Allan, Bernice, Candice and Derrick shared the cost of a present. The amount Allan paid is 1/14 of the total amount Bernice and Candice paid. The amount Bernice paid is 1/4 of the total amount Candice and Derrick paid. The amount Derrick paid is twice the amount Candice paid. Given that Candice paid $63 more than Allan, how much was the cost of the present?
Let's represent the amount Allan paid as A, the amount Bernice paid as B, the amount Candice paid as C, and the amount Derrick paid as D.
From the first piece of information, we know that:
A = 1/14(B + C)
From the second piece of information, we know that:
B = 1/4(C + D)
From the third piece of information, we know that:
D = 2C
Finally, from the fourth piece of information, we know that:
C = A + 63
We can use these equations to solve for the values of A, B, C, and D.
Substituting C = A + 63 into the first equation, we get:
A = 1/14(B + A + 63)
Multiplying both sides by 14, we get:
14A = B + A + 63
Simplifying, we get:
13A - B = 63
Substituting B = 1/4(C + D) and D = 2C, we get:
B = 1/4(C + 2C) = 3/4C
Substituting D = 2C into this equation, we get:
B = 3/4(2C) = 3/2C
Substituting these values into the equation 13A - B = 63, we get:
13A - 3/2C = 63
Substituting C = A + 63 into this equation, we get:
13A - 3/2(A + 63) = 63
Simplifying, we get:
19/2A = 189
Multiplying both sides by 2/19, we get:
A = 21
Substituting this value back into the equation C = A + 63, we get:
C = 84
Using D = 2C, we get:
D = 168
And using B = 3/2C, we get:
B = 126
Therefore, the total cost of the present is:
A + B + C + D = 21 + 126 + 84 + 168 = $399.
From the given information, we can write a system of equations:
A = (1/14)(B + C)
B = (1/4)(C + D)
D = 2C
C = A + $63
We want to find the total cost of the present, which is the sum of what each person paid:
Total cost = A + B + C + D
Substituting the equations for B and D in terms of C and solving for C, we get:
B = (1/4)(C + 2C) = (3/4)C
D = 2C
Therefore, A = (1/14)(B + C) = (1/14)(3/4C + C) = (11/56)C
Substituting the value of A in terms of C and simplifying, we get:
Total cost = A + B + C + D = (11/56)C + (3/4)C + C + 2C = (251/56)C
Finally, we know that Candice paid $63 more than Allan, so C = A + $63. Substituting this equation into the expression for the total cost, we get:
Total cost = (251/56)(A + $63)
We can see that the total cost will be minimized when A is minimized. Since A = (11/56)C, we want to minimize C. Since C is a positive quantity, the minimum value it can take is when C = $63. Therefore, the total cost is:
Total cost = (251/56)($63 + A) = (251/56)($63 + (11/56)($63)) = $279
Therefore, the cost of the present was $279.