If A : B = 1 : 2, B : C = 3 : 4 and C : D = 5 : 6, then find the value of D : C : B.

Answer

To solve the problem step by step, we will first express the ratios in terms of a common variable and then find the values of A, B, C, and D accordingly.

### Step-by-Step Solution:

1. **Given Ratios:**
- \( A : B = 1 : 2 \) implies \( \frac{A}{B} = \frac{1}{2} \) or \( A = \frac{1}{2}B \)
- \( B : C = 3 : 4 \) implies \( \frac{B}{C} = \frac{3}{4} \) or \( B = \frac{3}{4}C \)
- \( C : D = 5 : 6 \) implies \( \frac{C}{D} = \frac{5}{6} \) or \( C = \frac{5}{6}D \)

2. **Express B in terms of C:**
From \( B = \frac{3}{4}C \), we can express C in terms of B:
\[ C = \frac{4}{3}B \]

3. **Express D in terms of C:**
From \( C = \frac{5}{6}D \), we can express D in terms of C:
\[ D = \frac{6}{5}C \]

4. **Substituting C in terms of B into D:**
Now substitute \( C = \frac{4}{3}B \) into \( D = \frac{6}{5}C \):
\[ D = \frac{6}{5} \left(\frac{4}{3}B\right) = \frac{24}{15}B = \frac{8}{5}B \]

5. **Now we have expressions for A, B, C, and D in terms of B:**
- \( A = \frac{1}{2}B \)
- \( B = B \)
- \( C = \frac{4}{3}B \)
- \( D = \frac{8}{5}B \)

6. **Finding D : C : B:**
To find the ratio \( D : C : B \), we write:
\[ D : C : B = \frac{8}{5}B : \frac{4}{3}B : B \]
We can simplify this by eliminating B:
\[ D : C : B = \frac{8}{5} : \frac{4}{3} : 1 \]

7. **Finding a common denominator:**
The common denominator for \( \frac{8}{5} \), \( \frac{4}{3} \), and \( 1 \) is 15. We will convert each term:
\[ D = \frac{8}{5} = \frac{8 \times 3}{5 \times 3} = \frac{24}{15} \]
\[ C = \frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15} \]
\[ B = 1 = \frac{1 \times 15}{1 \times 15} = \frac{15}{15} \]

8. **Final Ratio:**
Thus, we have:
\[ D : C : B = 24 : 20 : 15 \]

### Final Answer:
The value of \( D : C : B \) is \( 24 : 20 : 15 \).