This problem asks us to find the value of an algebraic expression involving three variables, \(x\), \(y\), and \(z\), given their ratio in fractional form. To solve this, we first need to simplify the given ratio into whole numbers and then substitute these proportional values into the expression.

Ratio Simplification of x:y:z

The given ratio is \(x : y : z = \frac{1}{2} : \frac{1}{3} : \frac{1}{4}\).

To convert this fractional ratio into a simple whole number ratio, we need to find the Least Common Multiple (LCM) of the denominators (2, 3, and 4).

• The multiples of 2 are 2, 4, 6, 8, 10, 12, ...
• The multiples of 3 are 3, 6, 9, 12, ...
• The multiples of 4 are 4, 8, 12, ...

The LCM of 2, 3, and 4 is 12.

Now, multiply each part of the ratio by the LCM (12):

\(x : y : z = \left(\frac{1}{2} \times 12\right) : \left(\frac{1}{3} \times 12\right) : \left(\frac{1}{4} \times 12\right)\)

This simplifies to:

\(x : y : z = 6 : 4 : 3\)

Variable Representation from Ratio

From the simplified ratio \(x : y : z = 6 : 4 : 3\), we can represent \(x\), \(y\), and \(z\) in terms of a common proportionality constant, let's call it \(k\). This means:

• \(x = 6k\)
• \(y = 4k\)
• \(z = 3k\)

Here, \(k\) can be any non-zero real number. The actual value of \(k\) does not affect the final ratio or the value of the expression, as it will cancel out during the calculation.

Expression Evaluation for (x+z-y)/y

We need to find the value of the expression \(\frac{x + z - y}{y}\).

Substitute the proportional values of \(x\), \(y\), and \(z\) into the expression:

\(\frac{x + z - y}{y} = \frac{6k + 3k - 4k}{4k}\)

Combine the terms in the numerator:

\(\frac{6k + 3k - 4k}{4k} = \frac{(6 + 3 - 4)k}{4k}\)

Perform the arithmetic in the numerator:

\(\frac{(9 - 4)k}{4k} = \frac{5k}{4k}\)

Since \(k\) is a non-zero constant, it cancels out from the numerator and the denominator:

\(\frac{5k}{4k} = \frac{5}{4}\)

Convert the fraction to a decimal:

\(\frac{5}{4} = 1.25\)

Therefore, the value of \(\frac{x + z - y}{y}\) is 1.25.