The length and width of a rectangular display screen are in the ratio 5:2, respectively. When the length of the screen is increased by 25% while the width remains unchanged, the surface area of the screen increases by 160 sq m. Find the original length (in meters) of the display screen.

The correct answer is 40.

Step 1: Define the initial dimensions of the display screen using a common multiplier.
The original length and width of the display screen are given in the ratio 5:2.
Let the common multiplier be $x$.
Original length, $L=5x$
Original width, $W=2x$

Step 2: Calculate the original surface area.
The area of a rectangle is the product of its length and width.
$\text{Original Area}=L\times W=5x\times 2x=10x^2$

Step 3: Determine the new length and new surface area.
The length of the screen is increased by 25%, while the width remains unchanged.
New length, $L^{\prime }=5x+(0.25\times 5x)=1.25\times 5x=6.25x$
New width, $W^{\prime }=2x$
$\text{New Area}=L^{\prime }\times W^{\prime }=6.25x\times 2x=12.5x^2$

Step 4: Set up the equation using the given increase in surface area.
The surface area increases by 160 sq m.
$\text{Increase in Area}=\text{New Area}-\text{Original Area}$
$12.5x^2-10x^2=160$
$2.5x^2=160$

Step 5: Solve for $x$ and find the original length.
$x^2=\frac{160}{2.5}=64$
Taking the square root on both sides:
$x=\sqrt{64}=8$

Now, substitute $x=8$ to find the original length of the display screen:
$\text{Original length}=5x=5\times 8=40\text{meters}$

Thus, the original length of the display screen is 40 meters.