If 30% of (A+B) = 50% of (A-B), then what percent of B is A?

Options

A. a. 100%
B. b. 300%
C. c. 450%
D. d. 400%

Answer

To solve the problem step by step, let's start from the equation given in the question.

### Step 1: Write the Equation
We know from the question that:
\[ 30\% \text{ of } (A + B) = 50\% \text{ of } (A - B) \]

### Step 2: Convert Percentages to Fractions
Convert the percentages into fractions:
\[ \frac{30}{100} (A + B) = \frac{50}{100} (A - B) \]
This simplifies to:
\[ \frac{3}{10} (A + B) = \frac{1}{2} (A - B) \]

### Step 3: Eliminate the Fractions
To eliminate the fractions, we can multiply both sides by 20 (the least common multiple of 10 and 2):
\[ 20 \cdot \frac{3}{10} (A + B) = 20 \cdot \frac{1}{2} (A - B) \]
This simplifies to:
\[ 6(A + B) = 10(A - B) \]

### Step 4: Expand Both Sides
Now, expand both sides:
\[ 6A + 6B = 10A - 10B \]

### Step 5: Rearrange the Equation
Rearranging the equation to get all terms involving \(A\) on one side and all terms involving \(B\) on the other side gives us:
\[ 6A - 10A = -10B - 6B \]
This simplifies to:
\[ -4A = -16B \]

### Step 6: Simplify the Equation
Dividing both sides by -4 gives:
\[ A = 4B \]

### Step 7: Find the Percentage of A in terms of B
To find what percent of \(B\) is \(A\), we can express it as:
\[ \frac{A}{B} \times 100\% \]
Substituting \(A = 4B\):
\[ \frac{4B}{B} \times 100\% = 4 \times 100\% = 400\% \]

### Final Answer
Thus, \(A\) is **400%** of \(B\).

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