Question

Positive integer y is 50 percent of 50 percent of positive integer x, and y percent of x equals 100. What is the value of x?

• A. 50
• B. 100
• C. 200
• D. 1,000
• E. 2,000

Hint:

When translating "percent" into math, remember that "P percent" is equivalent to the decimal \(P/100\). Be systematic in converting each phrase into its mathematical equivalent before trying to solve.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires translating two English statements about percentages into algebraic equations and then solving the system of equations.
Step 2: Key Formula or Approach:
We will set up two equations based on the given information.
1. "y is 50 percent of 50 percent of x" translates to: \[ y = 0.50 \times (0.50 \times x) \] 2. "y percent of x equals 100" translates to: \[ \frac{y}{100} \times x = 100 \] Step 3: Detailed Explanation:
First, simplify the first equation: \[ y = 0.25x \] Now, simplify the second equation: \[ yx = 100 \times 100 \] \[ yx = 10000 \] We have a system of two equations: (1) \(y = 0.25x\) (2) \(yx = 10000\) Substitute the expression for y from equation (1) into equation (2): \[ (0.25x)x = 10000 \] \[ 0.25x^2 = 10000 \] To solve for \(x^2\), divide both sides by 0.25 (which is the same as multiplying by 4): \[ x^2 = \frac{10000}{0.25} = 10000 \times 4 = 40000 \] Now, take the square root of both sides. Since x is a positive integer: \[ x = \sqrt{40000} = 200 \] Step 4: Final Answer:
The value of x is 200. We can check this: if \(x=200\), then \(y = 0.25 \times 200 = 50\). Then, y percent of x is 50% of 200, which is \(0.50 \times 200 = 100\). This matches the given information.