(A) 2 students are sharing a loaf of bread. Student A eats half of the loaf, then student B eats half of what remains, then student A eats half of what remains, and so on. How much of the loaf will each student eat?
(B) 2 students are sharing a loaf of bread. Student A eats 2/3 of the loaf, then student B eats half of what remains, then student A eats 2/3 of what remains, then student B eats half of what remains, and so on. How much of the loaf will each student eat?
(3) 3 students decide to share a loaf of bread. Student A eats half of the loaf, passes what remains to student B who eats half, and then onto student C who eats half, and then back to student A who eats half, and so on. How much of the loaf will each student eat?
* calculus II - Reiny, Sunday, April 11, 2010 at 9:18pm
make 2 columns, marked A and B
A eats 1/2 leaving 1/2 -- B eats 1/4 leaving 1/4
A eats 1/8 th leaving 1/8th -- B eats 1/16 th, leaving 1/16 th
So A eats 1/2 + 1/8 + 1/32 + ...
a Geometric series with a=1/2, r = 1/4
S∞ = a/(1-r) = (1/2)/(1 - 1/4)
= (1/2)/(3/4) = 2/3
So A eats 2/3 of the loaf
The clearly B must eat the missing parts or 1/3.
Do the others the same way.
* calculus II - Harold, Wednesday, April 14, 2010 at 12:33am
All I need to know is how do you do part two (2)? Is there a summation formula that we can use?
To solve part (A) of the question, we need to see the pattern of how the loaf is being divided between the two students.
Student A eats half of the loaf, which leaves half of the loaf remaining. Then, student B eats half of what remains. This means B eats 1/2 * 1/2 = 1/4 of the original loaf.
Next, student A eats half of what remains after B's turn. This means A eats 1/2 * 1/4 = 1/8 of the original loaf.
This process continues indefinitely, with each student eating half of what remains after the other student's turn.
To find the total amount each student will eat, we need to find the sum of the infinite geometric series.
In this case, the first term (a) is 1/2 and the common ratio (r) is 1/4.
The sum of an infinite geometric series is given by the formula: S∞ = a / (1 - r).
Plugging in the values, we get: S∞ = (1/2) / (1 - 1/4) = (1/2) / (3/4) = 2/3.
So, student A will eat 2/3 of the loaf, and student B will eat the remaining 1/3 of the loaf.
Now, let's move on to part (B) of the question.
In this case, Student A eats 2/3 of the loaf first. This leaves 1/3 of the loaf remaining.
Then, student B eats half of what remains. Half of 1/3 is 1/6.
Next, student A eats 2/3 of what remains after B's turn. 2/3 of 1/6 is 2/18, which simplifies to 1/9.
This process continues indefinitely, with each student eating their respective portions based on the remaining fraction after the previous turn.
To find the total amount each student will eat, we need to find the sum of the infinite series.
Unfortunately, we cannot directly apply the same formula as in part (A) since the ratios are changing after each turn.
To solve this, we need to find a pattern in the terms and come up with a formula to find the nth term.
In this case, the pattern seems to be alternating between 2/3 and 1/3 for A's turn, and 1/6 and 1/9 for B's turn.
To find the nth term, we can use a recursive formula:
aₙ = aₙ₋₂ / 3 if n is even
aₙ = aₙ₋₂ / 2 if n is odd
Where a₀ is 2/3 and a₁ is 1/3.
Using this recursive formula, we can find the terms in the series as follows:
a₂ = 1/9
a₃ = 1/18
a₄ = 1/36
a₅ = 1/72
...
Now, the sum of the infinite series is given by the formula:
S∞ = a₀ + a₁ + a₂ + ...
To find the sum, we can use a calculator or a computer program to add up the terms until we see the series converge to a certain value.
For part (C) of the question, the process is similar to part (A), but now there are three students involved.
Student A eats half of the loaf, which leaves half of the loaf remaining. Then, student B eats half of what remains, which is 1/4 of the original loaf.
Next, student C eats half of what remains after B's turn, which is 1/8 of the original loaf.
Finally, the loaf goes back to student A, who eats half of what remains after C's turn, which is 1/16 of the original loaf.
This process continues indefinitely, with each student eating half of what remains after the previous turn.
To find the total amount each student will eat, we need to find the sum of the infinite geometric series.
Using the same formula as in part (A), with the first term (a) being 1/2 and the common ratio (r) being 1/2, we can calculate the sum of the series as:
S∞ = a / (1 - r) = (1/2) / (1 - 1/2) = (1/2) / (1/2) = 1.
So, each student will eat 1/3 of the loaf, which is equal to 1.