Which of the following given ratio of units of length gives the highest value?

Options

A. `frac{1Pm}{1fm}`
B. `frac{1hm}{1Gm}`
C. `frac{1dm}{1um}`
D. `frac{1Mm}{1nm}`

Answer

To determine which of the given ratios of units of length yields the highest value, we need to convert each unit to meters and then calculate the ratios. Let's go through the options step by step.

### Step 1: Convert Units to Meters

1. **Pico meter (pm)**:
- 1 pm = \(10^{-12}\) meters
2. **Fermi (fm)**:
- 1 fm = \(10^{-15}\) meters

**Ratio**:
\[ \frac{1 \text{ pm}}{1 \text{ fm}} = \frac{10^{-12}}{10^{-15}} = 10^{3} \]

### Step 2: Hectometer and Gigameter

1. **Hectometer (hm)**:
- 1 hm = \(10^{2}\) meters
2. **Gigameter (Gm)**:
- 1 Gm = \(10^{9}\) meters

**Ratio**:
\[ \frac{1 \text{ hm}}{1 \text{ Gm}} = \frac{10^{2}}{10^{9}} = 10^{-7} \]

### Step 3: Dekameter and Micrometer

1. **Dekameter (dam)**:
- 1 dam = \(10\) meters
2. **Micrometer (μm)**:
- 1 μm = \(10^{-6}\) meters

**Ratio**:
\[ \frac{1 \text{ dam}}{1 \text{ μm}} = \frac{10}{10^{-6}} = 10^{7} \]

### Step 4: Megameter and Nanometer

1. **Megameter (Mm)**:
- 1 Mm = \(10^{6}\) meters
2. **Nanometer (nm)**:
- 1 nm = \(10^{-9}\) meters

**Ratio**:
\[ \frac{1 \text{ Mm}}{1 \text{ nm}} = \frac{10^{6}}{10^{-9}} = 10^{15} \]

### Step 5: Compare All Ratios

Now we compare the results from each step:
- Ratio of pm to fm: \(10^{3}\)
- Ratio of hm to Gm: \(10^{-7}\)
- Ratio of dam to μm: \(10^{7}\)
- Ratio of Mm to nm: \(10^{15}\)

### Conclusion

The highest value among these ratios is \(10^{15}\) from the ratio of Megameter to Nanometer.

**Final Answer**: The fourth option (Megameter to Nanometer) gives the highest value.

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