Problem 8 - College Algebra

Question

Ben originally filled out 8 more applications than Henry. Then each boy filled out 3 additional applications, bringing the total to 28 . How many applications did each boy originally fill out?

Step-by-Step Solution

Answer

Henry originally filled 7 applications, Ben filled 15.

Step 1: Define Variables

Let \( x \) represent the number of applications Henry originally filled out. According to the problem, Ben filled out 8 more applications than Henry, so let \( x + 8 \) represent the number of applications Ben originally filled out.

Step 2: Create an Equation

Both boys filled out 3 more applications, so Henry filled out \( x + 3 \) applications in total and Ben filled out \( (x + 8) + 3 = x + 11 \) applications in total. Together, they filled out 28 applications. We can write the equation: \((x + 3) + (x + 11) = 28\).

Step 3: Simplify the Equation

Combine like terms in the equation: \(x + 3 + x + 11 = 2x + 14 = 28\).

Step 4: Solve for x

Subtract 14 from both sides of the equation to isolate the terms with \(x\): \(2x = 28 - 14\), which simplifies to \(2x = 14\). Divide both sides by 2 to solve for \(x\): \(x = 7\).

Step 5: Find Original Applications for Each Boy

Henry originally filled out \(x = 7\) applications. Ben filled out \(x + 8 = 7 + 8 = 15\) applications originally.

Key Concepts

Problem Solving

When faced with a mathematical word problem, the first step is understanding the situation. Identify what is being asked and determine what information is provided. In the problem involving Ben and Henry, we know a few key facts:

• Ben originally filled out 8 more applications than Henry.
• Each boy filled out 3 additional applications later on.
• After filling out additional applications, they completed 28 in total.

The goal is to figure out how many applications each boy originally filled out. To achieve this, break down the situation into manageable parts. Translate the word problem into a mathematical expression using known information, allowing you to find a logical solution efficiently.

Algebraic Expressions

An algebraic expression is crucial in converting word problems into solvable equations. In this case, we first define variables to symbolize unknown quantities. Here, let:

• \( x \) represent the number of applications Henry originally filled out.
• \( x + 8 \) represent the number of applications Ben originally filled out.

These expressions capture the relationship between Ben and Henry’s work.Next, incorporate changes to these original values. With each filling out 3 more applications, their totals become:

• Henry: \( x + 3 \)
• Ben: \( x + 11 \) (since \( (x + 8) + 3 = x + 11 \))

Crafting these expressions is essential, as they form the foundation of your equation, embodying the problem's context.

Equation Simplification

Simplifying an equation involves combining like terms and isolating the variable on one side. Starting from the equation formed by the expressions representing the boys' work:\[(x + 3) + (x + 11) = 28\]Combine like terms:\[x + x + 3 + 11 = 2x + 14\]This simplification reduces the complexity of your equation. Next, solve it by performing operations such as subtraction and division to isolate the variable:

• Subtract 14 from both sides: \(2x = 14\).
• Divide both sides by 2: \(x = 7\).

These steps reveal that Henry originally filled out 7 applications. Calculating Ben's using \( x + 8 \), we find he filled out 15. Simplifying equations is a powerful technique for solving problems efficiently.