The ratio of A's present age to B's present age is $3:4$, and the ratio of C's present age to B's present age is $5:8$. If the difference between the difference of C's and B's ages and the difference of B's and A's ages is 9 years, what was the average of their ages twelve years ago?

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**Solution:**

1. Let the present ages of A, B, and C be $3x$, $4x$, and $y$ respectively, based on the given ratios.

2. Given the ratio of C's age to B's age is $5:8$, so:
   $$\frac{y}{4x} = \frac{5}{8} \implies y = \frac{5}{8} 	imes 4x = \frac{5}{2}x$$

3. The difference between the difference of C's and B's ages and the difference of B's and A's ages is 9 years:
   $$ (y - 4x) - (4x - 3x) = 9 $$

4. Substitute $y = \frac{5}{2}x$:
   $$ \left( \frac{5}{2}x - 4x ight) - (4x - 3x) = 9 $$
   $$ \left( \frac{5}{2}x - 4x ight) - x = 9 $$

5. Simplify inside the parentheses:
   $$ \left( \frac{5}{2}x - \frac{8}{2}x ight) - x = 9 $$
   $$ \left( -\frac{3}{2}x ight) - x = 9 $$

6. Combine like terms:
   $$ -\frac{3}{2}x - x = 9 $$
   $$ -\frac{3}{2}x - \frac{2}{2}x = 9 $$
   $$ -\frac{5}{2}x = 9 $$

7. Multiply both sides by $-\frac{2}{5}$:
   $$ x = 9 	imes \left(-\frac{2}{5}ight) = -\frac{18}{5} $$

Since age cannot be negative, re-examine the sign in step 3:

Step 3 should be:
   $$ (y - 4x) - (4x - 3x) = 9 $$
   $$ (y - 4x) - (4x - 3x) = 9 $$
   $$ (y - 4x) - (4x - 3x) = 9 $$

Calculate the differences:
   $$ (y - 4x) - (4x - 3x) = (y - 4x) - (x) = 9 $$

Substitute $y = \frac{5}{2}x$:
   $$ \frac{5}{2}x - 4x - x = 9 $$
   $$ \frac{5}{2}x - 5x = 9 $$
   $$ \frac{5}{2}x - \frac{10}{2}x = 9 $$
   $$ -\frac{5}{2}x = 9 $$

Multiply both sides by $-\frac{2}{5}$:
   $$ x = 9 	imes \left(-\frac{2}{5}ight) = -\frac{18}{5} $$

Negative again, so check the order of differences carefully.

Try reversing the order:
   $$ (4x - y) - (4x - 3x) = 9 $$
   $$ (4x - y) - x = 9 $$

Substitute $y = \frac{5}{2}x$:
   $$ 4x - \frac{5}{2}x - x = 9 $$
   $$ 4x - \frac{5}{2}x - x = 9 $$
   $$ \left(4x - xight) - \frac{5}{2}x = 9 $$
   $$ 3x - \frac{5}{2}x = 9 $$
   $$ \frac{6}{2}x - \frac{5}{2}x = 9 $$
   $$ \frac{1}{2}x = 9 $$

Multiply both sides by 2:
   $$ x = 18 $$

8. Calculate present ages:
   - A's age = $3x = 3 	imes 18 = 54$ years
   - B's age = $4x = 4 	imes 18 = 72$ years
   - C's age = $\frac{5}{2}x = \frac{5}{2} 	imes 18 = 45$ years

9. Find their ages 12 years ago:
   - A: $54 - 12 = 42$ years
   - B: $72 - 12 = 60$ years
   - C: $45 - 12 = 33$ years

10. Calculate the average age 12 years ago:
    $$ \frac{42 + 60 + 33}{3} = \frac{135}{3} = 45 $$ years

**Answer:** 45 years

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