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### 2. Unit Rate for Canned Juice
The unit rate represents the number of cases of canned juice packaged per minute.
To find the unit rate for canned juice in cases per minute:
1. You start with an assumed packaging rate for canned juice, which is 1500 cans per hour.
2. Convert this hourly rate to a case rate, assuming 1 case contains 24 cans:
- [tex]\( \text{cases per hour} = \frac{1500 \, \text{cans per hour}}{24 \, \text{cans per case}} \)[/tex]
- [tex]\( \text{cases per hour} = 62.5 \, \text{cases per hour} \)[/tex]
3. Convert this hourly case rate to a minute rate by dividing by 60 (as there are 60 minutes in an hour):
- [tex]\( \text{cases per minute} = \frac{62.5 \, \text{cases per hour}}{60 \, \text{minutes per hour}} \)[/tex]
- [tex]\( \text{cases per minute} = 1.0416666666666667 \, \text{cases per minute} \)[/tex]
So, the unit rate for canned juice is approximately 1.0417 cases per minute.
### 3. Packaging Rate Comparison: Cans vs. Bottles
To determine whether cans or bottles are packaged at a faster rate, we compare their unit rates of packaging in cases per minute.
Cans:
- As found, the unit rate is approximately 1.0417 cases per minute.
Bottles:
- The assumed packaging rate for bottled juice is 1200 bottles per hour.
- Convert this hourly rate to a case rate, assuming 1 case contains 24 bottles:
- [tex]\( \text{cases per hour} = \frac{1200 \, \text{bottles per hour}}{24 \, \text{bottles per case}} \)[/tex]
- [tex]\( \text{cases per hour} = 50 \, \text{cases per hour} \)[/tex]
- Convert this hourly case rate to a minute rate:
- [tex]\( \text{cases per minute} = \frac{50 \, \text{cases per hour}}{60 \, \text{minutes per hour}} \)[/tex]
- [tex]\( \text{cases per minute} = 0.8333333333333334 \, \text{cases per minute} \)[/tex]
So, the unit rate for bottled juice is approximately 0.8333 cases per minute.
Comparison:
- Canned juice: 1.0417 cases per minute
- Bottled juice: 0.8333 cases per minute
Since 1.0417 cases per minute (canned juice) is greater than 0.8333 cases per minute (bottled juice), canned juice is packaged at a faster rate.
Therefore, cans are packaged more quickly than bottles.
### 2. Unit Rate for Canned Juice
The unit rate represents the number of cases of canned juice packaged per minute.
To find the unit rate for canned juice in cases per minute:
1. You start with an assumed packaging rate for canned juice, which is 1500 cans per hour.
2. Convert this hourly rate to a case rate, assuming 1 case contains 24 cans:
- [tex]\( \text{cases per hour} = \frac{1500 \, \text{cans per hour}}{24 \, \text{cans per case}} \)[/tex]
- [tex]\( \text{cases per hour} = 62.5 \, \text{cases per hour} \)[/tex]
3. Convert this hourly case rate to a minute rate by dividing by 60 (as there are 60 minutes in an hour):
- [tex]\( \text{cases per minute} = \frac{62.5 \, \text{cases per hour}}{60 \, \text{minutes per hour}} \)[/tex]
- [tex]\( \text{cases per minute} = 1.0416666666666667 \, \text{cases per minute} \)[/tex]
So, the unit rate for canned juice is approximately 1.0417 cases per minute.
### 3. Packaging Rate Comparison: Cans vs. Bottles
To determine whether cans or bottles are packaged at a faster rate, we compare their unit rates of packaging in cases per minute.
Cans:
- As found, the unit rate is approximately 1.0417 cases per minute.
Bottles:
- The assumed packaging rate for bottled juice is 1200 bottles per hour.
- Convert this hourly rate to a case rate, assuming 1 case contains 24 bottles:
- [tex]\( \text{cases per hour} = \frac{1200 \, \text{bottles per hour}}{24 \, \text{bottles per case}} \)[/tex]
- [tex]\( \text{cases per hour} = 50 \, \text{cases per hour} \)[/tex]
- Convert this hourly case rate to a minute rate:
- [tex]\( \text{cases per minute} = \frac{50 \, \text{cases per hour}}{60 \, \text{minutes per hour}} \)[/tex]
- [tex]\( \text{cases per minute} = 0.8333333333333334 \, \text{cases per minute} \)[/tex]
So, the unit rate for bottled juice is approximately 0.8333 cases per minute.
Comparison:
- Canned juice: 1.0417 cases per minute
- Bottled juice: 0.8333 cases per minute
Since 1.0417 cases per minute (canned juice) is greater than 0.8333 cases per minute (bottled juice), canned juice is packaged at a faster rate.
Therefore, cans are packaged more quickly than bottles.
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