(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?

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.

All I need to know is how do you do part two (2)? Is there a summation formula that we can use?

(4/9)^(n-.5)

Two students are sharing 1/2 of a bread. How much will each student get in fraction?

(A) In this scenario, let's assume that the loaf of bread is divided into 1 unit. Student A starts by eating half of the loaf, which is 1/2.

After student A eats half of the loaf, what remains is also 1/2 of the original loaf. Then, student B eats half of what remains, which is (1/2) * (1/2) = 1/4 of the original loaf.

Let's continue the pattern. After student B eats, what remains is also 1/4 of the original loaf. Student A then eats half of what remains, which is (1/2) * (1/4) = 1/8 of the original loaf.

This pattern keeps repeating. Each time, the remaining portion is halved, and the student taking their turn eats half of what remains. So, the amount of the loaf each student eats can be represented as follows:

Student A: 1/2 + 1/8 + 1/32 + 1/128 + ...

To calculate the total amount of the loaf student A will eat, we can use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

Where:
S is the sum of the series,
a is the first term, which is 1/2, and
r is the common ratio, which is 1/4.

Plugging in these values, we get:

S = (1/2) / (1 - 1/4)
= (1/2) / (3/4)
= (1/2) * (4/3)
= 4/6
= 2/3

Therefore, student A will eat 2/3 of the loaf, and since there are only 2 students, student B will eat the remaining 1/3 of the loaf.

(B) In this scenario, student A starts by eating 2/3 of the loaf, which is the same as (2/3) * 1 = 2/3 of the original loaf.

After student A eats 2/3 of the loaf, what remains is (1 - 2/3) = 1/3 of the original loaf. Then, student B eats half of what remains, which is (1/3) * (1/2) = 1/6 of the original loaf.

This pattern continues, with student A eating 2/3 of what remains each time, and student B eating half of what remains after student A. The total amount of the loaf each student eats can be represented as follows:

Student A: (2/3) + (2/3) * (1/6) + (2/3) * (1/6)^2 + (2/3) * (1/6)^3 + ...

To calculate the total amount of the loaf student A will eat, we can again use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

Where:
S is the sum of the series,
a is the first term, which is 2/3, and
r is the common ratio, which is 1/6.

Plugging in these values, we get:

S = (2/3) / (1 - 1/6)
= (2/3) / (5/6)
= (2/3) * (6/5)
= 12/15
= 4/5

Therefore, student A will eat 4/5 of the loaf, and student B will eat the remaining 1/5 of the loaf.

(3) In this scenario, let's assume that the loaf of bread is divided into 1 unit. Student A starts by eating half of the loaf, which is 1/2.

After student A eats half of the loaf, what remains is also 1/2 of the original loaf. Then, student B eats half of what remains, which is (1/2) * (1/2) = 1/4 of the original loaf.

After student B eats, what remains is again halved, so it becomes 1/4 of the original loaf. Then, student C eats half of what remains, which is (1/4) * (1/2) = 1/8 of the original loaf.

This pattern continues, with each student eating half of what remains after the previous student. The total amount of the loaf each student eats can be represented as follows:

Student A: 1/2 + 1/8 + 1/32 + 1/128 + ...

Again, to calculate the total amount of the loaf student A will eat, we can use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

Where:
S is the sum of the series,
a is the first term, which is 1/2, and
r is the common ratio, which is 1/2.

Plugging in these values, we get:

S = (1/2) / (1 - 1/2)
= (1/2) / (1/2)
= 1/2

Therefore, student A will eat 1/2 of the loaf, and since there are three students, student B and student C will each eat 1/4 of the loaf.