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Peter is 20 years old and Mary is 24 years old
Step-by-step explanation:
1. Mary is four years older than Peter:
M = P + 4
2. Peter’s age is five-sixths of Mary’s age:
[tex]P = \frac{5}{6}M[/tex]
We can substitute the first equation into the second equation to find the values of ( P ) and ( M ):
[tex]{ \text{Substitute }( M = P + 4 ) \: into \: ( P = \frac{5}{6}M ):}[/tex]
[tex]P = \frac{5}{6}(P + 4)[/tex]
[tex](\frac{5}{6})[/tex]
[tex]P = \frac{5}{6}P + \frac{5}{6} \cdot 4[/tex]
[tex]P = \frac{5}{6}P + \frac{20}{6}[/tex]
[tex]P = \frac{5}{6}P + \frac{10}{3}[/tex]
Next, subtract
[tex](\frac{5}{6}P)[/tex]
from both sides to isolate ( P ):
[tex]P - \frac{5}{6}P = \frac{10}{3}[/tex]
[tex]\frac{1}{6}P = \frac{10}{3}[/tex]
To solve for ( P ), multiply both sides by 6:
[tex]P = 6 \cdot \frac{10}{3}[/tex]
[tex]P = \boxed {20}[/tex]
Now that we know Peter's age (( P = 20 )), we can find Mary's age using ( M = P + 4 ):
[tex]M = 20 + 4[/tex]
[tex]M = \boxed {24}[/tex]