Solve the following mathematical word problem.

Two servers, X and Y, operate at 52% and 67% less capacity than the maximum capacity of a data center, respectively. Server X operates at 120 units above the standard threshold capacity, whereas Server Y operates at 30 units below the standard threshold capacity. If Server Z operates at 54% of the maximum capacity, by how many units does Z exceed the standard threshold capacity?

The correct answer is 180.

Let us break down the problem step-by-step to find the required capacity difference for Server Z.

First, let $M$ be the maximum capacity of the data center, and let $T$ be the standard threshold capacity.

We are given the operating capacities of Server X and Server Y in terms of $M$:

Server X operates at 52% less capacity than $M$:

$C_X=(1-0.52)M=0.48M$

Server Y operates at 67% less capacity than $M$:

$C_Y=(1-0.67)M=0.33M$

Next, we write the operating capacities of X and Y in terms of the standard threshold capacity $T$:

Server X operates at 120 units above threshold:

$C_X=T+120$

Server Y operates at 30 units below threshold:

$C_Y=T-30$

Equating the expressions for $C_X$ and $C_Y$:

$0.48M=T+120$

$0.33M=T-30$

Subtracting the second equation from the first to eliminate $T$:

$(0.48M-0.33M)=(T+120)-(T-30)$

$0.15M=150$

Solving for $M$:

$M=\frac{150}{0.15}=1000$

Now, we can find the standard threshold capacity $T$ using Server Y's capacity equation:

$T=0.33M+30=0.33\left(1000\right)+30=330+30=360$

Server Z operates at 54% of the maximum capacity $M$:

$C_Z=0.54\times 1000=540$

Finally, we calculate by how many units Server Z exceeds the standard threshold capacity $T$:

$C_Z-T=540-360=180$

Therefore, Server Z exceeds the standard threshold capacity by 180 units.