
Shortcut Trick
Simplify ratios by multiplying with the LCM of their denominators: A:B = 4:3, B:C = 3:5, and C:D = 10:9.
Since B is common (value 3), we combine the first two: A:B:C = 4:3:5.
To match C = 10 in the ratio C:D, multiply A:B:C by 2 to get 8:6:10.
Combine all parts: A:B:C:D = 8:6:10:9.
∴ The correct answer is 8 ∶ 6 ∶ 10 ∶ 9.

Alternate Method
Given: A ∶ B = 1/2 ∶ 3/8, B ∶ C = 1/3 ∶ 5/9, C ∶ D = 5/6 ∶ 3/4

Step 1: Simplify individual fractional ratios.
⇒ A ∶ B = (1/2) × 8 ∶ (3/8) × 8 = 4 ∶ 3
⇒ B ∶ C = (1/3) × 9 ∶ (5/9) × 9 = 3 ∶ 5
⇒ C ∶ D = (5/6) × 12 ∶ (3/4) × 12 = 10 ∶ 9
Step 2: Combine ratios using common terms.
⇒ Since B is 3 in both A ∶ B and B ∶ C, A ∶ B ∶ C = 4 ∶ 3 ∶ 5
Step 3: Align C ∶ D with A ∶ B ∶ C.
⇒ C is 5 in A ∶ B ∶ C and 10 in C ∶ D. Multiply A ∶ B ∶ C by 2.
⇒ A ∶ B ∶ C = (4 × 2) ∶ (3 × 2) ∶ (5 × 2) = 8 ∶ 6 ∶ 10
⇒ Final combined ratio A ∶ B ∶ C ∶ D = 8 ∶ 6 ∶ 10 ∶ 9
∴ The correct answer is 8 ∶ 6 ∶ 10 ∶ 9.

Additional Information
Simplifying Fractional Ratios
To simplify a ratio like a/b ∶ c/d, multiply both terms by the LCM of the denominators (b and d) to convert them into integers.
Ratio Equalization
To combine two ratios (A:B and B:C), find the LCM of the common term (B) and multiply each ratio by a factor that makes the value of B identical in both.
Compound Ratio
The compound ratio of (a:b), (c:d), and (e:f) is found by multiplying the antecedents and consequents: (a × c × e) ∶ (b × d × f).