If 5% of (P + Q) = 20% of (P - Q), then P is what percentage of Q?

Options

A. 133.33
B. 166.66
C. 150
D. 171.33

Answer

To solve the problem step by step, let's start with the given equation:

**Step 1: Write the given equation.**
We are given that:
\[ 5\% \text{ of } (P + Q) = 20\% \text{ of } (P - Q) \]

**Step 2: Convert percentages to fractions.**
We can express the percentages as fractions:
\[ \frac{5}{100} (P + Q) = \frac{20}{100} (P - Q) \]
This simplifies to:
\[ \frac{1}{20} (P + Q) = \frac{1}{5} (P - Q) \]

**Step 3: Eliminate the fractions by multiplying through by 20.**
To eliminate the fractions, multiply both sides by 20:
\[ P + Q = 4(P - Q) \]

**Step 4: Expand the right side of the equation.**
Expanding the right side gives:
\[ P + Q = 4P - 4Q \]

**Step 5: Rearrange the equation.**
Now, let's rearrange the equation to isolate terms involving \(P\) and \(Q\):
\[ P + Q + 4Q = 4P \]
This simplifies to:
\[ P + 5Q = 4P \]

**Step 6: Move all terms involving \(P\) to one side.**
Rearranging gives us:
\[ 5Q = 4P - P \]
\[ 5Q = 3P \]

**Step 7: Solve for the ratio of \(P\) to \(Q\).**
Now, we can express \(P\) in terms of \(Q\):
\[ P = \frac{5}{3} Q \]

**Step 8: Find \(P\) as a percentage of \(Q\).**
To find \(P\) as a percentage of \(Q\), we use the formula:
\[ \text{Percentage} = \left(\frac{P}{Q}\right) \times 100 \]
Substituting \(P = \frac{5}{3} Q\):
\[ \text{Percentage} = \left(\frac{\frac{5}{3} Q}{Q}\right) \times 100 \]
This simplifies to:
\[ \text{Percentage} = \frac{5}{3} \times 100 = \frac{500}{3} \]

**Step 9: Convert to a mixed number.**
Calculating \(\frac{500}{3}\) gives:
\[ 500 \div 3 = 166.66\overline{6} \]
So, \(P\) is approximately \(166.67\%\) of \(Q\).

**Final Result:**
Thus, \(P\) is \(166.67\%\) of \(Q\).

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