Question 233181
A fish has a four-inch long head. Its tail is as long as its head, plus 1/4 the length of its body. Its body makes up 3/4 of its total length. What is the length of the fish.


Let H represent the length of the head.
Let T represent the length of the tail.
Let B represent the length of the body.
Let F represent the length of the fish.


H = 4
T = H + (1/4) * B
B = (3/4) * F
F = H + B + T


First thing to do is replace H with 4 in the equation for T.


You get T = 4 + (1/4) * B


Next thing to do is replace H and T in the equation for B.


B = (3/4) * F


Since F = (H + B + T), this means that:


B = (3/4) * (H + T + B)


Since H = 4, this means that:


B = (3/4) * (4 + T + B)


Since T = (4 + (1/4)*B), this means that:


B = (3/4) * (4 + (4 + (1/4)*B) + B).


Next thing to do is solve for B.


Simplify the expression by completing the indicated operations and removing parentheses starting with the innermost parentheses and working out.


B = (3/4) * (4 + (4 + (1/4)*B) + B) becomes:


B = (3/4) * (4 + 4 + (1/4)*B + B) becomes:


B = (3/4) * (8 + (1/4)*B + (4/4)*B) becomes:


B = (3/4) * (8 + ((5/4) * B)) becomes:


B = ((3/4)*8) + ((3/4) * (5/4) * B) becomes:


B = (24/4) + ((15/16) * B).


Subtract (15/16)*B from both sides to get:


B - (15/16)*B = (24/4)


Factor out the B to get:


B * (1-(15/16) = (24/4)


Simplify to get:


B * (1/16) = (24/4)


Multiply both sides of the equation by (16/1)) to get:


B = (24/4) * (16/1) = (24*16)/(4*1) = (24*4) = 96


T = H + (1/4)*B = 4 + 24 = 28


F = H + T + B = 4 + 28 + 96 = 128.


The body = 96 inches in length.


The head = 4 inches in length.


The tail = 28 inches in length


The fish = 128 inches in length.


The Tail is equal to the head plus 1/4 of the body which equals 4 + 96/4 which e43quals 4 + 24 which equals 28 so this part of the original problem statement is correct.


The Body equals 3/4 of the fish = 3/4 * 128 = 96 so this part of the original problem statement is correct.


Looks like you got it right.


Your answer to the question is that the fish is 128 inches in length.