Problem 65 - Elementary Algebra

Question

If the ratio of the length of a rectangle to its width is \(\frac{5}{2}\), and the width is 24 centimeters, find its length.

Step-by-Step Solution

Answer

The length of the rectangle is 60 cm.

Step 1: Interpret the Ratio

The problem states that the ratio of the length of the rectangle to its width is \( \frac{5}{2} \). This means that for every 2 units of width, there are 5 units of length.

Step 2: Set Up the Proportion Equation

Using the given ratio \( \frac{5}{2} \), and knowing the width is 24 cm, set up the equation \( \frac{\text{Length}}{24} = \frac{5}{2} \).

Step 3: Solve for the Length

Cross-multiply to get \(2 \times \text{Length} = 5 \times 24\).Simplifying, we have \(2 \times \text{Length} = 120\).

Step 4: Isolate the Length

Divide both sides by 2 to solve for Length: \(\text{Length} = \frac{120}{2}\).

Step 5: Calculate the Length

Perform the division to find the length: \(\text{Length} = 60\).

Key Concepts

Proportion Equation

A proportion equation is a statement that two ratios are equal. In a proportion, you often see this showing up as two fractions set equal to each other. Here is a simple way to understand it:
Imagine you are comparing two different sets of values, saying they have the same relationship. For example, if you have a ratio of boys to girls as 3 to 4, and you want to create another group with the same ratio, you'd use a proportion equation to find out how many girls should be there if you have 6 boys.

In our rectangle problem, the proportions help compare the length and width of the rectangle using the ratio \( \frac{5}{2} \). It means that the length is 5 parts when the width is 2 parts. Since we know the width is 24 cm, we set up the proportion equation \( \frac{\text{Length}}{24} = \frac{5}{2} \) to find the unknown length.

Cross-Multiplication

Cross-multiplication is a method used to solve proportion equations, and it helps to find an unknown in a fraction. It's a valuable trick that simplifies operating with two ratios. Let's break it down a bit more.
When you have a proportion, such as \( \frac{5}{2} = \frac{\text{Length}}{24} \), you "cross-multiply" by multiplying the numerator of each fraction by the denominator of the other fraction. So here, multiply 5 by 24 and 2 by the Length.

• This gives: \( 2 \times \text{Length} = 5 \times 24 \)
• Simplifying this, you get \( 2 \times \text{Length} = 120 \)

With cross-multiplication, you step closer toward unraveling the mystery of the unknown (Length in this case), making it a super helpful tool for ratio and proportion problems.

Rectangle Dimensions

Rectangles are one of the simplest yet most common shapes in geometry. To know about rectangle dimensions means understanding its fundamental properties: length and width.
Every rectangle has two pairs of opposite sides that are equal and parallel. In this specific problem, the dimensions are tied together through a ratio, meaning there's a specific relationship between the length and width.

• The length refers to the longer side of the rectangle.
• The width is the shorter side, in usual scenarios.

Here, the width is given as 24 cm. Knowing this, and using the ratio \( \frac{5}{2} \), we applied proportion and cross-multiplication techniques to find the length, which calculated as 60 cm. Understanding dimensions not only helps in solving math problems but also is crucial in real-life applications, like design and engineering. Knowing how to manipulate and calculate rectangle dimensions can make projects more efficient.