Suppose we know that Angela is thirty years younger than her father. Also, that in ten years, Angela will be half the age of her father at that time. Let’s determine Angela’s current age, and her father’s current age.
First, we ask "what quantities are we interested in, but are unknown?". In this example, we are interested in Angela’s current age and her father’s current age. To help us write these algebraically, let’s give these quantities variable names (and we need to be specific about what these variables stand for, so that readers, and ourselves, do not get confused.)
Let
\(A := \) Angela’s current age (in years), and let
\(F := \) Angela’s father’s current age (in years).
Since Angela is thirty years younger than her father, we need to think about how to translate this to an algebraic statement. We can think of trying to reword this sentence to have the concept of equality in it. For example, we could say "Angela’s age plus thirty years is the same age as her father" or "If you subtract thirty years from her father’s age, it will be Angela’s age" or something else. Using the first rewording, we can make the algebraic equation
\begin{equation*}
A+30 = F.
\end{equation*}
For the second sentence, we need to be careful; remember that this holds in ten years time. So Angela’s age in ten years is
\(A+10\) and her father’s is
\(F+10\text{.}\) Now we know that in ten years, Angela will be half the age of her father, which means if you double her age then, it will be equal to her father’s age. So we can write
\begin{equation*}
2 \times (A+10) = F+10.
\end{equation*}
Now, in our problem both equations are things that are true, so we can use both at the same time. Here we can use substitution to replace the
\(F\) in the second equation with
\(A+30\) since these quantities equal in the first equation. This way, we will end up with an equation with only one unknown/variable, which we can solve using the properties of arithmetic of equal expressions.
\begin{equation*}
2 \times (A+10) = (A+30)
\end{equation*}
Expanding the bracket on the left hand side:
\begin{equation*}
2A+20 = A+30
\end{equation*}
Subtracting
\(A\) and
\(20\) from both sides (this will have our variable only on one side and our constants only on the other)
\begin{equation*}
A = 10
\end{equation*}
So we know that Angela’s current age is
\(10\) years old. We can then use the first equation to find her father’s current age using substitution again:
\begin{equation*}
(10) + 30 = F
\end{equation*}
\begin{equation*}
40 = F
\end{equation*}
and thus Angela’s father’s current age is
\(40\) years old.
