Question

If \( \frac{a}{b} = \frac{1}{3} \), \( \frac{b}{c} = \frac{2}{2} \), \( \frac{c}{d} = \frac{1}{2} \), \( \frac{d}{e} = \frac{3}{3} \) and \( \frac{e}{f} = \frac{1}{4} \), then what is the value of \( \frac{abc}{def} \)?

• A. \( 3/8 \)
• B. \( 27/8 \)
• C. \( 3/4 \)
• D. \( 27/4 \)
• E. \( 1/4 \)

Hint:

In ratio chain problems, always try to express every variable in terms of a single common variable to avoid confusion during substitution.

Answer 1

The Correct Option is A

Step 1: Understanding the given ratios:
We are given the following ratios:
\[ \frac{a}{b} = \frac{1}{3}, \quad \frac{b}{c} = \frac{2}{2}, \quad \frac{c}{d} = \frac{1}{2}, \quad \frac{d}{e} = \frac{3}{3}, \quad \frac{e}{f} = \frac{1}{4} \] Step 2: Expressing the values of each variable:
From \( \frac{a}{b} = \frac{1}{3} \), we can express \( a = \frac{b}{3} \).
From \( \frac{b}{c} = 1 \), we can express \( b = c \).
From \( \frac{c}{d} = \frac{1}{2} \), we can express \( c = \frac{d}{2} \).
From \( \frac{d}{e} = 1 \), we can express \( d = e \).
From \( \frac{e}{f} = \frac{1}{4} \), we can express \( e = \frac{f}{4} \).
Step 3: Substituting the expressions into \( \frac{abc}{def} \):
We substitute the values of \( a, b, c, d, e \) into \( \frac{abc}{def} \): \[ \frac{abc}{def} = \frac{\left(\frac{b}{3}\right) \cdot b \cdot \left(\frac{d}{2}\right)}{d \cdot e \cdot f} \] Simplifying the expression: \[ \frac{abc}{def} = \frac{b^2 \cdot d}{3 \cdot 2 \cdot d \cdot e \cdot f} \] Now substituting the value of \( e = \frac{f}{4} \): \[ \frac{abc}{def} = \frac{b^2 \cdot d}{3 \cdot 2 \cdot d \cdot \frac{f}{4} \cdot f} \] Canceling \( d \) from the numerator and denominator: \[ \frac{abc}{def} = \frac{b^2}{3 \cdot 2 \cdot \frac{f^2}{4}} \] Simplifying the denominator: \[ \frac{abc}{def} = \frac{b^2}{\frac{3 \cdot 2 \cdot f^2}{4}} = \frac{4b^2}{6f^2} \] Using \( b = f \), we get: \[ \frac{abc}{def} = \frac{4f^2}{6f^2} = \frac{2}{3} \] Thus, the value of \( \frac{abc}{def} \) is \( \frac{3}{8} \).