Problem 47 - Prealgebra

Question

Tate's flower garden has a perimeter of 25 feet. He plans to add 2 feet 9 inches to the width and 3 feet 9 inches to the length. What is the new perimeter in feet?

Step-by-Step Solution

Answer

The new perimeter is 38 feet.

Step 1: Convert Inches to Feet

First, convert inches to feet for ease of calculation. Since there are 12 inches in a foot, 9 inches is equivalent to \( \frac{9}{12} \) feet, which simplifies to \( \frac{3}{4} \) feet (or 0.75 feet).

Step 2: Calculate Original Dimensions

The original perimeter is 25 feet. For a rectangle, the perimeter \( P = 2(l + w) \). To find the sum \( l + w \), divide the perimeter by 2: \( l + w = \frac{25}{2} = 12.5 \) feet.

Step 3: Calculate New Width and Length Addition in Feet

According to the problem, Tate is adding 2 feet 9 inches to the width and 3 feet 9 inches to the length. This translates to adding 2.75 feet for width and 3.75 feet for length.

Step 4: Calculate New Dimensions

The new width \( w' = w + 2.75 \) and the new length \( l' = l + 3.75 \). These new dimensions need to be used to find the new perimeter.

Step 5: Calculate New Perimeter

The new perimeter \( P' = 2(l' + w') \). Substitute \( l' = l + 3.75 \) and \( w' = w + 2.75 \) into this formula: \[ P' = 2((12.5 - w + 2.75) + w + 3.75) \]. Simplifying gives: \[ P' = 2(12.5 + 6.5) = 2(19) = 38 \] feet.

Key Concepts

Unit Conversion

Understanding unit conversion is essential, especially when dealing with mixed units, such as feet and inches. When converting inches to feet, remember that:

• 1 foot is equal to 12 inches.
• This means to convert inches to feet, divide the number of inches by 12.
• For example, 9 inches can be converted to feet by calculating \( \frac{9}{12} = 0.75 \) feet.

Breaking down units is important for simplifying calculations and ensuring accuracy in measurements. By converting all measurements to the same unit, you can add them more easily and arrive at correct solutions.

Rectangle Dimensions

A rectangle's dimensions are a fundamental concept in geometry. The perimeter \( P \) of a rectangle is calculated using the formula:

• \( P = 2(l + w) \)
• Here, \( l \) represents the length, and \( w \) represents the width.

Knowing these dimensions allows you to understand how much space the rectangle covers or defines its boundary. Understanding the relationship between perimeter and dimensions enables you to solve problems involving changes to a rectangle, such as increasing length or width.

Addition of Measurements

Adding measurements often involves working with units such as feet and inches together. Here's how to add these measurements effectively:

• Ensure all units are consistent before adding. Convert all measurements to the same unit.


• For example, adding 2 feet 9 inches involves converting 9 inches to feet. This becomes 2.75 feet, making addition straightforward.


• Next, use simple addition to find new dimensions: add the converted measurements to the existing length and width.

By converting units beforehand, you avoid confusion and make the arithmetic smooth and error-free.

Mathematical Simplification

Mathematical simplification is a crucial step in solving problems efficiently. Simplification helps reduce complex equations into manageable forms making it easier to understand the problem at hand and solve it.

• In this case, the solution involved simplifying fractions, such as converting \( \frac{9}{12} \) to \( \frac{3}{4} \).


• It also involved arithmetic operations like adding numbers and factoring to simplify the perimeter formula.

By breaking computations into smaller pieces and combining like terms, you achieve a concise answer. This concept is not only helpful in geometry but in all areas of math.