Andrea is 6 years younger than Tony. In two years, Tony will be three times as old as Andrea. Let a be Tony's current age. Which of these linear equations can be used to find Andrea's and Tony's current ages?

Question

A. a + 2 = 3(a - 6 + 2)
B. a + 2 = 3(a + 6 + 2)
C. a = 3(a + 2)
D. 3(a + 2) = a - 6 + 2

Answer 1

Let Andrea's current age be represented as \( a - 6 \) since she is 6 years younger than Tony.

In two years, Tony's age will be \( a + 2 \) and Andrea's age will be \( (a - 6) + 2 = a - 4 \).

According to the problem, in two years, Tony will be three times as old as Andrea. We can express this as an equation:

\[ a + 2 = 3((a - 6) + 2) \]

Now, simplifying the right side of the equation:

\[ a + 2 = 3(a - 4) \]

This can be rewritten as:

\[ a + 2 = 3a - 12 \]

Now, re-arranging the equation gives:

\[ 2 + 12 = 3a - a \] \[ 14 = 2a \] \[ a = 7 \]

So Tony's current age \( a = 7 \) and Andrea's current age is \( 7 - 6 = 1 \).

Looking at the answer choices, we started with the equation:

\[ a + 2 = 3(a - 6 + 2) \]

This matches the first option.

Thus, the correct answer is:

A. \( a + 2 = 3(a - 6 + 2) \)

Answer 2

are you sure this is correct

Answer 3

Let's go through the problem carefully once more to ensure we find the correct equation.

1. Identifying the Variables:

   • Let \( a \) be Tony's current age.
   • Therefore, Andrea's current age is \( a - 6 \) (since Andrea is 6 years younger than Tony).

2. Aging Process:

   • In 2 years, Tony will be \( a + 2 \).
   • In 2 years, Andrea will be \( (a - 6) + 2 = a - 4 \).

3. Setting Up the Equation:

   • According to the problem, in 2 years, Tony's age will be three times Andrea's age: \[ a + 2 = 3(a - 4) \]

4. Solve the Equation:

   • Distributing on the right: \[ a + 2 = 3a - 12 \]
   • Rearranging gives: \[ 2 + 12 = 3a - a \] \[ 14 = 2a \] \[ a = 7 \]

So, Tony's current age is 7 and Andrea's current age is \( 7 - 6 = 1 \).

Now let's compare the derived equation to the given choices:

1. Choice A: \( a + 2 = 3(a - 6 + 2) \)

   • This translates to: \( a + 2 = 3(a - 4) \).
   • This is indeed correct.

2. Choice B: \( a + 2 = 3(a + 6 + 2) \)

   • This does not match the setup.

3. Choice C: \( a = 3(a + 2) \)

   • This doesn't fit either.

4. Choice D: \( 3(a + 2) = a - 6 + 2 \)

   • This does not correspond to the aging setup.

Thus, the correct equation that accurately reflects the relationships described in the problem is indeed:

A. \( a + 2 = 3(a - 6 + 2) \)

Thank you for your patience!