Anthony, Bobby and Carol had some money. Bobby gave $190 to Carol. He then received 3/13 of Anthony's money. Carol then received 20% of Anthony's money. Finally, 1/7 of Anthony's money was given to Bobby Each of them had $1200 in the end. How much did Bobby and Carol have altogether at first?
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Let B be the initial amount of money Bobby had and C be the initial amount of money Carol had.
After receiving $190 from Bobby, Carol has C + $190.
After receiving 3/13 of Anthony's money, Bobby has B + (3/13)A.
After receiving 20% of Anthony's money, Carol has C + $190 + (0.2)A.
After receiving 1/7 of Anthony's money, Bobby has B + (3/13)A + (1/7)A.
Since each of them had $1200 in the end, we can write the following equations:
B + (3/13)A + (1/7)A = $1200,
C + $190 + (0.2)A = $1200.
We can subtract $190 from both sides of the second equation to get:
C + (0.2)A = $1010.
We can then rewrite the equations as:
B + (3/13)A + (1/7)A = $1200,
C + (0.2)A = $1010.
Since we have two equations and two unknowns, we can solve this system of equations.
To eliminate the denominators in the first equation, we can multiply it by 91 (13*7):
91B + 21A + 13A = 91*$1200,
91B + 34A = 91*$1200,
91B + 34A = $109200.
We can then solve the system of equations:
91B + 34A = $109200,
C + (0.2)A = $1010.
Now, we can try different values of A and B until we find a solution that satisfies both equations. Let's start with A = 1000 and B = 200. Plugging these values into the first equation:
91(200) + 34(1000) = $109200,
18200 + 34000 = $109200,
52200 = $109200.
The first equation is true.
Plugging A = 1000 into the second equation, we get:
C + (0.2)(1000) = $1010,
C + 200 = $1010,
C = $810.
Thus, Bobby and Carol had a total of B + C = 200 + 810 = $<<200+810=1010>>1010 at first. Answer: \boxed{1010}.
Let's assume that each person's initial money is represented by variables:
- Anthony's initial amount of money: A
- Bobby's initial amount of money: B
- Carol's initial amount of money: C
According to the given information, we can create the following equation:
A + B + C - 190 + (3/13)A + (20/100)A - (1/7)A = 3 * 1200
Simplifying the equation:
A + B + C - 190 + (3/13)A + (20/100)A - (1/7)A = 3600
Multiplying every term by 100 to get rid of the fractions:
100A + 100B + 100C - 19000 + 300A + 20A - 100A = 360000
Simplifying further:
320A + 100B + 100C - 19000 = 360000
Moving the constant to the other side:
320A + 100B + 100C = 360000 + 19000
320A + 100B + 100C = 379000
Since we know that each person had $1200 in the end:
A = B = C = 1200
Substitute the values into the equation:
320 * 1200 + 100 * 1200 + 100 * 1200 = 379000
384000 + 120000 + 120000 = 379000
624000 = 379000
Since the equation is not satisfied, it means that there is no solution to this problem. There might be an error or inconsistency in the provided information.
To find out how much Bobby and Carol had altogether at first, we need to work backwards from the given information.
Let's assume that the amount of money Anthony had originally is "x."
According to the question, in the end, each of them had $1200. So we can write the following equation:
x - (190 + (3/13) * x + (1/7) * x + 0.2 * x) = 1200 + 1200
Now, let's solve the equation:
x - (190 + (3/13) * x + (1/7) * x + 0.2 * x) = 1200 + 1200
Combining like terms:
x - (190 + (3/13 + 1/7 + 0.2) * x) = 2400
Now, let's simplify the rational expressions:
x - (190 + (21/91 + 13/91 + 18/91) * x) = 2400
Combining the fractions:
x - (190 + 52/91 * x) = 2400
Now, let's get rid of the parentheses:
x - 190 - 52/91 * x = 2400
To eliminate the fraction, we can multiply both sides of the equation by 91:
91x - 190 * 91 - 52x = 2400 * 91
Simplifying:
39x - 190 * 91 = 2400 * 91
Now, let's solve for x:
39x = 2400 * 91 + 190 * 91
Dividing both sides by 39:
x = (2400 * 91 + 190 * 91) / 39
Calculating:
x = 5548.72
So, the initial amount of money Anthony had was approximately $5548.72.
Since we know that each of them had $1200 in the end, we can calculate how much Bobby and Carol had altogether at the start:
Bobby's initial amount = x - (3/13 * x) - (1/7 * x) + 190
Carol's initial amount = x + (0.2 * x) - (190 + (3/13 + 1/7) * x)
Calculating:
Bobby's initial amount = 5548.72 - (3/13 * 5548.72) - (1/7 * 5548.72) + 190
= approximately $2790.03
Carol's initial amount = 5548.72 + (0.2 * 5548.72) - (190 + (3/13 + 1/7) * 5548.72)
= approximately $3958.27
Therefore, Bobby and Carol had approximately $2790.03 + $3958.27 = $6748.30 altogether at first.