Two cogged wheels of which one has 32 cogs and other 54 cogs, work into each other. If the latter turns 80 times in three quarters of a minute, how often does the other turn in 8 seconds?

Options

A. 48
B. 24
C. 38
D. None of these

Answer

To solve the problem, we need to find out how many times the first cogged wheel (with 32 cogs) turns when the second cogged wheel (with 54 cogs) turns 80 times in three quarters of a minute. We will then calculate how often the first wheel turns in 8 seconds.

### Step 1: Determine the time duration in seconds
Three quarters of a minute is equal to:
\[ \text{Time} = \frac{3}{4} \times 60 \text{ seconds} = 45 \text{ seconds} \]

**Hint:** Remember that 1 minute = 60 seconds, so you can convert minutes to seconds by multiplying by 60.

### Step 2: Calculate the number of turns of the second wheel in 1 second
The second wheel turns 80 times in 45 seconds. To find out how many times it turns in 1 second, we divide the total turns by the total time:
\[ \text{Turns per second} = \frac{80 \text{ turns}}{45 \text{ seconds}} \approx 1.78 \text{ turns/second} \]

**Hint:** To find the rate of something per unit time, divide the total amount by the total time.

### Step 3: Calculate the number of turns of the second wheel in 8 seconds
Now, we can find out how many times the second wheel turns in 8 seconds:
\[ \text{Turns in 8 seconds} = 1.78 \text{ turns/second} \times 8 \text{ seconds} \approx 14.22 \text{ turns} \]

**Hint:** To find the total number of occurrences over a period, multiply the rate by the time period.

### Step 4: Determine the ratio of the cogs
The first wheel has 32 cogs and the second wheel has 54 cogs. The ratio of the number of cogs is:
\[ \text{Ratio} = \frac{32 \text{ cogs}}{54 \text{ cogs}} = \frac{16}{27} \]

**Hint:** The ratio of two quantities can be found by dividing one by the other.

### Step 5: Calculate the number of turns of the first wheel
Since the wheels are interlocked, the number of turns of the first wheel can be calculated using the ratio of the cogs:
\[ \text{Turns of first wheel} = \text{Turns of second wheel} \times \frac{32}{54} \]
Substituting the turns of the second wheel:
\[ \text{Turns of first wheel} = 14.22 \times \frac{32}{54} \approx 8.43 \text{ turns} \]

**Hint:** When two gears are connected, the turns of one can be calculated using the ratio of their cogs.

### Final Answer
The first cogged wheel turns approximately **8.43 times** in 8 seconds.

---