Linda is twice as old as Vera. Tanya is four less than four times the age of Linda. Their total age is two more than nine times the age of Vera. How old is Tanya?

1 answer

Let \( V \) represent Vera's age. Then, Linda's age \( L \) can be expressed as:

\[ L = 2V \]

Tanya's age \( T \) can be expressed as:

\[ T = 4L - 4 \]

Substituting the expression for Linda's age into Tanya's age gives:

\[ T = 4(2V) - 4 = 8V - 4 \]

The total age of Linda, Vera, and Tanya can be expressed as:

\[ L + V + T \]

Substituting the expressions we have for \( L \) and \( T \):

\[ L + V + T = 2V + V + (8V - 4) \] \[ = 2V + V + 8V - 4 = 11V - 4 \]

According to the problem statement, this total age is also equal to two more than nine times Vera's age:

\[ 11V - 4 = 9V + 2 \]

To solve for \( V \), we first isolate \( V \) by moving the terms involving \( V \) to one side:

\[ 11V - 9V = 2 + 4 \] \[ 2V = 6 \] \[ V = 3 \]

Now that we have Vera's age, we can find Linda's age:

\[ L = 2V = 2(3) = 6 \]

Next, we can find Tanya's age using the expression for \( T \):

\[ T = 8V - 4 = 8(3) - 4 = 24 - 4 = 20 \]

Thus, Tanya's age is:

\[ \boxed{20} \]

To verify, we can check the total age calculation:

  • Vera's age: \( 3 \)
  • Linda's age: \( 6 \)
  • Tanya's age: \( 20 \)

Total age = \( 3 + 6 + 20 = 29 \).

Now check the condition \( 9V + 2 \):

\[ 9(3) + 2 = 27 + 2 = 29 \]

Since both totals match, the calculations confirm that Tanya's age is indeed \( \boxed{20} \).