Question

p\(\%\) of a number P is q\(\%\) more than r\(\%\) of the number R. If the difference between P and R is r\(\%\) of R and if the sum of P and R is 210, then which of the following statements is always true?

• A. P = 110; R = 100
• B. P = 220; R = 200
• C. P = 3300; R = 3000
• D. All of the above

Answer 1

The Correct Option is A

The correct option is (A): P = 110; R = 100
Let's break down the information given in the problem step by step to find the correct relationship between \( P \) and \( R \).

Given Information:
1. \( p\% \) of \( P \) is \( q\% \) more than \( r\% \) of \( R \).
2. The difference between \( P \) and \( R \) is \( r\% \) of \( R \).
3. The sum of \( P \) and \( R \) is 210.

 Equations Derived:
1. From the first point:
  \[\frac{p}{100} P = \frac{r}{100} R + \frac{q}{100} \left( \frac{r}{100} R \right)\]
  This simplifies to:
  \[\frac{p}{100} P = \frac{(r + qr/100)}{100} R\]
  or:
  \[pP = (r + \frac{qr}{100}) R\]

2. From the second point:
  \[P - R = \frac{r}{100} R\]
  Rearranging gives:
  \[P = R + \frac{r}{100} R = R \left(1 + \frac{r}{100}\right)\]

3. From the third point:
  \[P + R = 210\]

Solving the System of Equations:
Now we have:
1. \( P = R \left(1 + \frac{r}{100}\right) \)
2. \( P + R = 210 \)

Substituting the first equation into the second gives:
\[R \left(1 + \frac{r}{100}\right) + R = 210\]
This simplifies to:
\[R \left(2 + \frac{r}{100}\right) = 210\]
Thus:
\[R = \frac{210}{2 + \frac{r}{100}}\]

Now substituting \( R \) back into the equation for \( P \):
\[P = \frac{210}{2 + \frac{r}{100}} \left(1 + \frac{r}{100}\right)\]

Exploring Possible Statements:
Now, let’s test the provided options:

1. \( P = 110; R = 100 \):
  \( P + R = 110 + 100 = 210 \) (Correct)
   \( P - R = 110 - 100 = 10 \) which equals \( \frac{r}{100} R \) (Check required)

2. \( P = 220; R = 200 \):
   \( P + R = 220 + 200 = 420 \) (Incorrect)

3. \( P = 3300; R = 3000 \):
  \( P + R = 3300 + 3000 = 6300 \) (Incorrect)

4. All of the above (not applicable due to earlier checks).

 Conclusion:
From the valid checks, the only true option is \( P = 110 \) and \( R = 100 \). Therefore, the correct answer is:

P = 110; R = 100.