Question

If \(\frac{c-a}{c-b}=2 \quad, \quad \frac{5 b-5 a}{c-a}=\) A. $$0.5$$ B. $$1$$ C. $$1.5$$ D. $$2$$ E. $$2.5$$

Short answer

The value of \(\frac{5 b-5 a}{c-a}\) is 2.5.

Step-by-step solution

- Understand the given equation

The equation provided is \(\frac{c-a}{c-b}=2\). Let's isolate \(c-a\) to express it in terms of \(c-b\).

- Solve for \(c-a\)

From the given equation, \(\frac{c-a}{c-b}=2\), multiply both sides by \(c-b\) to get: \(c-a = 2(c-b)\).

- Simplify the equation

Distribute the 2 on the right side of the equation: \(c-a = 2c - 2b\). Next, isolate \(c-a\) by subtracting \(c\) from both sides: \(-a = c - 2b\). Finally, add \(a\) to both sides to express \(c\): \(c = 2b - a\).

- Substitute \(c\) in the second expression

Given the second fraction \(\frac{5b - 5a}{c - a}\), substitute \(c = 2b - a\) into the denominator: \(\frac{5b - 5a}{(2b - a) - a}\).

- Simplify the denominator

Simplify the expression in the denominator: \(\frac{5b - 5a}{2b - a - a} = \frac{5b - 5a}{2b - 2a}\).

- Factor and simplify the fraction

Factor out the common terms in the numerator and the denominator: \(\frac{5(b - a)}{2(b - a)}\). The \(b - a\) terms cancel out, leaving: \(\frac{5}{2}\).

Key concepts

Algebraic Manipulation

Algebraic manipulation involves performing operations that alter the form of an equation, without changing its value. The goal is often to isolate a specific variable or to simplify the expression.
In this exercise, we started with the equation \(\frac{c-a}{c-b}=2\). The first step was to isolate \(c-a\) by multiplying both sides of the equation by \(c-b\). This gives us \(c-a = 2(c-b)\).

Next, we distributed 2, resulting in \(c-a = 2c - 2b\). To further simplify, we rearranged terms and solved for \(c\). The steps are straightforward: subtract \(c\) from both sides and add \(a\) to both sides, giving us: \(c = 2b - a\). This step-by-step manipulation helps in substituting back into the other parts of the problem.

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. It often involves factoring out common terms.
In this problem, after substituting \(c = 2b - a\), we simplified the fraction \(\frac{5b - 5a}{(2b - a) - a}\). First, we simplified the denominator to \(2b - 2a\).

Then we factored out the common terms in both the numerator and the denominator, seeing that \((b - a)\) is present in both the numerator and the denominator. This allowed us to cancel out these common terms, simplifying the fraction to \(\frac{5}{2}\).
Fraction simplification is a valuable skill, especially for exams like the GMAT, as it can drastically simplify complex problems.

GMAT Exam Preparation

Preparing for the GMAT exam requires a solid understanding of various mathematical concepts, including algebra and fractions.
Solving problems like the one presented here helps build crucial problem-solving skills. The key to effective GMAT preparation is practice and understanding the underlying principles of the problems.

• Practice regularly with a variety of problems.
• Understand the reasoning behind each step.
• Learn to recognize patterns and common problem types.

By consistently working on these skills, you'll enhance your ability to tackle GMAT math problems efficiently and accurately.