The ratio of the present age of P to Q six years ago was $8:7$. The present age of R is twice the present age of Q. If six years from now the ratio of the age of P to R is $11:23$, then find the present age of R (in years).

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Accepted Answer

**Solution:**

1. Let the present ages of P, Q, and R be $P$, $Q$, and $R$ respectively.

2. Given that the ratio of the present age of P to Q six years ago was $8:7$.
   - So, $(P - 6) : (Q - 6) = 8 : 7$
   - This can be written as:
     $$\frac{P - 6}{Q - 6} = \frac{8}{7}$$

3. Cross-multiply to get:
   $$7(P - 6) = 8(Q - 6)$$
   $$7P - 42 = 8Q - 48$$
   $$7P - 8Q = -6 \quad \quad (1)$$

4. Given that the present age of R is twice the present age of Q:
   $$R = 2Q \quad \quad (2)$$

5. Six years hence, the ratio of the age of P to R is $11:23$:
   $$\frac{P + 6}{R + 6} = \frac{11}{23}$$

6. Cross-multiplied:
   $$23(P + 6) = 11(R + 6)$$
   $$23P + 138 = 11R + 66$$
   $$23P - 11R = -72 \quad \quad (3)$$

7. Substitute $R = 2Q$ from equation (2) into equation (3):
   $$23P - 11(2Q) = -72$$
   $$23P - 22Q = -72 \quad \quad (4)$$

8. Now solve the system of equations (1) and (4):
   $$7P - 8Q = -6$$
   $$23P - 22Q = -72$$

9. Multiply equation (1) by 11 and equation (4) by 4 to align coefficients of $Q$:
   $$77P - 88Q = -66$$
   $$92P - 88Q = -288$$

10. Subtract the first from the second:
    $$(92P - 88Q) - (77P - 88Q) = -288 - (-66)$$
    $$15P = -222$$
    $$P = \frac{-222}{15} = -14.8$$

This negative value indicates a miscalculation. Let's re-check step 9.

11. Recalculate step 9:
    - Multiply equation (1) by 11:
      $$7P 	imes 11 = 77P$$
      $$-8Q 	imes 11 = -88Q$$
      $$-6 	imes 11 = -66$$

- Multiply equation (4) by 8:
      $$23P 	imes 8 = 184P$$
      $$-22Q 	imes 8 = -176Q$$
      $$-72 	imes 8 = -576$$

12. Subtract equation (1) multiplied by 11 from equation (4) multiplied by 8:
    $$184P - 176Q = -576$$
    $$77P - 88Q = -66$$

Multiply the second equation by 2:
    $$154P - 176Q = -132$$

Now subtract:
    $$(184P - 176Q) - (154P - 176Q) = -576 - (-132)$$
    $$30P = -444$$
    $$P = \frac{-444}{30} = -14.8$$

Again negative. This suggests an error in the initial setup.

13. Re-examining step 2:
    The ratio given is "The ratio of the present age of P to Q six years ago was 8:7".
    This means:
    $$\frac{P}{Q - 6} = \frac{8}{7}$$
    or
    $$\frac{P - 6}{Q - 6} = \frac{8}{7}$$?

The question states "The ratio of the present age of P to Q six years ago was 8:7".
    This means P's present age to Q's age six years ago is 8:7:
    $$\frac{P}{Q - 6} = \frac{8}{7}$$

14. Using this corrected ratio:
    $$7P = 8(Q - 6)$$
    $$7P = 8Q - 48$$
    $$7P - 8Q = -48 \quad \quad (1')$$

15. Now use equation (4) again:
    $$23P - 22Q = -72 \quad \quad (4)$$

16. Multiply (1') by 11:
    $$77P - 88Q = -528$$

Multiply (4) by 4:
    $$92P - 88Q = -288$$

17. Subtract the first from the second:
    $$(92P - 88Q) - (77P - 88Q) = -288 - (-528)$$
    $$15P = 240$$
    $$P = \frac{240}{15} = 16$$

18. Substitute $P = 16$ into (1'):
    $$7(16) - 8Q = -48$$
    $$112 - 8Q = -48$$
    $$-8Q = -160$$
    $$Q = 20$$

19. From (2), $R = 2Q = 2 	imes 20 = 40$

**Answer:** The present age of R is $40$ years.

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