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kevincan
480 bottles of wine, each weighing 1 1/4 kilogram, are to be packed into boxes. Because of shipping restrictions, the total weight of the wine in each box must be between 10 and 20 kilograms, inclusive.

If each box must contain the same number of bottles, how many possible values are there for the number of bottles that each box can contain?

A. 1
B. 2
C. 3
D. 4
E. 5
480 bottles of wine, each weight 1.25 kg.

The weight in each box is between 10 and 20 inclusive.

So, if we divide the weight by 1.25 kg, we can get the total bottles per box.

(10/1.25) <= n <= (20/1.25)

8 <= n < = 16

So, n should be a factor of 480.

480 = 6*8*2*5 = 2*3*8*2*5

Possible values are : 8

(2*5)=10

(2*3*2) = 12

(3*5) = 15

(2*8) = 16

So, the list of values for number of bottles in a box = (8,10,12,15,16. Totally 5 values.

Option E
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Hi suntprovident,

You're missing n = 8 bottles per box!

Let's check: each bottle weighs 1 and 1/4 kg, which is 5/4 kg.

If a box has 8 bottles, the total weight is 8 × 5/4 = 10 kg.

The problem says the weight must be between 10 and 20 kilograms, INCLUSIVE. That means 10 kg is allowed. So 8 bottles giving exactly 10 kg is a valid option.

Let's also verify 8 divides 480: 480 ÷ 8 = 60 boxes. Yes, it divides evenly.

So the complete list of valid values is:
- 8 bottles → 10 kg ✓
- 10 bottles → 12.5 kg ✓
- 12 bottles → 15 kg ✓
- 15 bottles → 18.75 kg ✓
- 16 bottles → 20 kg ✓

That's 5 values, giving us answer E.

It looks like you may have started checking from 10 bottles instead of working out the minimum from the weight constraint. The key step is to first solve the inequality: 10 ≤ n × (5/4) ≤ 20, which gives 8 ≤ n ≤ 16. Then find which divisors of 480 fall in that range.

General principle: when a problem says "inclusive," always check the boundary values — they're often the ones that are easy to overlook.

Answer: E
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