Problem 18 - Elementary and Intermediate Algebra

Question

Solve each of these number problems. See Example \(1 .\) If a number is added to the numerator of \(\frac{5}{7},\) and twice as much is subtracted from the denominator, the result is \(8 .\) Find the number.

Step-by-Step Solution

Answer

The number is 3.

Step 1: Understand the Problem

We are given a fraction \( \frac{5}{7} \) and we need to add a number \( x \) to the numerator and subtract twice this number from the denominator to make the fraction equal to 8. This means we can express the condition as \( \frac{5+x}{7-2x} = 8 \).

Step 2: Set Up the Equation

The problem tells us that after making modifications to the fraction \( \frac{5}{7} \), the result should be 8. We can set up the equation based on this: \( \frac{5+x}{7-2x} = 8 \).

Step 3: Solve the Equation

Multiply both sides of the equation \( \frac{5+x}{7-2x} = 8 \) by \(7-2x\) to get rid of the fraction: \(5+x = 8(7-2x)\). Simplify the right side: \(5+x = 56 - 16x\). Move all terms involving \(x\) to one side to solve for \(x\): \(x + 16x = 56 - 5\). Simplifying gives \(17x = 51\), therefore \(x = 3\).

Step 4: Verification

Substitute \(x = 3\) back into the modified fraction to verify: Add 3 to the numerator and subtract 6 from the denominator, giving \( \frac{8}{1} = 8 \), which matches the desired result.

Key Concepts

Fraction Manipulation

Fractions are a way to express one quantity as a part of another. They consist of a numerator and a denominator. Understanding how to manipulate fractions is crucial in solving many algebraic problems. In this exercise:

• We start with the fraction \( \frac{5}{7} \).
• We then modify this fraction to meet certain conditions given in the problem statement.

To increase familiarity with these operations, remember that adding a number to the numerator is like increasing the share you have in the whole, while subtracting from the denominator is like dividing your share into smaller pieces. These processes can significantly change the value of a fraction. It's important to methodically adjust one part of the fraction at a time, observing how each change affects the overall value.

Equation Solving Steps

When solving problems involving algebraic equations derived from fractions, a systematic approach is necessary:- **Understand the Problem:** Clearly grasp what changes need to be made to the original fraction. For example, here we modify \( \frac{5}{7} \) into a new expression.- **Translate into an Equation:** This means translating the problem conditions into a mathematical equation. Thus, the problem transforms to \( \frac{5+x}{7-2x} = 8 \).- **Clear the Fraction:** Multiply both sides of the equation by the denominator, \( 7-2x \), to eliminate the fraction and simplify the equation into one involving basic algebra.- **Solve for the Variable:** Combine and isolate terms to solve for the unknown variable, \( x \).- **Verify the Solution:** Always substitute back into the original modified fraction to ensure the solution meets the conditions laid out in the problem.

Numerator and Denominator Adjustment

An adjustment to either part of a fraction, whether a numerator or a denominator, significantly affects its value. In this problem:

• Adding the number \( x \) to the numerator results in \( 5 + x \).
• Subtracting twice this amount, \( 2x \), from the denominator transforms it to \( 7 - 2x \).

Each operation directly influences the value of the fraction, changing it to achieve a resultant value of 8. Adjusting the numerator increases the value if it becomes larger, while reducing the denominator increases the fraction's value more dramatically. Understanding these principles enables you to correctly maneuver through similar fraction-based algebraic equations.