Based on the mathematical statements, Zoe is 56 years old and Luke is 66 years old, now
<h3>How to determine how old they are now?</h3>
From the question, we have the following statements that can be used in our computation:
<u>30 years ago</u>
Zoe was 2/3 as old as Luke
<u>18 years ago</u>
Zoe was 5/6 as old as Luke
Let their present ages be represented as
Zoe = x
Luke = y
So, we have the following representations
<u>30 years ago</u>
Zoe was 2/3 as old as Luke
x - 36 = 2/3(y - 36)
<u>18 years ago</u>
Zoe was 5/6 as old as Luke
x - 18 = 5/6(y - 18)
So, we have the following system of equations
x - 36 = 2/3(y - 36)
x - 18 = 5/6(y - 18)
Make x the subject in x - 18 = 5/6(y - 18)
x = 5/6(y - 18) + 16
Substitute x = 5/6(y - 18) + 16 in x - 36 = 2/3(y - 36)
5/6(y - 18) + 16 - 36 = 2/3(y - 36)
Open the brackets
5/6y - 15 + 16 - 36 = 2/3y - 24
Evaluate the like terms
5/6y - 35 = 2/3y - 24
Multiply through by 6
5y - 210 = 4y - 144
Evaluate the like terms
y = 66
Substitute y = 66 in x = 5/6(y - 18) + 16
x = 5/6(66 - 18) + 16
Evaluate
x = 56
Recall that
Zoe = x
Luke = y
So, we have
Zoe = x = 56
Luke = y = 66
Hence, they are 56 and 66 years, now
Read more about equations at
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, thus guaranteeing that every possible rational zero is an integer, which means that your friend is correct.