Introduce x into a word problem and create an equation

Question

Jessica is six times as old as Sophie, who is two years older than Lucas. In six years, Jessica will be four times as old as Lucas. How old is Jessica?

I've been stuck on this question for awhile, I cant properly form the linear equation to be able to solve it correctly. I've tried $$6x + 2 = 6 \cdot 4$$ but I am not sure if that's correct

Answer 1

You have 3 equations and 3 unknowns

Jessica is six times as old as Sophie, $$J=6S$$ who is two years older than Lucas. $$S=L+2$$

In six years, Jessica will be four times as old as Lucas. $$ J+6=4(L+6) $$

Answer 2

It is always a good idea to define your variables when using them in an answer.

Let $x$ be Jessica's age. Let $y$ be Sophie's age. Let $z$ be Lucas' age.

Rewriting the english statements as mathematics gives:

$x=6y$ <-- Jessica is six times as old as Sophie.

$y=z+2$ <-- Sophie is two years older than Lucas.

$x+6=4(z+6)$ <-- In six years Sophie will be $x+6$. In six years Lucas will be $z+6$.

We can substitute Sophie's age $y$ into the first equation to get: $x=6(z+2)$.

Then we could add six to it to get: $x+6=6(z+2)+6$.

This makes the left hand side the same as the last equation so:

$6(z+2)+6=4(z+6)$

Expanding:

$6z+12+6=4z+24$

$6z+18=4z+24$

$2z = 6$

$z=3$

So Lucas is 3 years old. Therefore Sophie is 5 years old. And so Jessica is 30 years old.

We can check the solution by looking at the ages in six years time: Lucas will be 9 years old and Sophie will be 36 years old which is four times Lucas' age so we know we got the right answer.

Answer 3

When I was in grade 10 I used to find problems like these confusing sometimes ... then I discovered subscripts.

Another thing that I realized was that word problems are grammar problem, not math problems.

Consider

Jessica is six times as old as Sophie, who is two years older than Lucas.

In this sentence the adjective clause "who is two years older than Lucas" refers to Sophie. Buy Substituting "Sophie" for "who" we can split the sentence into two sentences.

Jessica is six times as old as Sophie. Sophie is two years older than Lucas.

The sentence

In six years, Jessica will be four times as old as Lucas.

refers to Jessica and Lucas in the future, so in the problem, there are two Jessicas and two Lucases: an old one and a young one. So, I'm going to change the wording a bit.

Old Jessica is six years older than young Jessica. Old Lucas is six years older than young Lucas. Old Jessica is four times as old as old Lucas.

Putting it all together we have

Jessica is six times as old as Sophie. Sophie is two years older than Lucas. Old Jessica is six years older than young Jessica. Old Lucas is six years older than young Lucas. Old Jessica is four times as old as old Lucas.

Which translates into $$\begin{array}{lll} x&=&J\\ J&=&6S\\ S&=&L+2\\ J_{old}&=&J+6\\ L_{old}&=&L+6\\ J_{old}&=&4L_{old}\\ \end{array}$$ Now, all we have to do is solve for $x$.