5. dishwasher detergent is sold in individual packs. it is sold in 20-, 60-, and 90-pack containers.
a. which container do you think has the lowest unit rate of dollars per pack? why?
b. the 20 - pack container sells for $5.49. what is the unit rate in dollars per pack? round your answer to the nearest cent.
the 60 - pack container sells for $10.97. what is the unit rate in dollars per pack? round your answer to the nearest cent.
the 90 - pack container sells for $18.95. what is the unit rate in dollars per pack? round your answer to the nearest cent.
which container has the lowest unit rate? how does this compare with your answer in part (a)?

Question

Accepted Answer

### Part a
# Brief Explanations:
Typically, in retail, larger quantity packages (like the 90 - pack here) often have a lower unit rate (dollars per pack) because companies offer volume discounts. This is based on the idea of economies of scale - the cost per unit tends to decrease as the quantity of units increases, as fixed costs (like packaging, production setup) are spread over more units. So we would expect the 90 - pack container to have the lowest unit rate initially, but we will confirm with calculations in later parts.
# Answer:
We would think the 90 - pack container has the lowest unit rate. This is because of the concept of volume discounts (economies of scale), where the cost per unit often decreases as the quantity of units in a package increases, as fixed costs are spread over more units.

### Part b
# Explanation:
## Step 1: Recall the formula for unit rate
The unit rate (dollars per pack) is calculated by dividing the total cost by the number of packs. The formula is $Unit\ Rate=\frac{Total\ Cost}{Number\ of\ Packs}$.
## Step 2: Substitute the values for the 20 - pack
For the 20 - pack container, the total cost is $\$5.49$ and the number of packs is 20. So we calculate $\frac{5.49}{20}$.
## Step 3: Perform the division
$\frac{5.49}{20} = 0.2745$ dollars per pack.
## Step 4: Round to the nearest cent
To round to the nearest cent (hundredth of a dollar), we look at the thousandth place. The number $0.2745$ has a 4 in the thousandth place, which is less than 5, so we round down. So $0.2745\approx0.27$ dollars per pack.
# Answer:
The unit rate for the 20 - pack container is $\$0.27$ per pack.

### Part (for 60 - pack, assuming the question number was missed but following the pattern)
# Explanation:
## Step 1: Use the unit rate formula
Again, $Unit\ Rate=\frac{Total\ Cost}{Number\ of\ Packs}$. Here, the total cost is $\$10.97$ and the number of packs is 60.
## Step 2: Perform the division
Calculate $\frac{10.97}{60}\approx0.1828$ dollars per pack.
## Step 3: Round to the nearest cent
Looking at the thousandth place (2), which is less than 5, we round down. So $0.1828\approx0.18$ dollars per pack.
# Answer:
The unit rate for the 60 - pack container is $\$0.18$ per pack.

### Part (for 90 - pack, assuming the question number was missed but following the pattern)
# Explanation:
## Step 1: Use the unit rate formula
$Unit\ Rate=\frac{Total\ Cost}{Number\ of\ Packs}$. Here, the total cost is $\$18.95$ and the number of packs is 90.
## Step 2: Perform the division
Calculate $\frac{18.95}{90}\approx0.2106$ dollars per pack. Wait, no, wait, let's recalculate: $\frac{18.95}{90}\approx0.2106$? Wait, no, $18.95\div90\approx0.2106$? Wait, no, 90*0.21 = 18.9, so 18.95 - 18.9 = 0.05, so 0.21+0.05/90≈0.21 + 0.00055≈0.2106. Wait, but when we round to the nearest cent, we look at the thousandth place (0), so we round to $0.21$? Wait, no, wait, maybe I made a mistake. Wait, $18.95\div90$: 90*0.21 = 18.9, 18.95 - 18.9 = 0.05, so 0.05/90≈0.00055, so total is≈0.21055, so to the nearest cent (hundredth), we look at the thousandth digit which is 0, so we round down to $0.21$? Wait, no, wait, 0.21055, the hundredth place is 1, the thousandth is 0, so yes, $0.21$ when rounded to the nearest cent? Wait, no, wait, 0.21055 is approximately 0.21 when rounded to the nearest cent? Wait, no, 0.21055: the first decimal is tenths (2), second is hundredths (1), third is thousandths (0). So we round to the hundredth place, so since the thousandth is 0 < 5, we keep the hundredth as 1. So unit rate is≈$\$0.21$ per pack? Wait, no, wait, maybe I miscalculated. Wait, 18.95 divided by 90: let's do this division properly. 90 goes into 18.95 zero times. 90 goes into 189 two times (2*90 = 180), subtract 180 from 189, we get 9. Bring down the 5 to make 95. 90 goes into 95 once (1*90 = 90), subtract 90 from 95, we get 5. Add a decimal point and a zero, make it 50. 90 goes into 50 zero times. Add another zero, make it 500. 90 goes into 500 five times (5*90 = 450), subtract 450 from 500, we get 50. So we see a repeating pattern. So 18.95÷90 = 0.210555... So to the nearest cent (two decimal places), we look at the third decimal place, which is 0, so we round to 0.21. Wait, but earlier for the 60 - pack: total cost $10.97, number of packs 60. So 10.97÷60≈0.1828, which is≈0.18 when rounded to the nearest cent (since the third decimal is 2 < 5, so 0.18). Wait, I think I messed up the 90 - pack calculation earlier. Let's recalculate 18.95÷90:

