R is elder to Q?

Given:

(i) The present age of P is half of the age of R twelve years hence.

(ii) Age of P six years hence is equal to the sum of $\frac{2}{3}$ of the present age of R and $\frac{1}{5}$ of the present age of Q.

(iii) P is 12 years elder to Q’s daughter and Q is 18 years elder to his daughter.

Which of the following statements is correct?

Question

R is elder to Q?

Given:

(i) The present age of P is half of the age of R twelve years hence.

(ii) Age of P six years hence is equal to the sum of $\frac{2}{3}$ of the present age of R and $\frac{1}{5}$ of the present age of Q.

(iii) P is 12 years elder to Q’s daughter and Q is 18 years elder to his daughter.

Which of the following statements is correct?

Accepted Answer

**Solution:**

Let the present ages of P, Q, and R be $p$, $q$, and $r$ respectively.

1. From statement (i):
   $$ 2p = r + 12 $$

2. From statement (ii):
   Age of P six years hence is $p + 6$.
   Given,
   $$ p + 6 = \frac{2}{3}r + \frac{1}{5}q $$

3. From statement (iii):
   Let the age of Q's daughter be $a$.
   Then,
   $$ p = a + 12 $$
   $$ q = a + 18 $$

From these,
   $$ q - p = (a + 18) - (a + 12) = 6 $$
   So,
   $$ q = p + 6 $$

4. Using $q = p + 6$ in equation from (ii):
   $$ p + 6 = \frac{2}{3}r + \frac{1}{5}q = \frac{2}{3}r + \frac{1}{5}(p + 6) $$

Rearranging,
   $$ p + 6 - \frac{1}{5}p - \frac{6}{5} = \frac{2}{3}r $$
   $$ \left(1 - \frac{1}{5}ight)p + \left(6 - \frac{6}{5}ight) = \frac{2}{3}r $$
   $$ \frac{4}{5}p + \frac{24}{5} = \frac{2}{3}r $$

5. From (i), express $r$ in terms of $p$:
   $$ r = 2p - 12 $$

6. Substitute $r$ into the above equation:
   $$ \frac{4}{5}p + \frac{24}{5} = \frac{2}{3}(2p - 12) $$
   $$ \frac{4}{5}p + \frac{24}{5} = \frac{4}{3}p - 8 $$

7. Multiply both sides by 15 to clear denominators:
   $$ 12p + 72 = 20p - 120 $$

8. Rearranging:
   $$ 72 + 120 = 20p - 12p $$
   $$ 192 = 8p $$
   $$ p = 24 $$

9. Find $r$:
   $$ r = 2p - 12 = 2 	imes 24 - 12 = 48 - 12 = 36 $$

10. Find $q$:
    $$ q = p + 6 = 24 + 6 = 30 $$

Since $r = 36$ and $q = 30$, clearly $r > q$, so R is elder than Q.

**Conclusion:**

- Using only (ii) and (iii) together, we can conclude that R is elder than Q.
- Statements (i) and (iii) alone or (iii) alone are insufficient.

**Answer:** Only (ii) & (iii) together

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