Math Problem?

Question

show work

1) To concerve the contents of a 16-ounce bottle of tonic, a castaway adopts the following procedure. On the first day he drinks 1 oz. of the tonic and then refills the bottle with rain water. On the second day, he drinks 2 oz. of the mixture and then refills the bottl ewith more rain water. On the thrid day, he drinks 3 oz. of the mixture and again refills the bottle with more water. This procedure continues until the bottl eis empty. How many ounces of water did he drink when the bottle is completely empty?

2) This time the castaway has a smaller bottle that contains only five ounce of tonic. Using the same procedure as above, on which day does he drink the most tonic?

Answer 1

PROBLEM 1:
You just have to figure out how much water in total is added because in the end he will drink all the tonic plus all the added water.

On day 1 he adds 1 oz.
On day 2 he adds 2 oz.
On day 3 he adds 3 oz.
...
On day 14 he adds 14 oz.
On day 15 he adds 15 oz.
On day 16 he drinks everything and is done.

So you just need the sum of 1 through 15.

A quick formula for this is n(n+1)/2 = 15 * 16 / 2 = 120 oz.

PROBLEM 2:
For this part of the problem, we have to think about the amount of tonic that remains each time.

Day 1: 1 oz of 100% tonic = 1 oz.
That leaves 4 oz. of tonic or an 80% tonic solution.

Day 2: 2 oz of 80% tonic = 1.6 oz.
That leaves 2.4 oz. of tonic or a 48% tonic solution

Day 3: 3 oz of 48% tonic = 1.44 oz.
That leaves 0.96 oz. of tonic or a 19.2% tonic solution

Day 4: 4 oz of 19.2% tonic = 0.768 oz.
That leaves 0.192 oz. of tonic or a 3.84% tonic solution

Day 5: 5 oz of 3.84% tonic = 0.192 oz.
That leaves no tonic

You can easily see that on day 2 he drank the most tonic.

As a double check, let's make sure he drank 5 oz. of tonic:
1 + 1.6 + 1.44 + 0.768 + 0.192 = 5 oz.

The answer for problem 2 is Day 2.

Answer 2

He's drinking one ounce equal to the day from which he started. It's a 1:1 ratio. So after 16 days, he should be completely out of tonic. That means that on the 16th day, he drank 15oz of water, on the 15th day, he drank 14oz of water, and on the first day, zero ounces of water. Make a table like this:
Day oz. of water
16 - 15
15 - 14
14 -13
13 - 12
.
.
.
1 - 0

Then add up everything in the oz. column to find out how much water he drank.

The answer to the second question is he drank the same amount of tonic each day until the bottle was empty.

I hope this helps, it was a quick answer off the top of my head.

Answer 3

q1
he drank up to day 16.

in all he drank
=1+2+3+...+16=17*8=136 oz

only 16 of this is tonic

water drank=136-16=120 oz

q2
on day 1
1*(5/5)=1 oz tonic

day 2
2*(5-1)/5=8/5 oz tonic

day 3
3*(5-1-8/5)/5
=3*(12/5)/5=36/25 oz tonic

day 4
4*(5-1-8/5-36/25)/5
=4*24/125
=96/125 oz tonic

day 5
=5-1-8/5-36/25-96/125
=(500-200-180-96)/125
=24/125 oz tonic

most tonic consumed
=day 2
=1.6 oz tonic

Answer 4

1)1+2+...+15 = (1/2)(1+15)(15) = 120 oz
Ideas: You subtract 16 oz from the total 1+2+...+16.

2)
Day 1: 1 oz tonic
Day 2: (80%)(2) = 1.6 oz tonic (maximum)
Day 3: (2.4/5)(3) = 1.2 oz tonic
Day 4: (1.2/5)(4) = .96 oz tonic
Day 5: .24 oz tonic