# Thread: figuring out the age

1. ## figuring out the age

Maria is 6 times as old as Tina. In 20 years, Maria will be only twice as old as Tina. How old is Maria now?

2. Originally Posted by Judi
Maria is 6 times as old as Tina. In 20 years, Maria will be only twice as old as Tina. How old is Maria now?
I find the easiest way to do a problem is to write everything mathematically.

$\displaystyle M=\text{Maria}$
$\displaystyle T=\text{Tina}$

Next, write out what you know.

Right now Maria is 6 times as old as Tina, therefore...$\displaystyle M=6T$

In 20 years Maria will only be twice as old as Tina, therefore...$\displaystyle M+20=2(T+20)$

Now we solve both equations for $\displaystyle T$

$\displaystyle M=6T\quad\rightarrow\quad \frac{M}{6}=T$

$\displaystyle M+20=2(T+20)\quad\rightarrow\quad M+20=$$\displaystyle 2T+40\quad\rightarrow\quad M+20-$$\displaystyle 40=2T\quad\rightarrow\quad \frac{M-20}{2}=T$

Now we combine the two equations...

$\displaystyle \frac{M}{6}=\frac{M-20}{2}$ multiply by 6

$\displaystyle M=\frac{6(M-20)}{2}$ extend...

$\displaystyle M=\frac{3\cdot\not2(M-20)}{\not2}$ multiply by 3

$\displaystyle M=3M-60$ subtract $\displaystyle 3M$ from both sides...

$\displaystyle M-3M=-60\quad\rightarrow\quad-2M=-60\quad\rightarrow\quad \boxed{M=30}$

Voila!

3. Thank you,
Judi

4. Hello, Judi!

Here's another approach . . . with one variable.

Maria is 6 times as old as Tina.
In 20 years, Maria will be only twice as old as Tina.
How old is Maria now?

Let $\displaystyle x$ = Tina's age.

Since Maria's age is 6 times Tina's age,
. . then $\displaystyle 6x$ = Maria's age.

In 20 years, they will both be 20 years older.
. . Tina will be $\displaystyle x + 20$ years old.
. . Maria will be $\displaystyle 6x + 20$ years old.

In the future, Maria's age is twice Tina's age.
. . . . . . . . . . . . . .$\displaystyle \downarrow\qquad\;\downarrow\quad\downarrow\qquad \,\downarrow$
. . . . . . . . . . . .$\displaystyle 6x + 20 \;\;=\;\;2\cdot (x + 20)$

And there is your equation . . . go for it!

5. You guys are awesome!!!