How is this word problem converted into algebra?

Question

Yvonne is 12. Xavier is twice as old as Yvonne was when Xavier was as old as Yvonne is now. How old is Xavier?

The answer given is: Xavier is 16.

How is this word problem converted into algebra? What is your reasoning to arrive at such algebra equations?

This "Puzzle" comes from the book "Theta Mathematics" by David Barton. 2011 edition. Page 224. This is not a homework problem. I just study maths for the fun of it.

Answer 1

Xavier is some age $X$, Yvonne some age $Y$. When Xavier was as old as Yvonne is now, he was 12. So Xavier's age is 12 plus some other number of years, say $X = 12 + a$ years. So when Xavier was 12, that was $a$ many yeas ago, meaning Yvonne was $12 - a$ years old at the time.

So Xavier is twice as old as that, says the question. Therefore $$ X = 2 (12 - a) $$ but we know already that $X = 12 + a$, so $$ 12 + a = 2 (12 - a) \\ \implies a = 12 - 2a \\ \implies 3a = 12 \\ \implies a = 4 $$

So $X = 12 + a = 12 + 4$, meaning $X = 16$.

Answer 2

The sentences can be written as such $$Y=12\\X=2(12-(X-12))\\X=2(24-X)\\3X=48$$

Reasoning: $$\begin{array}{|c|c|} \hline \text{Text}& \text{Equation} \\ \hline \text{Xavier is} & X= \\ \hline \text{twice as old} & 2 \\ \hline \text{as Yvonne was} & (12 \\ \hline \text{when Xavier was as old as Yvonne is now} & - (X-12)) \\ \hline \end{array}$$

Answer 3

The idea is to choose your variables wisely: choose variables according to the entities mentioned in the wording. Then translate word by word to equations. After that stop thinking and start running your algorithm you learned for solving these equations.

Another approach would be with three variables, where the terms are closer to the original phrasing and where you need to "think" less but "calculate" more. Generaly fact: if a person is born in year $b$, then in some other year $a$ the person is aged $a - b$.

Let $X$ and $Y$ be the years of birth of Xaver and Yvonne respectively. And let $N$ be the year now.

The we have: Yvonne is 12 now: $$ N - Y = 12 \Leftrightarrow Y = N - 12 $$

The year "when Xavier was as old as Yvonne is now" is $$ X + 12 $$ In this year, Yvonne was $$ (X + 12) - Y $$ "Now, Xavier is twice as old as Yvonne was back then" $$ N - X = 2 \cdot ( X + 12 - Y) $$

And from this position on, we only need to calculate: $$ N -X = 2\cdot X + 24 - 2\cdot Y $$ Resolving this for the birth year of Xaver: $$3\cdot X = 24 - 2\cdot Y - N = 24 - 2\cdot (N - 12) - N \\ = 24 - 2\cdot N + 24 -N = 48 - 3\cdot N \\ \Leftrightarrow X = 16 - N \\ \Leftrightarrow N - X = 16 $$ So Xaver is indeed 16.

Though shorter solutions were posted already, I hope you can see the translation quite clear here.

Answer 4

Pictorially, just for fun

Same number of years have passed between "now" and "some years back", so

$$12-x=2x-12$$

$$3x=24$$ $$x=8$$

Thus, Xavier is $2x=2\times8=16$ years old now and we are comparing their age now and $4$ years back.