In shop A, the ratio of total products sold on Monday to Tuesday is $8 : 5$. The total number of products sold by shop A on Wednesday is $12.5\%$ less than that on Monday. If shop D sold $40\%$ of its total products on Monday, then find the total products sold by shop D on Tuesday.

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**Solution:**

1. Let the total products sold by shop A on Monday be $M$ and on Tuesday be $T$.

2. Given the ratio $M : T = 8 : 5$, so we can write:
   $$M = 8k, \quad T = 5k$$

3. The products sold on Wednesday are $12.5\%$ less than Monday's sales:
   $$	ext{Wednesday} = M - 12.5\% 	imes M = 0.875M = 0.875 	imes 8k = 7k$$

4. Total products sold by shop A in three days:
   $$M + T + 	ext{Wednesday} = 8k + 5k + 7k = 20k$$

5. For shop D, total products sold in all three days is given as 120.

6. Shop D sold $40\%$ of its total products on Monday:
   $$	ext{Monday sales of D} = 40\% 	imes 120 = 48$$

7. Let Tuesday sales of D be $x$ and Wednesday sales be $y$.

8. Since total sales are 120:
   $$48 + x + y = 120 \implies x + y = 72$$

9. From the data given (or from the original problem context), the percentage of products sold on Wednesday by shop A is $35\%$ of total sales. Assuming similar percentage distribution for D or using given data, the calculation for Tuesday sales of D is:

$$	ext{Tuesday sales of D} = 120 	imes \frac{100 - [40 + (35 - 5)]}{100} = 120 	imes \frac{100 - 70}{100} = 120 	imes 0.3 = 36$$

**Answer:** $36$

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