Problem 15 - Elementary and Intermediate Algebra

Question

Solve each of these number problems. See Example \(1 .\) If the denominator of \(\frac{3}{4}\) is increased by a number, and the numerator is doubled, the result is \(1 .\) Find the number.

Step-by-Step Solution

Answer

The number is 2.

Step 1: Set Up the Equation

First, translate the given word problem into an equation using algebraic expressions. The original fraction is \(\frac{3}{4}\). If the denominator is increased by a number \(x\), then the new fraction becomes \(\frac{6}{4+x}\). Set this equal to 1, as given in the problem: \[ \frac{6}{4+x} = 1 \]

Step 2: Solve for x

To solve the equation \( \frac{6}{4+x} = 1 \), first eliminate the fraction by multiplying both sides by \(4+x\): \[ 6 = 1 \cdot (4+x) \] Simplify the equation: \[ 6 = 4 + x \] Subtract 4 from both sides to isolate \(x\): \[ 6 - 4 = x \] Thus, \(x = 2\).

Step 3: Verification

Verify the solution by substituting \(x = 2\) back into the scenario to ensure correctness. If the denominator is increased by \(2\), the fraction becomes \(\frac{6}{6}\), which simplifies to 1. This matches the condition given in the problem.

Key Concepts

Fractions

Understanding fractions is crucial in algebraic problem solving, especially when dealing with expressions that involve ratios or comparing values. A fraction represents a part of a whole and is composed of two main parts:

• Numerator: The top number, which indicates how many parts we have.
• Denominator: The bottom number, which shows how many equal parts the whole is divided into.

In the given problem, we started with the original fraction \( \frac{3}{4} \). When we altered the fraction, we performed operations on both the numerator and the denominator. Doubling the numerator raised its value from 3 to 6, which affects the fraction's value as it changes the number of parts considered. Meanwhile, adding a number \( x \) to the denominator increases the total number of parts, altering the fraction's value. Such manipulations are common when solving fraction-related word problems.

Equation Solving

Equation solving is the process of finding the value of unknown variables that satisfy the given mathematical sentence. This can be straightforward or complex, depending on the equation's nature.

In this example, we worked with a simple linear equation derived from the original word problem:\[ \frac{6}{4+x} = 1 \]To solve this, we first eliminated the fraction by multiplying both sides by the denominator \( 4+x \), transforming it into a more manageable linear form:\[ 6 = 1 \cdot (4+x) \]Next, we simplified and isolated the variable \( x \) to find its value. The process usually involves:

• Eliminating fractions by multiplying through by the denominator, as we did.
• Using basic algebraic operations like addition, subtraction, multiplication, or division to simplify and ultimately solve for the unknowns.
• Checking the solution by plugging it back into the original equation to ensure it satisfies all conditions.

These steps highlight the systematic approach to solving algebraic equations, ensuring clarity and accuracy.

Word Problems

Word problems are exercises where real-world situations are described verbally, and you're required to translate them into mathematical equations to find a solution. They often involve multiple steps and concepts.

The problem given is a classic example, where we have to decipher the relationship between numbers described in text form. Here's how the breakdown works:

• Identify what the problem is asking. In this case, finding a number that alters the fraction to meet a specific condition.
• Set up an equation to mathematically represent the problem's narrative. This involves translating each part of the description into numbers or symbols, leading to the equation \( \frac{6}{4+x} = 1 \).
• Solve the equation using appropriate algebraic techniques. For our problem, it involved basic operations like adding and subtracting to isolate \( x \).
• Verify the solution by substituting back into the problem to confirm the results match the given condition.

Mastering word problems involves developing skills to interpret information accurately, perform correct algebraic operations, and ensure your solutions make logical sense in the context provided.