18.95 ÷ 90:

= (18.95) / 90

= 0.210555...

So to the nearest cent (hundredth), we round 0.210555... to 0.21 (since the thousandth digit is 0, which is less than 5, so we don't round up the hundredth digit).

Wait, but let's do the 60 - pack correctly:

Total cost = $10.97, number of packs = 60.

Unit rate = 10.97 / 60 ≈ 0.182833...

Rounding to the nearest cent (hundredth), we look at the thousandth digit (2), which is less than 5, so we round to 0.18.

Ah, I see, I made a mistake earlier with the 90 - pack. Let's correct that.

So for the 60 - pack:

## Step 1: Apply the unit rate formula
$Unit\ Rate=\frac{10.97}{60}$
## Step 2: Perform the division
$\frac{10.97}{60}\approx0.1828$
## Step 3: Round to the nearest cent
The thousandth digit is 2, which is less than 5, so we round down to $0.18$ dollars per pack.

# Explanation (for 60 - pack)
## Step 1: Recall the formula for unit rate
The unit rate (dollars per pack) is calculated by dividing the total cost by the number of packs. The formula is $Unit\ Rate=\frac{Total\ Cost}{Number\ of\ Packs}$.
## Step 2: Substitute the values for the 60 - pack
For the 60 - pack container, the total cost is $\$10.97$ and the number of packs is 60. So we calculate $\frac{10.97}{60}$.
## Step 3: Perform the division
$\frac{10.97}{60}\approx0.1828$ dollars per pack.
## Step 4: Round to the nearest cent
To round to the nearest cent (hundredth of a dollar), we look at the thousandth place. The number $0.1828$ has a 2 in the thousandth place, which is less than 5, so we round down. So $0.1828\approx0.18$ dollars per pack.
# Answer:
The unit rate for the 60 - pack container is $\$0.18$ per pack.

### Part (for 90 - pack)
# Explanation:
## Step 1: Apply the unit rate formula
The unit rate (dollars per pack) is calculated by dividing the total cost by the number of packs. The formula is $Unit\ Rate=\frac{Total\ Cost}{Number\ of\ Packs}$.
## Step 2: Substitute the values for the 90 - pack
For the 90 - pack container, the total cost is $\$18.95$ and the number of packs is 90. So we calculate $\frac{18.95}{90}$.
## Step 3: Perform the division
$\frac{18.95}{90}\approx0.2106$ dollars per pack.
## Step 4: Round to the nearest cent
To round to the nearest cent (hundredth of a dollar), we look at the thousandth place. The number $0.2106$ has a 0 in the thousandth place, which is less than 5, so we round down. So $0.2106\approx0.21$ dollars per pack.
# Answer:
The unit rate for the 90 - pack container is $\$0.21$ per pack.

### Part (Which container has the lowest unit rate?)
# Brief Explanations:
We compare the unit rates:
- 20 - pack: $\$0.27$ per pack
- 60 - pack: $\$0.18$ per pack
- 90 - pack: $\$0.21$ per pack

By comparing these values, we can see that the 60 - pack container has the lowest unit rate ($\$0.18$ per pack), which is different from our initial thought in part (a) where we assumed the 90 - pack would be the lowest. This shows that our initial assumption based on volume alone was incorrect, and we need to calculate the actual unit rates to determine the best deal.
# Answer:
The 60 - pack container has the lowest unit rate. The 20 - pack has a unit rate of $\$0.27$ per pack, the 60 - pack has $\$0.18$ per pack, and the 90 - pack has $\$0.21$ per pack. Comparing these, $\$0.18$ is the lowest, so the 60 - pack has the lowest unit rate. This differs from our initial thought in part (a) (where we thought the 90 - pack would be lowest), showing the importance of calculating actual unit rates rather than just relying on the volume of the pack.

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QUESTION IMAGE

Question

Explanation:

Part a

Step 1: Recall the formula for unit rate

Step 2: Substitute the values for the 20 - pack

Step 3: Perform the division

Step 4: Round to the nearest cent

Step 1: Use the unit rate formula

Step 2: Perform the division

Step 3: Round to the nearest cent

Answer:

Part b