Question

If \(2x = 5\) and \(3y = 8\), then \( \frac{4x}{9y} = \)

• A. \( \frac{5}{18} \)
• B. \( \frac{5}{16} \)
• C. \( \frac{5}{12} \)
• D. \( \frac{5}{8} \)
• E. \( \frac{5}{4} \)

Hint:

The second method (manipulating the expression) is often faster and less prone to errors with complex fractions. Look for ways to form the given expressions (like \(2x\) and \(3y\)) within the expression you need to evaluate.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is an algebra problem where we need to evaluate an expression involving variables \(x\) and \(y\), given two equations for those variables.
Step 2: Key Formula or Approach:
There are two common methods: 1. Solve for \(x\) and \(y\) individually from the given equations and substitute their values into the expression. 2. Manipulate the expression to be evaluated so that it contains the terms \(2x\) and \(3y\), and then substitute their given values directly.
Step 3: Detailed Explanation:
Method 1: Solve for x and y first From \(2x = 5\), we can solve for \(x\): \[ x = \frac{5}{2} \] From \(3y = 8\), we can solve for \(y\): \[ y = \frac{8}{3} \] Now substitute these values into the expression \( \frac{4x}{9y} \): \[ \frac{4x}{9y} = \frac{4(\frac{5}{2})}{9(\frac{8}{3})} \] Simplify the numerator and the denominator: Numerator: \( 4 \times \frac{5}{2} = \frac{20}{2} = 10 \) Denominator: \( 9 \times \frac{8}{3} = \frac{72}{3} = 24 \) So the expression is: \[ \frac{10}{24} \] Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2: \[ \frac{10 \div 2}{24 \div 2} = \frac{5}{12} \] Method 2: Manipulate the expression The expression is \( \frac{4x}{9y} \). We can rewrite the numerator and denominator to use the given terms \(2x\) and \(3y\). Numerator: \( 4x = 2 \times (2x) \) Denominator: \( 9y = 3 \times (3y) \) So, the expression becomes: \[ \frac{4x}{9y} = \frac{2 \times (2x)}{3 \times (3y)} \] Now, substitute the given values \(2x=5\) and \(3y=8\): \[ \frac{2 \times (5)}{3 \times (8)} = \frac{10}{24} \] Simplify the fraction: \[ \frac{10}{24} = \frac{5}{12} \] Step 4: Final Answer:
Both methods yield the result \( \frac{5}{12} \).