Question

If \( x : y : z = \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \), what is the value of \( \frac{x + z - y}{y} \)?

• A. 0.75
• B. 1.25
• C. 2.25
• D. 3.25

Hint:

When dealing with ratios, express each term in terms of a common constant, then simplify the given expression step-by-step.

Answer 1

The Correct Option is B

We are given the ratio \( x : y : z = \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \), which means we can express the values of \( x \), \( y \), and \( z \) in terms of a common variable. Let’s solve for the ratio using the simplest method:
Step 1: Expressing the variables in terms of a common constant
We can write the ratios as:
\[ x = \frac{1}{2}k, \quad y = \frac{1}{3}k, \quad z = \frac{1}{4}k \] where \( k \) is a constant.
Step 2: Substituting into the given expression
We are asked to find the value of \( \frac{x + z - y}{y} \). Substituting the values of \( x \), \( y \), and \( z \) into this expression:
\[ \frac{x + z - y}{y} = \frac{\frac{1}{2}k + \frac{1}{4}k - \frac{1}{3}k}{\frac{1}{3}k} \] Step 3: Simplifying the expression
First, simplify the numerator:
\[ \frac{1}{2}k + \frac{1}{4}k - \frac{1}{3}k = \left(\frac{2}{4} + \frac{1}{4} - \frac{1}{3}\right)k \] \[ = \left(\frac{3}{4} - \frac{1}{3}\right)k \] To subtract the fractions, get a common denominator:
\[ = \left(\frac{9}{12} - \frac{4}{12}\right)k = \frac{5}{12}k \] Now, the expression becomes:
\[ \frac{\frac{5}{12}k}{\frac{1}{3}k} \] Step 4: Final simplification
Simplify the fraction:
\[ \frac{\frac{5}{12}k}{\frac{1}{3}k} = \frac{5}{12} \times \frac{3}{1} = \frac{15}{12} = 1.25 \] Thus, the value of \( \frac{x + z - y}{y} \) is 1.25, making (B) the correct answer.