RE: Thinking Skills (A/ASLevel) Please Help!
#6
Requirements:
 Buy as many cards as you can
 Have a mixture of designs
 Have no more than 4 cards of any one design
Facts:
 Each design occurs only in one type of pack
 Each pack of cards has the same number of cards for each design [e.g., the pack of 20 cards with 4 designs has 5 cards of each design]
 Packs of a type are identical
Bear in mind the requirements, especially that you cannot have more than 4 cards of any one design, and the facts, especially that each pack of cards has the same number of cards per design. This allows us to eliminate several items.
"Number of designs: 2; number of cards in pack: 8; price: 0.40" will be notated as 2/8/0.40.
2/8/0.40 has 4 cards of each design, so it's OK. (8/2 = 4; if the result >4 then we can't use it)
2/16/0.80 has 8 cards of each design, so we can't use it. (16/2 = 8)
Using this method we eliminate the following:
2/16/0.80 (8)
4/20/1.40 (5)
4/40/1.60 (10)
2/10/0.50 (5)
4/32/2.00 (8)
8/40/2.40 (5)
[Total items: 6]
And we are left to use these:
2/8/0.40 (4)
5/20/1.00 (4)
4/12/0.80 (3)
5/10/1.30 (2)
[Total items: 4]
We have eliminated 6 items and kept 4, so that all 10 items in the table are accounted for.
Now to blow up that table for easy reference:
2 designs in 8 cards for 40c
5 designs in 20 cards for 100c
4 designs in 12 cards for 80c
5 designs in 10 cards for 130c
In total, that is 50 cards for a total of 350 cents.
According to this, the average price (350/50) of each card will be 7 cents so someone else should check over this as I must have missed something.
Answer key says 8 cents.
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#10
Start with the simplest part. In July he paid $50 and used 400 units. That's $0.125 per unit.
At that rate,
500 units is $62.50,
1000 units is $125,
and 300 units is $37.50.
This will let us better refer to the values in each of the possible answers.
Now let's look at March. He used 800 units and paid $92.50.
At the rate from July, 800 units would cost $100 (800 * .125). So there is clearly a lower rate after a certain amount of units.
According to the answers, the lower rate may begin after 300, 500, or 1000 units.
Since we can see that there is a lower rate at some point, answer C is eliminated.
Since we can see that the lower rate kicked in by the time we reached 800 units, answer B is eliminated.
Answer D is hard to test because we don't know what the fixed charge would be, and since the 300 unit threshold is below our reference point of $50 for 400 units, it's difficult to find where the lower rate would begin.
So in hopes of getting lucky, we will test answer A first.
At the rate of $0.125 per unit, the first 500 units (of the 800 on the March bill) would cost $62.50.
That leaves us with 300 units to account for, costing a total of $30 (92.50  62.50 = 30)
30/300 = $0.1
This tells us that the standard rate is $0.125, and the reduced rate, which begins after 500 units, is $0.1.
Now for the final test.
He paid $132.50 in January for 1200 units.
The first 500 units are billed at $0.125 = $62.50
The other 700 units are billed at $0.1 = $70.00
$70.00 + $62.50 = $132.50
Thus the rule is proven and answer is A in agreement with the answer key.
If this did not resolve as expected, then we would be left to assume that the correct answer is D.
I have to come back later for the rest.
