Adam is 3 times as old as Cynthia and Fred is 16 years younger than Adam. One year ago, Adam's age was twice the sum of Cynthia's and Fred's age. Find their present age.
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Adam is 3 times as old as Cynthia or A = 3C.
Fred is 16 years younger than Adam or F = A - 16.
One year ago, Adam's age was twice the sum of Cynthia's and Fred's age or A - 1 = 2(C + F).
Find their present age.
Making the appropriate substitutions and solving yields C = 11, A = 33 and F = 17.
Let's break down the problem step by step.
Step 1: Create variables for their ages.
Let's assume that Cynthia's age is C, Adam's age is A, and Fred's age is F.
Step 2: Translate the given information into equations.
According to the problem, Adam is 3 times as old as Cynthia, so we can write:
A = 3C (equation 1)
Also, Fred is 16 years younger than Adam, which can be written as:
F = A - 16 (equation 2)
Step 3: Express the problem in terms of one variable.
We are given that one year ago, Adam's age was twice the sum of Cynthia's and Fred's age. So one year ago, Adam's age would be A - 1, Cynthia's age would be C - 1, and Fred's age would be F - 1.
The equation can be written as:
A - 1 = 2(C - 1 + F - 1)
Step 4: Simplify the equation.
Expand the equation on the right side:
A - 1 = 2C - 2 + 2F - 2
A - 1 = 2C + 2F - 4
A = 2C + 2F - 3 (equation 3)
Step 5: Solve the equations.
Now, we have a system of three equations:
A = 3C (equation 1)
F = A - 16 (equation 2)
A = 2C + 2F - 3 (equation 3)
Substitute equation 1 into equation 3 to eliminate A:
3C = 2C + 2F - 3
C = 2F - 3 (equation 4)
Substitute equation 1 in equation 2 to eliminate A:
F = 3C - 16 (equation 5)
Substitute equation 4 into equation 5:
F = 3(2F - 3) - 16
F = 6F - 9 - 16
F = 6F - 25
25 = 6F - F
25 = 5F
F = 5
Substitute F = 5 in equation 4:
C = 2(5) - 3
C = 7
Substitute C = 7 in equation 1:
A = 3(7)
A = 21
So, the present ages are Cynthia = 7 years, Adam = 21 years, and Fred = 5 years.