# Christian is $3$ times as old as Marie. Marie is $12$ years younger - Is there a simple method for elementary school student?

Christian is 3 times as old as Marie. Marie is $12$ years younger than Christian. How old is Christian?

The method that I know requires $2$ variables and $2$ equations.

\begin{align}C &= 3M \\ M &= C - 12 \end{align}

Substitute first equation into second equation and solve

\begin{align}M &= 3M - 12\\ 12 &= 2M \\ M &= 6 \text{ years} \\ C &= 3M ( 3 \times 6 ) = 18\text{ years}\end{align}

Is there a simpler way to explain this to a $5$th grader?

He still hasn't learned math equations with $2$ variables or how to substitute first equation into second equation.

I would consider the picture below:

Drawing $3$ blocks for age of Christian an a single block for the age of Marie.

Marie is $12$ years younger, so each block represent $6$ years.

Hence, Marie is $6$ years old and Christina is $3\times 6=18$ years old.

• Thank you so much for this solution.I was hoping there is an age appropriate method.This really helps. – Pearl Feb 18 '18 at 7:34
• you are welcome. glad it helps. – Siong Thye Goh Feb 18 '18 at 7:38

One idea that comes to mind is making a list of ages that are twelve years apart and then seeing which pair is the first to have one age be three times the other, like this: $$\begin{array}{c|lcr} \text{Marie's age} & \text{Christian's age} \\ \hline 1 & \qquad\ \ \ 13 \\ 2 & \qquad\ \ \ 14 \\ 3 & \qquad\ \ \ 15 \\ 4 & \qquad\ \ \ 16 \\ 5 & \qquad\ \ \ 17 \\ 6 & \qquad\ \ \ 18 \end{array}$$

Suppose Marie were $\,1\,$ year old. Then Christian would be $\,3 \cdot 1 = 3\,$ years old, and the difference between their ages would be $\,3 - 1 = \color{blue}{2}\,$ years. But the actual difference is $\,12 = \color{red}{6} \cdot \color{blue}{2}\,$ years, so Marie must actually be $\,\color{red}{6} \cdot 1 = 6\,$ years old, and then Christian is $\,6\cdot 3 =18\,$.

The above obviously plays on the linearity of the problem, without spelling it out as such. It is important to teach that such tricks do not always work, yet elementary school students can quickly develop an intuition of sorts for linearity, even if it's not formally called by that name.

Let $M$ be Marie's age.

We know Christian is $3$ times as old as Marie, so Christian's age is $3M.$

But we also know Marie is $12$ years younger than Christian. That is, $M = 3M - 12.$

Now you have an equation in one variable. Solve it. That gives you $M.$ Then multiply by $3,$ because Christian's age is $3M.$

Of course you can assign a new symbol $C$ to represent Christian's age, in addition to finding that Christian's age is $3M,$ if the additional symbol makes it easier for you to understand the method of solving the problem. But there is no requirement to assign such a symbol.