The ratio of time taken by A and B to covera certain distance is 7 : 9. The ratio of their respective speed will be:

Options

A. `7:9`
B. `7:3`
C. `3:7`
D. `9:7`

Answer

To find the ratio of the speeds of A and B based on the given ratio of their times, we can follow these steps:

### Step 1: Understand the relationship between speed and time
We know that speed is inversely proportional to time when the distance is constant. This means that if one person takes more time to cover the same distance, they will be slower.

### Step 2: Write down the given ratio of time
The ratio of the time taken by A and B is given as:
\[ \text{Time taken by A} : \text{Time taken by B} = 7 : 9 \]

### Step 3: Express the speeds in terms of time
Let the time taken by A be \( 7x \) and the time taken by B be \( 9x \) for some constant \( x \).

### Step 4: Use the formula for speed
Since speed is equal to distance divided by time, we can express the speeds of A and B as follows:
- Speed of A: \( \text{Speed}_A = \frac{D}{7x} \)
- Speed of B: \( \text{Speed}_B = \frac{D}{9x} \)

Where \( D \) is the distance covered.

### Step 5: Find the ratio of speeds
Now, we can find the ratio of their speeds:
\[ \text{Speed}_A : \text{Speed}_B = \frac{D}{7x} : \frac{D}{9x} \]
This simplifies to:
\[ \text{Speed}_A : \text{Speed}_B = \frac{1}{7} : \frac{1}{9} \]

### Step 6: Invert the ratio
To find the ratio of speeds, we invert the ratio of times:
\[ \text{Speed}_A : \text{Speed}_B = 9 : 7 \]

### Final Answer
Thus, the ratio of the speeds of A and B is:
\[ 9 : 7 \]

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