To find \( TL \) given \( WT = 53 \) and \( LW = 45 \), we can use the relationships we have.
Let's denote the values of each variable as follows:
- \( T \) for \( T \)
- \( W \) for \( W \)
- \( L \) for \( L \)
From the equations given, we know:
- \( W \cdot T = 53 \) (equation 1)
- \( L \cdot W = 45 \) (equation 2)
We want to find \( T \cdot L \) (equation 3).
Step 1: Solve for \( W \)
From equation 1, we can express \( W \) in terms of \( T \): \[ W = \frac{53}{T} \]
Step 2: Substitute \( W \) into equation 2
Now we substitute \( W \) from equation 1 into equation 2:
\[ L \cdot \left( \frac{53}{T} \right) = 45 \]
Step 3: Rearranging the equation
Multiply both sides by \( T \) to eliminate the fraction: \[ L \cdot 53 = 45T \]
Now we can solve for \( L \): \[ L = \frac{45T}{53} \]
Step 4: Finding \( T \cdot L \)
Now we find \( T \cdot L \) using the expression we derived: \[ T \cdot L = T \cdot \left( \frac{45T}{53} \right) \]
This simplifies to: \[ T \cdot L = \frac{45T^2}{53} \]
Step 5: Express \( TL \) in simpler terms
Since we don't have a specific numerical value for \( T \), \( TL \) in terms of \( T \) is expressed as: \[ TL = \frac{45T^2}{53} \]
Unfortunately, without an additional relationship or more information about \( T \) or \( L \), we cannot find a specific numerical value for \( TL \) directly.
However, we can check if there's a way to eliminate \( T \) or to calculate \( TL \) using the equations derived.
To compute \( TL \):
- If we multiply the product of the two equations together: \[ (WT) \cdot (LW) = (TL)(W) \]
Focusing on \( TL \): \[ (WT)(LW) = 53 \cdot 45 \]
\[ TL \cdot W = 53 \cdot 45 \]
Now to find \( W \):
- Using the expression for \( W \): \[ TL \cdot \left(\frac{53}{T}\right) = 53 \cdot 45 \]
Conclusion
Thus, we arrive at: \[ TL = \frac{53 \cdot 45 \cdot T}{53} \] This gives: \[ TL = 45T \] At this point, without loss of generality, we derived the result \( TL \) expressed in terms of \( T \). However, the exercise does not give a specific numerical value, confirming once again that direct computation of \( TL \) theoretically leads us to a relationship dependent on \( T \) without identifiable actual values.
Since the initial problem doesn't provide extra conditions or values, a complete numeric answer for \( TL \) is not quantifiable under current details in this approach, but the derived relationship suffices for analytical perspectives.