If u : v : w = 3 : 6 : 8, then what will be the ratio of (u/v) : (v/w) : (w/u)?

Options

A. 8 : 6 : 3
B. `8 : 4 : 3`
C. `5 : 9 : 12`
D. `6 :9 : 32`

Answer

To solve the problem, we need to find the ratio of (u/v) : (v/w) : (w/u) given that u : v : w = 3 : 6 : 8.

### Step-by-step Solution:

1. **Express u, v, and w in terms of a common variable:**
Since u : v : w = 3 : 6 : 8, we can let:
- u = 3k
- v = 6k
- w = 8k
where k is a positive constant.

2. **Calculate u/v:**
\[ \frac{u}{v} = \frac{3k}{6k} = \frac{3}{6} = \frac{1}{2} \]

3. **Calculate v/w:**
\[ \frac{v}{w} = \frac{6k}{8k} = \frac{6}{8} = \frac{3}{4} \]

4. **Calculate w/u:**
\[ \frac{w}{u} = \frac{8k}{3k} = \frac{8}{3} \]

5. **Now, we have the ratios:**
- u/v = 1/2
- v/w = 3/4
- w/u = 8/3

6. **Express these ratios with a common denominator:**
The least common multiple (LCM) of the denominators (2, 4, 3) is 12. We will convert each fraction to have a denominator of 12:
- \(\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}\)
- \(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)
- \(\frac{8}{3} = \frac{8 \times 4}{3 \times 4} = \frac{32}{12}\)

7. **Now, we can write the ratios as:**
\[ \frac{u}{v} : \frac{v}{w} : \frac{w}{u} = 6 : 9 : 32 \]

8. **Simplify the ratio:**
The ratio 6 : 9 : 32 can be simplified by dividing each term by 3:
\[ 2 : 3 : \frac{32}{3} \]
However, since 32 does not divide evenly by 3, we keep it as is.

9. **Final Ratio:**
The final ratio of (u/v) : (v/w) : (w/u) is:
\[ 2 : 3 : 32 \]