Paula, Dianna and Annie are three friends. Paula is twice as old as Dianna, and in 2 years she will be three times as old as Annie. If Dianna is 5 years older than Annie, how old is Paula?
please use algebra to justify your answer.
p = 2d
p+2 = 3(a+2)
d = a+5
p+2 = 3(d-5+2) = 3(p/2 - 3)
p+2 = 3/2 p - 9
p/2 = 11
p = 22
To solve this problem algebraically, let's assign variables to the ages of the three friends. Let's say:
Let A be Annie's age
Let D be Dianna's age
Let P be Paula's age
Based on the given information, we can establish the following equations:
1. Paula is twice as old as Dianna:
P = 2D
2. In 2 years, Paula will be three times as old as Annie:
P + 2 = 3(A + 2)
3. Dianna is 5 years older than Annie:
D = A + 5
Now, we can substitute the value of D from equation 3 into equations 1 and 2:
From equation 1:
P = 2(A + 5)
From equation 2:
2(A + 5) + 2 = 3(A + 2)
Simplifying equation 2:
2A + 10 + 2 = 3A + 6
2A + 12 = 3A + 6
Bringing like terms together:
3A - 2A = 12 - 6
A = 6
Now, substitute the value of A back into equation 3 to find D:
D = A + 5
D = 6 + 5
D = 11
And finally, substitute the value of D into equation 1 to find P:
P = 2D
P = 2 * 11
P = 22
Therefore, Paula is 22 years old.