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 Cythia and Freds age. Find the present ages.
I have
a=3c f=a-16 a-1=2(c-1 + f-1)
and I don't understand what comes next and why
a=3c
f=a-16
a-1=2(c-1 + f-1)
so far, so good. Now start substituting. First, get rid of a:
f = 3c-16
3c-1 = 2(f+c-2)
Now get rid of the f:
3c-1 = 2((3c-16)+c-2)
3c-1 = 2(4c-18)
3c-1 = 8c - 36
5c = 35
c = 7
so, f = 3c-16 = 5
and a = 3c = 21
To find the present ages of Adam, Cynthia, and Fred, we can use the given information and solve the system of equations.
Let's start with the equations you already have:
1) a = 3c (Adam is 3 times as old as Cynthia)
2) f = a - 16 (Fred is 16 years younger than Adam)
3) a - 1 = 2(c - 1 + f - 1) (One year ago, Adam's age was twice the sum of Cynthia and Fred's age)
We can proceed as follows:
From equation 2), we can replace 'a' in equation 3) with 3c (using equation 1)):
3c - 1 = 2(c - 1 + f - 1)
Simplifying equation 3):
3c - 1 = 2c + 2f - 4
Rearranging the terms:
3c - 2c + 1 = 2f - 4
Simplifying further:
c + 1 = 2f - 4
Now, let's express 'f' (Fred's age) in terms of 'c' (Cynthia's age). From equation 2):
f = a - 16
f = 3c - 16
We can substitute this expression for 'f' into the equation above:
c + 1 = 2(3c - 16) - 4
Expanding and simplifying:
c + 1 = 6c - 32 - 4
c + 1 = 6c - 36
Now, let's solve for 'c' by getting all the 'c' terms on one side of the equation:
c - 6c = -36 - 1
-5c = -37
Dividing both sides by -5:
c = -37 / -5
c = 7.4
Since ages are typically represented as whole numbers, we can round 'c' to the nearest whole number, which is 7. Therefore, Cynthia's present age is 7 years.
Using equation 1):
a = 3c
a = 3 * 7
a = 21
So, Adam's present age is 21 years.
Using equation 2):
f = a - 16
f = 21 - 16
f = 5
Therefore, Fred's present age is 5 years.
To summarize:
Adam's present age is 21 years,
Cynthia's present age is 7 years,
and Fred's present age is 5 years.