Three friends Alex, Ben, and Casey presented a young mathematician with a puzzle. They told him that the sum of
their ages add up to 98 years. Double Alex’s age and add to Ben’s gives 84 years. The sum of three times Alex’s age
and twice Casey’s age is 163. Find the ages of Alex, Ben and Casey
The sum of their ages add up to 98 years --> a+b+c=98
Double Alex’s age and add to Ben’s gives 84 years
2a + b = 84
The sum of three times Alex’s age and twice Casey’s age is 163
---> 3a + 2c = 163
form 2a+b = 84 ---> b = 84-2a
I would now plug that into the first equation, to have only a and c
the third equation has only a and c
solve those, then back-substitute
To find the ages of Alex, Ben, and Casey, we can set up a system of equations based on the given information.
Let's use the variables:
A = Alex's age
B = Ben's age
C = Casey's age
Based on the information given, we can set up the following equations:
Equation 1: A + B + C = 98 (The sum of their ages is 98)
Equation 2: 2A + B = 84 (Double Alex's age and add to Ben's gives 84 years)
Equation 3: 3A + 2C = 163 (The sum of three times Alex's age and twice Casey's age is 163)
We now have a system of equations. We can solve them simultaneously to find the values of A, B, and C.
One way to solve this system of equations is by substitution:
From Equation 2, we can isolate B:
B = 84 - 2A
Substituting this value of B into Equation 1, we get:
A + (84 - 2A) + C = 98
84 - A + C = 98
C = 14 + A
Now we can substitute these expressions for B and C into Equation 3:
3A + 2(14 + A) = 163
3A + 28 + 2A = 163
5A = 163 - 28
5A = 135
A = 27
Now that we know A = 27, we can substitute it back into the expressions for B and C:
B = 84 - 2A = 84 - 2(27) = 84 - 54 = 30
C = 14 + A = 14 + 27 = 41
Therefore, the ages of Alex, Ben, and Casey are:
Alex = 27 years old
Ben = 30 years old
Casey = 41 years old