# How is this word problem converted into algebra?

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.

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 $X = 16$.

• I was typing an absolutely identical answer (same notations $X,Y,a$) when your answer appeared ! Cheers Commented Jan 4, 2015 at 7:40
• @ClaudeLeibovici It's amazing how such brilliant discoveries are sometimes made simultaneously.
– user147263
Commented Jan 22, 2015 at 14:27

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.

• Wow, that graphic is awesome! Commented Jan 5, 2015 at 8:13
• @Blakes7, thank you very much :) Commented Jan 5, 2015 at 16:01
• This is really a very nice approach ! May I ask how it came to your mind ? Cheers :-) Commented Jan 22, 2015 at 16:29
• @ClaudeLeibovici, Bonjour! Thank you very much Claude, it feels good when your effort is noticed and appreciated. I have a habit of thinking/showing/representing things/events/situations with analogies, may be visualising problem/solution pictorially is the result/extension/by-product of this habit. After drawing the diagram I wondered what is the shortest and simplest way to solve it?, and diagram itself gave me the method. :) bonne journée :) Commented Jan 23, 2015 at 8:19

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

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

• Ha. Downvote? Is my method incorrect? (seriously) Commented Jan 4, 2015 at 8:09
• Your method is correct, but the question asks for your reasoning to arrive at the algebraic equations and how it is converted into algebra. If I have misinterpreted when I should down vote an answer then I can reverse it if you like? Commented Jan 4, 2015 at 8:32
• I like the table - shows exactly what to write while reading the sentence. +1 Commented Jan 4, 2015 at 17:07

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.