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?
C. 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?
CAN YOU HELP PLEASE
1 at start
a eats 2/3 ** 1/9 *** 1/54
b eats 1/6 ** 1/36 ** 1/108
first line is geometric series
2/3 + 2/3(1/6) + (2/3)(1/6)^2 ...
An = 2/3 +(1/6)^n-1
sum of that series = a/(1-r)
= (2/3)/(1-1/6) = 2/3 /5/6 = 4/5
try similar for the rest
Sure! Let's go step by step to figure out how much of the loaf each student will eat in each scenario.
Scenario B:
1. Student A eats 2/3 of the loaf.
2. The remaining 1/3 of the loaf is passed to student B.
3. Student B eats half of what remains, which is 1/2 * 1/3 = 1/6 of the original loaf.
4. The remaining 1/6 of the loaf is passed back to student A.
5. Student A eats 2/3 of what remains, which is 2/3 * 1/6 = 2/18 = 1/9 of the original loaf.
6. The process continues alternately between student A and student B.
To determine the total amount consumed by each student, we can use a pattern. Notice that every time we go back and forth between student A and student B, the amount eaten decreases by a factor of 3.
So, student A will consume 2/3 + 1/9 + (1/9) / 3 + ((1/9) / 3) / 3 + ... which can be simplified as:
2/3 + 1/9 + 1/27 + 1/81 + ...
This is an infinite geometric series with a common ratio of 1/3 and a first term of 2/3. The sum of an infinite geometric series can be calculated using the formula: S = a / (1 - r), where "a" is the first term and "r" is the common ratio.
Plugging in the values, we get:
S = (2/3) / (1 - 1/3)
S = (2/3) / (2/3)
S = 1
Therefore, in Scenario B, student A will eat the entire loaf.
Now let's move on to Scenario C:
1. Student A eats half of the loaf.
2. The remaining half is passed to student B.
3. Student B eats half of what remains, which is 1/2 * 1/2 = 1/4 of the original loaf.
4. The remaining 1/4 of the loaf is passed to student C.
5. Student C eats half of what remains, which is 1/2 * 1/4 = 1/8 of the original loaf.
6. The process continues alternately between student A, student B, and student C.
To determine the total amount consumed by each student, we can again use a pattern. Notice that every time we go through all three students, the amount eaten decreases by a factor of 2.
So, student A will consume 1/2 + (1/8) / 2 + ((1/8) / 2) / 2 + ... which can be simplified as:
1/2 + 1/16 + 1/32 + 1/64 + ...
This is another infinite geometric series with a common ratio of 1/2 and a first term of 1/2. Using the sum formula, we get:
S = (1/2) / (1 - 1/2)
S = (1/2) / (1/2)
S = 1
Therefore, in Scenario C, each student will eat the entire loaf.
I hope this clarifies how much of the loaf each student will eat in both scenarios!