Compound Ratio Calculation

This solution explains how to find the compound ratio for the given set of ratios: $45 : 75$, $3 : 5$, $51 : 68$, and $256 : 81$. The compound ratio is calculated by multiplying the first terms together and the second terms together.

Compound Ratio Definition

The compound ratio of several ratios $a : b$, $c : d$, $e : f$, ... is given by $(a \times c \times e \times ...) : (b \times d \times f \times ...)$.

Ratio Calculation Steps

1. Simplify individual ratios:

   • $45 : 75$ simplifies to $\frac{45}{75} = \frac{3 \times 15}{5 \times 15} = \frac{3}{5}$.
   • $3 : 5$ is already in simplest form, $\frac{3}{5}$.
   • $51 : 68$ simplifies to $\frac{51}{68} = \frac{3 \times 17}{4 \times 17} = \frac{3}{4}$.
   • $256 : 81$ is $\frac{256}{81}$.

2. Multiply the simplified ratios:

   Compound Ratio = $\frac{3}{5} \times \frac{3}{5} \times \frac{3}{4} \times \frac{256}{81}$

3. Calculate the product of numerators and denominators:

   Numerator Product = $3 \times 3 \times 3 \times 256 = 27 \times 256$

   Denominator Product = $5 \times 5 \times 4 \times 81 = 25 \times 4 \times 81 = 100 \times 81 = 8100$

   Compound Ratio = $\frac{27 \times 256}{8100}$

4. Simplify the resulting fraction:

   Divide numerator and denominator by common factors.

   Notice that $81 = 3 \times 27$. So, $\frac{27}{8100} = \frac{1}{3 \times 100} = \frac{1}{300}$.

   The ratio becomes $\frac{1 \times 256}{300} = \frac{256}{300}$.

   Simplify $\frac{256}{300}$ by dividing both by 4:

   $\frac{256 \div 4}{300 \div 4} = \frac{64}{75}$.

Resulting Compound Ratio

The compound ratio is $\frac{64}{75}$. This corresponds to Option A.