Problem 44 - Elementary Algebra

Question

What number must be subtracted from the numerator and denominator of \(\frac{29}{31}\) to produce a fraction equivalent to \(\frac{11}{12}\) ?

Step-by-Step Solution

Answer

Subtract 7 from both numerator and denominator.

Step 1: Define the Problem

We need to find a number, say \(x\), that when subtracted from both the numerator (29) and the denominator (31) of the fraction \( \frac{29}{31} \), results in a fraction equivalent to \( \frac{11}{12} \).

Step 2: Set Up an Equation

We can set up the equation based on the problem's condition: \( \frac{29-x}{31-x} = \frac{11}{12} \). This equation comes from the requirement that the modified fraction equals \( \frac{11}{12} \).

Step 3: Cross-Multiply to Eliminate Fractions

Cross-multiply to eliminate the fraction: \[(29-x) \times 12 = 11 \times (31-x)\].This step results in an equation without fractions, allowing us to solve for \(x\).

Step 4: Simplify the Equation

Expand both sides of the equation: \[12(29-x) = 11(31-x)\]Which simplifies to \[348 - 12x = 341 - 11x\].

Step 5: Solve for x

Re-arrange the equation to isolate \(x\): \[348 - 341 = 12x - 11x\].Simplifying gives: \[7 = x\].Thus, the number to be subtracted is 7.

Key Concepts

Understanding Fraction Equivalence

Fraction equivalence is a fundamental concept in algebra. It helps us identify when two fractions represent the same value, even if their numerators and denominators differ. A fraction, like \(\frac{a}{b}\), is equivalent to another fraction, \(\frac{c}{d}\), if the relationship \(a \times d = b \times c\) holds true.
This equation reflects the cross-multiplication principle, which confirms that two fractions are equal. Being comfortable with fraction equivalence is essential for solving many algebraic problems, especially those involving the modification of values in the numerator and denominator.

• Ensure both fractions represent the same value.
• Understand the relationship between numerators and denominators.
• Use equivalence to compare or manipulate fractions easily.

By mastering fraction equivalence, you can seamlessly navigate through problems involving comparisons or transformations of fractions.

Mastering the Art of Solving Equations

Solving equations involves finding the value that satisfies a given mathematical statement. In our exercise, we are tasked with finding a number \(x\) that, when subtracted from both the numerator and denominator, results in a fraction that matches another specific fraction.
The process begins by setting up the equation derived from the problem statement: \(\frac{29-x}{31-x} = \frac{11}{12}\). This involves:

• Identifying the parts of the equation.
• Setting equal the parts that need to be compared or adjusted.
• Employing algebraic techniques to isolate the unknown value.

Understanding how to handle equations is crucial in algebra. It provides a structured approach to solving problems by simplifying complex expressions and isolating variables.

The Power of Cross-Multiplication

Cross-multiplication is a powerful tool in algebra that eliminates fractions from equations, making them easier to solve. It involves multiplying diagonally across an equation where two fractions are set equal.
For the equation \(\frac{29-x}{31-x} = \frac{11}{12}\), cross-multiplying gives us:

• \((29-x) \times 12 = 11 \times (31-x)\)

This results in the simplified form \(348 - 12x = 341 - 11x\), allowing us to continue solving for \(x\) without the complexities of fractions.
Cross-multiplication is especially effective because:

• It transforms a problem with fractions into a simpler linear equation.
• Reduces the equation to basic arithmetic, facilitating easier manipulation.
• Provides a clear pathway towards solving an equation quickly.

By utilizing cross-multiplication, you can tackle problems that might initially seem intimidating, turning them into straightforward exercises in algebra.