Problem 17 - Intermediate Algebra

Question

Kitchen Drawers. A woodworker wants to put two partitions crosswise in a drawer that is 28 inches deep, as shown below. He wants to place the partitions so that the spaces created increase by 3 inches from front to back. If the thickness of each partition is \(\frac{1}{2}\) inch, how far from the front end should he place the first partition? (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The first partition should be placed 12 inches from the front.

Step 1: Define the Variables

Let \( x \) be the distance from the front to the first partition. Then, the distance from the first to the second partition will be \( x + 3 \) inches as per the problem statement which says the spaces increase by 3 inches.

Step 2: Set Up the Equation

The woodworker has a total space of 28 inches divided as \( x + (x + 3) + \text{{partition thicknesses}} = 28 \) inches. Since each partition is \( \frac{1}{2} \) inch thick, there are two partitions contributing a total of \( 1 \) inch of thickness. Thus, the equation to solve is: \( x + (x + 3) + 1 = 28 \).

Step 3: Simplify the Equation

Combine like terms: \( x + x + 3 + 1 = 28 \) becomes \( 2x + 4 = 28 \).

Step 4: Solve for \( x \)

Isolate \( x \) by subtracting 4 from both sides: \( 2x = 24 \). Then divide by 2: \( x = 12 \).

Step 5: Verify the Solution

Check that the partition placements satisfy all conditions. The first partition is at 12 inches, the second at \( 12 + 3 + \text{{partition thickness}} = 15 + \text{{partition thickness}} \). Total space with thickness: \( 12 + (15 + 0.5) + 0.5 = 28 \), confirming the math is correct.

Key Concepts

Equations

Equations are like statements where we want to make two things equal using mathematical expressions. Think of them as the balance scales of math. Everything on one side of the scale must equal what is on the other side. In algebra, equations often help us find unknown values that make an expression true.
In this exercise, the equation is used to balance the total length of the drawer's internal dividers and the total space available, which is 28 inches. Our equation: \[x + (x + 3) + 1 = 28\] This helps us find out the exact placement of the partitions.

• Each part of the equation represents a segment of the drawer with added space for partition thickness.
• Using equations helps in systematically solving problems by providing a structure to arrange thoughts and processes.

Variable Definition

In algebra, variables are like little placeholders. They represent unknowns that we need to find or solve for. To keep everything organized and understandable, always define what your variables mean at the start.
For this problem, the variable \( x \) is defined as the distance from the front of the drawer to the first partition.

• This definition allows us to create expressions, like \( x + 3 \), for the distance from the first to second partition.
• Using variables is crucial, as it simplifies the problem into manageable parts.

Defining variables lays the groundwork for converting a word problem into a mathematical one, paving the way for using equations to solve it.

Problem Solving Steps

Problem-solving in algebra involves following specific steps to arrive at the correct solution. It's like having a set of instructions to assemble something.
Here's the breakdown:

• Step 1: Define your variables clearly. Determine what \( x \) and any other variables mean in the context of the problem.
• Step 2: Set up an equation based on the problem's conditions. This equation should incorporate all known values and relationships, like distances and partition thicknesses.
• Step 3: Simplify the equation. Combine like terms and make the equation as simple as possible.
• Step 4: Solve the equation by isolating the variable. Perform operations to find the value of your unknowns.
• Step 5: Always verify your solution. Check that it fits all the problem's given conditions.

These steps are helpful because they ensure that every part of the process is thorough and considered, making it much easier to tackle complex problems.

Geometry in Algebra

Sometimes, algebra involves solving problems with a geometric context like locating positions or measurements. This approach is called geometry in algebra.
In this exercise, we use geometry to understand and solve the spatial arrangement inside a drawer with partitions. Geometry helps in structuring the algebraic equations by considering the distances and separations between objects.

• The whole idea revolves around understanding how spaces increase and incorporating physical thicknesses into our algebraic expressions.
• Using geometry in algebra allows for practical, real-world applications, making abstract equations more relatable and understandable.

When geometric concepts meet algebra, it opens the door to solving spatial word problems effectively by measuring, calculating, and verifying geometric properties.