The ratio of the average weight of A and B to the average weight of A and C is $4:5$. Given that the weight of C is $40 ext{ kg}$ and the weights of A and B are in the ratio $7:5$, find the average weight of A, B, and C (in kg).

Question

The ratio of the average weight of A and B to the average weight of A and C is $4:5$. Given that the weight of C is $40 	ext{ kg}$ and the weights of A and B are in the ratio $7:5$, find the average weight of A, B, and C (in kg).

Accepted Answer

**Solution:**

1. Let the weights of A and B be $7x$ and $5x$ respectively, since their ratio is $7:5$.

2. The average weight of A and B is:
   $$\frac{7x + 5x}{2} = \frac{12x}{2} = 6x$$

3. The average weight of A and C is:
   $$\frac{7x + 40}{2}$$

4. According to the problem, the ratio of these averages is $4:5$, so:
   $$\frac{6x}{\frac{7x + 40}{2}} = \frac{4}{5}$$

5. Simplify the equation:
   $$\frac{6x}{\frac{7x + 40}{2}} = \frac{4}{5} \implies \frac{6x 	imes 2}{7x + 40} = \frac{4}{5}$$
   $$\frac{12x}{7x + 40} = \frac{4}{5}$$

6. Cross-multiply:
   $$5 	imes 12x = 4 	imes (7x + 40)$$
   $$60x = 28x + 160$$

7. Bring like terms together:
   $$60x - 28x = 160$$
   $$32x = 160$$

8. Solve for $x$:
   $$x = \frac{160}{32} = 5$$

9. Find the weights:
   - $A = 7x = 7 	imes 5 = 35 	ext{ kg}$
   - $B = 5x = 5 	imes 5 = 25 	ext{ kg}$
   - $C = 40 	ext{ kg}$ (given)

10. Calculate the average weight of A, B, and C:
    $$\frac{35 + 25 + 40}{3} = \frac{100}{3} = 33.33 	ext{ kg}$$

**Answer:** $33.33$

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