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?

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- Jul 6th 2006, 01:09 PMJudifiguring 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?

- Jul 6th 2006, 01:23 PMQuickQuote:

Originally Posted by**Judi**

first, define your variables...

$\displaystyle M=\text{Maria}$

$\displaystyle T=\text{Tina}$

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! - Jul 6th 2006, 02:00 PMJudi
Thank you,

Judi - Jul 6th 2006, 02:02 PMSoroban
Hello, Judi!

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

Quote:

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!

- Jul 6th 2006, 02:11 PMJudi
You guys are awesome!!!