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If cacb=2,5b5aca= A. 0.5 B. 1 C. 1.5 D. 2 E. 2.5

Short Answer

Expert verified
The value of 5b5aca is 2.5.

Step by step solution

01

- Understand the given equation

The equation provided is cacb=2. Let's isolate ca to express it in terms of cb.
02

- Solve for ca

From the given equation, cacb=2, multiply both sides by cb to get: ca=2(cb).
03

- Simplify the equation

Distribute the 2 on the right side of the equation: ca=2c2b. Next, isolate ca by subtracting c from both sides: a=c2b. Finally, add a to both sides to express c: c=2ba.
04

- Substitute c in the second expression

Given the second fraction 5b5aca, substitute c=2ba into the denominator: 5b5a(2ba)a.
05

- Simplify the denominator

Simplify the expression in the denominator: 5b5a2baa=5b5a2b2a.
06

- Factor and simplify the fraction

Factor out the common terms in the numerator and the denominator: 5(ba)2(ba). The ba terms cancel out, leaving: 52.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 cacb=2. The first step was to isolate ca by multiplying both sides of the equation by cb. This gives us ca=2(cb).

Next, we distributed 2, resulting in ca=2c2b. 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=2ba. 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=2ba, we simplified the fraction 5b5a(2ba)a. First, we simplified the denominator to 2b2a.

Then we factored out the common terms in both the numerator and the denominator, seeing that (ba) is present in both the numerator and the denominator. This allowed us to cancel out these common terms, simplifying the fraction to 52.
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.

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