Problem 41 - Elementary Algebra

Question

The numerator of a fraction is 8 less than the denominator. The fraction in its simplest form is \(\frac{5}{6}\). Find the fraction.

Step-by-Step Solution

Answer

The original fraction is \( \frac{40}{48} \).

Step 1: Understanding the Relationship

Let the denominator of the fraction be \( x \). The numerator is 8 less than this, so it is \( x - 8 \). The fraction is then \( \frac{x-8}{x} \).

Step 2: Simplifying the Fraction

We know the fraction reduces to \( \frac{5}{6} \). This gives us the equation:\[ \frac{x-8}{x} = \frac{5}{6} \]

Step 3: Cross-Multiplying to Solve for x

Cross-multiply to eliminate the fraction:\[ 6(x - 8) = 5x \]This simplifies to:\[ 6x - 48 = 5x \]

Step 4: Solving for the Denominator

Subtract \( 5x \) from both sides to isolate \( x \):\[ 6x - 5x = 48 \]This simplifies to:\[ x = 48 \]

Step 5: Finding the Numerator

Substitute \( x = 48 \) back into the expression for the numerator, \( x - 8 \). Thus:\[ x - 8 = 48 - 8 = 40 \]

Step 6: Presenting the Original Fraction

Now, the original fraction is \( \frac{40}{48} \). To check:Simplify \( \frac{40}{48} \) by dividing both the numerator and the denominator by 8:\[ \frac{40}{48} = \frac{5}{6} \]This confirms the solution.

Key Concepts

Cross Multiplication

Whenever you encounter a problem that involves fractions, especially in solving equations, you can make use of a technique called cross multiplication. This method helps you solve for unknowns efficiently by eliminating fractions altogether.
This is achieved by multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa.

For example, consider the equation \( \frac{x-8}{x} = \frac{5}{6} \)
By cross multiplying, you will get:

• Multiply the numerator of the first fraction (\(x-8\)) with the denominator of the second fraction (6).
• Multiply the numerator of the second fraction (5) with the denominator of the first fraction (\(x\)).

Thus, the resulting equation is:\[ 6(x - 8) = 5x \]
With cross multiplication, you can easily remove the fractions and work with a standard algebraic equation, making it simpler to solve for the variable.

Simplifying Fractions

Simplifying fractions is a crucial step in many problems as it allows for clearer and more manageable results. The goal is to reduce the fraction to its lowest terms, where the numerator and the denominator have no common factors other than 1.
In our problem, the fraction is originally expressed as \( \frac{40}{48} \).
To simplify:

• Find the greatest common divisor (GCD) of 40 and 48, which is 8.
• Divide both the numerator (40) and the denominator (48) by this GCD.

This results in:\[ \frac{40 \div 8}{48 \div 8} = \frac{5}{6} \]
By simplifying, it's easy to see that the fraction reduces to \( \frac{5}{6} \), confirming that both original values were correct.
Simplifying helps prevent errors and ensures that your final result is as straightforward as possible.

Algebraic Expressions

Algebraic expressions are composed of numbers, variables, and arithmetic operations and are fundamental in formulating equations to describe real-world relationships.
In dealing with the original exercise, setting up an algebraic expression laid the foundation for solving the problem.
Initially, we defined our fraction with:

• The denominator as \( x \).
• The numerator as \( x - 8 \), since the numerator is 8 less than the denominator.

This formed the expression \( \frac{x-8}{x} \), which equates to the simplified form of the fraction \( \frac{5}{6} \).
Using algebraic expressions, this relationship was translated into the equation:\[ \frac{x-8}{x} = \frac{5}{6} \]
Understanding how to construct such expressions is vital as it allows for structured problem-solving, bridging the gap between word problems and mathematical equations.