Question

A table is four times as long as it is wide. If it were 6 ft shorter and 6 ft​ wider, it would be square​ (with all sides​ equal). How long and how wide is the​ table?

Answer 1

Let the width of the table be denoted as \( w \). Then, since the table is four times as long as it is wide, the length \( l \) can be expressed as:

\[
l = 4w
\]

According to the problem, if the table were 6 feet shorter in length and 6 feet wider, it would become a square. This can be expressed with the equation:

\[
l - 6 = w + 6
\]

Now, substituting the expression for length \( l \):

\[
4w - 6 = w + 6
\]

To solve for \( w \), first, rearrange the equation:

\[
4w - w = 6 + 6
\]

This simplifies to:

\[
3w = 12
\]

Now, divide both sides by 3:

\[
w = 4
\]

Now that we have the width, we can find the length:

\[
l = 4w = 4 \times 4 = 16
\]

Therefore, the dimensions of the table are:

- Width: \( 4 \) ft
- Length: \( 16 \) ft

To verify, if the length is reduced by 6 ft and the width is increased by 6 ft:

- New Length: \( 16 - 6 = 10 \) ft
- New Width: \( 4 + 6 = 10 \) ft

Since the new dimensions are equal, the table indeed would be square in this scenario.

Thus, the dimensions of the table are:
- **Length: 16 ft**
- **Width: 4 ft**