The combined age of a father and his two sons is 50 years. Four years ago the father was eight times as old as his older son. Four years from now he will be 4 times as old as his younger son. Find the age of each at present.
I found that the ages are 36, 8, 6
But I want to learn how to find it out by using mathematical expressions.
If the sons are age a and b, with a<b,
f+a+b = 50
f-4 = 8(b-4)
f+4 = 4(a+4)
16
To find the ages of the father and his two sons, we can use algebraic expressions and solve the given problem step by step. Let's start by assigning variables.
Let:
- The father's present age be represented by F
- The older son's present age be represented by S
- The younger son's present age be represented by Y
From the problem statement, we are given the following information:
1) The combined age of the father and his two sons is 50 years:
F + S + Y = 50
2) Four years ago, the father was eight times as old as his older son:
F - 4 = 8(S - 4)
3) Four years from now, the father will be four times as old as his younger son:
F + 4 = 4(Y + 4)
Now we have a system of three equations with three unknowns. We can solve this system of equations using any method we prefer.
Method 1: Substitution
We can start by solving equation 2 for F in terms of S:
F = 8S - 28
Substituting this value of F into equations 1 and 3:
1) (8S - 28) + S + Y = 50
2) 9S + Y - 28 = 50
9S + Y = 78
3) (8S - 28) + 4 = 4(Y + 4)
8S - 24 = 4Y + 16
8S - 4Y = 40
We now have a new system of equations:
9S + Y = 78 -- Equation 4
8S - 4Y = 40 -- Equation 5
Next, we can solve this system of equations using any appropriate method, such as substitution or elimination.
Substituting Y from equation 4 into equation 5:
8S - 4(78 - 9S) = 40
8S - 312 + 36S = 40
44S - 312 = 40
44S = 352
S = 8
Substituting back the value of S into equation 4:
9(8) + Y = 78
72 + Y = 78
Y = 6
Finally, we can substitute the values of S and Y into equation 1 to find F:
F + 8 + 6 = 50
F = 36
Therefore, the present ages are:
- Father's age (F) = 36
- Older son's age (S) = 8
- Younger son's age (Y) = 6
So, your answer is correct. The father is 36 years old, the older son is 8 years old, and the younger son is 6 years old.