The correct answer is option (D):
50
Let's break down this problem step-by-step. We are given that 201 one-rupee coins are distributed among 8 kids (A, B, C, D, E, F, G, and H) in arithmetic progression. This means the number of coins each child receives increases or decreases by a constant amount.
Let 'a' be the number of coins received by A (the first term), and let 'd' be the common difference. Then the number of coins received by the kids will be:
A: a
B: a + d
C: a + 2d
D: a + 3d
E: a + 4d
F: a + 5d
G: a + 6d
H: a + 7d
The sum of these coins must equal 201. The sum of an arithmetic series is given by the formula: S = (n/2) * [2a + (n-1)d], where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
In our case, S = 201 and n = 8. Substituting these values into the formula, we get:
201 = (8/2) * [2a + (8-1)d]
201 = 4 * [2a + 7d]
201/4 = 2a + 7d
50.25 = 2a + 7d
Since the number of coins must be a whole number, a and d must be integers (or a fraction that ultimately leads to a whole number sum). Looking at the equation 50.25 = 2a + 7d, we know that 2a must produce an even whole number, and since 50.25 is not even, 7d must have a fractional portion that adds up to 0.25. 7d will result in .25 when d = 0.25, and that gives us a whole number of coins.
Let's assume the question meant that the common difference is a fraction.
If d = 0.25, then
50. 25 = 2a + 7(0.25)
51. 25 = 2a + 1.75
52. 50 = 2a
a = 24.25
So the coins received would be
A: 24.25
B: 24.5
C: 24.75
D: 25
E: 25.25
F: 25.5
G: 25.75
H: 26
Now, we need to find the total number of coins received by C and F.
C receives a + 2d coins, which is 24.25 + 2 * 0.25 = 24.75 + 0.5 = 25.5
F receives a + 5d coins, which is 24.25 + 5 * 0.25 = 24.25 + 1.25 = 25.5
The sum for C and F = 25.5 + 25.5 = 51
This gives us the answer close to 50. Since we are working with whole rupees, let's explore how a and d could be whole numbers. Consider that the sum of the coins for each kid is 201, and there are 8 kids, so the average amount is 201/8 = 25.125. This tells us the number of coins should be around 25 per kid.
Since it has to be arithmetic progression, let's consider cases where d is not a fraction. If d = 1, then the amount of coins is
A: a
B: a+1
C: a+2
D: a+3
E: a+4
F: a+5
G: a+6
H: a+7
8a + 28 = 201
8a = 173
a = 21.625
Consider d = 2
A: a
B: a+2
C: a+4
D: a+6
E: a+8
F: a+10
G: a+12
H: a+14
8a + 56 = 201
8a = 145
a = 18.125
Try setting the number of coins for each kid to integers with the number centered around the average of 201/8 = 25.125
Consider C = 23 and F = 27
If C = 23, then a+2d = 23.
If F = 27, then a+5d = 27
Subtracting, we get 3d = 4, so d = 4/3.
Let's consider that the difference needs to be a whole number for this to work as the question is phrased. Since the average is 25.125, the central values are the two middle kids C and F. If C = 22 and F = 28, the difference would be 28 - 22 = 6, and the average would be 3d = 6, so d = 2.
So C = a + 2d
F = a + 5d
F - C = 3d
Since the answer choices are 24, 36, 48, 50, 64, it makes sense that the answer has to be a whole number. Let's consider that there's an error in the wording. The question asked for C+F. If the numbers are meant to be whole numbers:
Let C and F be integers.
C: a + 2d
F: a + 5d
C + F = 2a + 7d
If we try to assume that the progression is nearly centered around the average, and C and F have a difference of 3 steps, we consider that the answer is close to 50. We need C + F to result in the answer, since the others are far off.
If C is one above the average, and F is one below the average, then the average would be 25.125. Since the numbers must be integers, consider:
A=22, B=23, C=24, D=25, E=26, F=27, G=28, H=29. The total is 204.
A=21, B=23, C=25, D=27, E=29, F=31, G=33, H=35. The total is 224.
Let's adjust and consider that the values of the coins are close to 25.
Consider the case where C and F are close to 25, such that they add up to 50.
C = 23, F = 27
27 - 23 = 4 coins
A=19, B=21, C=23, D=25, E=27, F=29, G=31, H=33. The sum of all is 208.
A=18, B=20, C=22, D=24, E=26, F=28, G=30, H=32. Sum is 200. close.
A=17, B=19, C=21, D=23, E=25, F=27, G=29, H=31. Sum is 192.
We want C + F, which is 22 + 28 = 50.
So let's find the values:
a = 18, and d = 2.
C: 18 + 2 * 2 = 22
F: 18 + 5 * 2 = 28.
C + F = 22 + 28 = 50.
The question might be formulated with a very slight error, the total should be 200 instead of 201 to be exact. Since we want an answer, the closest would be 50.
Final Answer: The final answer is $\boxed{50}$