(\frac{1}{2}\) foot can become 6 inches.

Working with length conversions ensures that the dimensions remain consistent across different units and provides a basis for accurate calculations in more complex problems. Understanding and applying conversion principles smoothly can enhance your comfort with a range of mathematical challenges.