use a trial and error strategy to solve this problem.two years ago sam was two-thirds of pam's age. in three years sam will be three-fourths of pam's age. in 20 years the sum of their ages will be 69. how old is sam?
2 Years ago:
Pam: X yrs. old.
Sam: 2x/3 yrs. old.
3 yrs. later:
Pam: x+3 yrs. old.
Sam: 2x/3 + 3 yrs. old
2x/3 + 3 = 3/4(x+3).
2x/3 + 3 = 3x/4+9/4
8x + 36 = 9x + 27.
X = 9.
So Pam is 3 yrs. older than Sam.
In 20 yrs.:
x + (x+3) = 69.
2x = 66.
X = 33.
X+3 = 36.
Today:
Sam: 33 - 20 = 13 yrs. old.
To solve this problem using a trial and error strategy, we will try different values for Sam's age until we find the one that satisfies all the given conditions.
Let's start by setting up the equations based on the given information:
1) Two years ago, Sam was two-thirds of Pam's age:
Sam's age 2 years ago = Sam's age - 2
Pam's age 2 years ago = Pam's age - 2
So, we can write the equation as: Sam's age - 2 = (2/3) * (Pam's age - 2)
2) In three years, Sam will be three-fourths of Pam's age:
Sam's age in 3 years = Sam's age + 3
Pam's age in 3 years = Pam's age + 3
So, we can write the equation as: Sam's age + 3 = (3/4) * (Pam's age + 3)
3) In 20 years, the sum of their ages will be 69:
Sam's age in 20 years = Sam's age + 20
Pam's age in 20 years = Pam's age + 20
So, we can write the equation as: Sam's age + Pam's age + 20 = 69
Now, we can use trial and error to find the value of Sam's age that makes all the equations hold true:
Let's assume Sam's age is 30:
1) Sam's age - 2 = (2/3) * (Pam's age - 2)
30 - 2 = (2/3) * (Pam's age - 2)
28 = (2/3) * (Pam's age - 2)
Pam's age - 2 = (3/2) * 28
Pam's age - 2 = 42
Pam's age = 42 + 2 = 44
2) Sam's age + 3 = (3/4) * (Pam's age + 3)
30 + 3 = (3/4) * (44 + 3)
33 = (3/4) * 47
33 = 35.25 (Not true)
Since the equation does not hold true, we need to try a different value for Sam's age.
Let's assume Sam's age is 25:
1) Sam's age - 2 = (2/3) * (Pam's age - 2)
25 - 2 = (2/3) * (Pam's age - 2)
23 = (2/3) * (Pam's age - 2)
Pam's age - 2 = (3/2) * 23
Pam's age - 2 = 34.5 (Not a whole number)
Again, the equation does not hold true, so we need to try another value.
Let's assume Sam's age is 20:
1) Sam's age - 2 = (2/3) * (Pam's age - 2)
20 - 2 = (2/3) * (Pam's age - 2)
18 = (2/3) * (Pam's age - 2)
Pam's age - 2 = (3/2) * 18
Pam's age - 2 = 27
Pam's age = 27 + 2 = 29
2) Sam's age + 3 = (3/4) * (Pam's age + 3)
20 + 3 = (3/4) * (29 + 3)
23 = (3/4) * 32
23 = 24 (Not true)
Once again, the equation does not hold true, so we need to try another value.
Let's assume Sam's age is 15:
1) Sam's age - 2 = (2/3) * (Pam's age - 2)
15 - 2 = (2/3) * (Pam's age - 2)
13 = (2/3) * (Pam's age - 2)
Pam's age - 2 = (3/2) * 13
Pam's age - 2 = 19.5 (Not a whole number)
Again, the equation does not hold true, so we need to try another value.
Let's assume Sam's age is 10:
1) Sam's age - 2 = (2/3) * (Pam's age - 2)
10 - 2 = (2/3) * (Pam's age - 2)
8 = (2/3) * (Pam's age - 2)
Pam's age - 2 = (3/2) * 8
Pam's age - 2 = 12
Pam's age = 12 + 2 = 14
2) Sam's age + 3 = (3/4) * (Pam's age + 3)
10 + 3 = (3/4) * (14 + 3)
13 = (3/4) * 17
13 = 12.75 (Not true)
Once again, the equation does not hold true, so we need to try another value.
Let's assume Sam's age is 5:
1) Sam's age - 2 = (2/3) * (Pam's age - 2)
5 - 2 = (2/3) * (Pam's age - 2)
3 = (2/3) * (Pam's age - 2)
Pam's age - 2 = (3/2) * 3
Pam's age - 2 = 4.5 (Not a whole number)
This value also does not satisfy the equation.
Therefore, after trying different values, we see that we cannot find a whole number value for Sam's age that satisfies all the given conditions. It is possible that there is no whole number solution to this problem.