mathematicians!?

Question

Help please! I need a solution for the following puzzle:

The first day one girl went fishing, she caught exactly 31416 pounds of fish.

The next day, she caught exactly 40% of what she caught the day before.

She continued fishing every day. And every day she caught exactly 40% of what she caught the previous day.

If she continued fishing every day for twenty years, how many pounds of fish, in total, would she catch? Please round to the nearest pound.

Answer 1

Are you familiar with geometric progressions?
anyway check this out
http://en.wikipedia.org/wiki/Geometric_progression

so what's happening is:

40% of A is just A*0.4 right?

so if A is the catch of day 1 (A = 31416)

Day 2 Catch: A*0.4
Day 3 Catch: A* 0.4*0.4 = A* (0.4)^2
Day 4 Catch: A* 0.4*0.4*0.4 = A*(0.4)^3
etc.
Day n Catch: A*(0.4)^(n-1)

the catch everyday is the nth element in the geometric progression, where the common ratio is 0.4 and the initial number is 31416.

Now there is a formula for getting the sum of all the terms from the first to the nth term. ( I won't explain how the formula came to be... just check the wiki )

If a(n) = A * r^(n-1)

Summation of ( a (n) ) with n from 1 to k =
A*(1-r^(n+1))/(1-r)

We have:

If a(n) = 31416 * (0.4) ^ (n-1)

then summation of ( a(n) ) with n from 1 to 365*20-5 =

31416 ( 1 - (0.4) ^ (7295) ) / (1-0.4)
= 31416 ( 1 - (0.4) ^21) / (0.6)
= 31416*5/3*(1-0) of course we can just treat (0.4)^7295 as 0
=52360

In summary, she caught a total of 52360 pounds.

Answer 2

Let there be 365 days in each yr. So total days for which she fished = 365 x 20 = 7300

The pounds of fish caught by her in 1st, 2nd, 3rd, ... days are
31416, 12566.4, 5026.56, ...

This is a Geometric Progression (GP) with 1st term (a) = 31416 , common ratio (r) = 0.4 and no. of terms (n) = 7300.

Sum of n terms of a GP = a (1 - r^n) / (1 - r)
= 31416 (1 - 0.4^7300) / (1 - 0.4) = 31416 (1) / 0.6 = 52360
[Here - 0.4^7300 is infinitely small no. as compared to 1 and can be considered as 0.]
Hence she caught 52360 pounds of fish in 20 yrs.

Hope this helps.

Answer 3

Supposing that there 365 days per year
F(total)=(31416)*(1+0.4+0.4^2++++++0.4^7299)
As 0.4^7299 is too small for any calculator you can take the second bracket as the sum to infinity of a geometric serie with r=0.4
this sum ia 1/0.6
so the girl fished 31416/06=52360 pounds

Answer 4

A similar question to this was posted recently.

This is a geometric series.
31416 * (1 + 0.4 + 0.4^2 + ....)
= 31416 * 1 / (1 - 0.4)
= 52360 lbs.

It is the sum to infinity because 20 years is long enough for 0.4^n to become negligible.

Answer 5

My first thought grew to become into they have been a base-7 society considering there have been 7 symbols (a,b,e,l,n,p,r) of their "huge sort" device. despite the fact that, whilst i attempted to come back up with values for each letter I have been given caught. including a 5 digit huge sort to a 4 digit huge sort could basically effect in a 6 digit huge sort that starts off with 10...., yet that could propose that A = 0, B = a million. yet that now makes APPLE start up with a nil and that messes each little thing up. i'm going to could think of a sprint extra in this...

Answer 6

In 20 years you will have 5 leap years, so she went fishing for 7305 days. In total, she caught 52360 pounds, exactly.

Answer 7

uhm you need a formula to solve this crap. i think its about variation. too lazy to solve. sorry. and i forgot the formula. that was like 2 years ago O_O