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

Answer: (A)

P = 110; R = 100

Answer 2

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.