The time by my watch is 10 minutes to 7, find the angle between the hour hand and the minute hand.

Options

A. `110^@`
B. `120^@`
C. `105^@`
D. `95^@`

Answer

To find the angle between the hour hand and the minute hand when the time is 10 minutes to 7 (which is 6:50), we can follow these steps:

### Step 1: Understand the time
The time given is 10 minutes to 7, which translates to 6:50.

### Step 2: Calculate the position of the hour hand
The hour hand moves at a rate of 30 degrees per hour (360 degrees/12 hours). Additionally, it moves 0.5 degrees per minute (30 degrees/60 minutes).

- **Position of the hour hand at 6:50:**
\[ \text{Position of hour hand} = (6 \text{ hours} \times 30 \text{ degrees/hour}) + (50 \text{ minutes} \times 0.5 \text{ degrees/minute}) \]
\[ = 180 \text{ degrees} + 25 \text{ degrees} = 205 \text{ degrees} \]

### Step 3: Calculate the position of the minute hand
The minute hand moves at a rate of 6 degrees per minute (360 degrees/60 minutes).

- **Position of the minute hand at 6:50:**
\[ \text{Position of minute hand} = 50 \text{ minutes} \times 6 \text{ degrees/minute} = 300 \text{ degrees} \]

### Step 4: Calculate the angle between the hour hand and the minute hand
To find the angle between the two hands, we take the absolute difference between their positions.

- **Angle between the hands:**
\[ \text{Angle} = |300 \text{ degrees} - 205 \text{ degrees}| = 95 \text{ degrees} \]

### Step 5: Determine the smaller angle
Since the angle between the two hands can be measured in two ways (the larger angle and the smaller angle), we need to ensure we have the smaller angle.

- The smaller angle is already calculated as 95 degrees, which is less than 180 degrees.

### Final Answer
The angle between the hour hand and the minute hand at 6:50 is **95 degrees**.

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