If one star equals four circles and three circles equal four diamonds, then what is the ratio of star to diamond?

### Concept & Algebraic Substitution When given word equations linking different entities, convert them into algebraic equations. To find the direct relationship between the first and last entity, create fractional ratios and multiply them together. $$ \frac{\text{Star}}{\text{Diamond}} = \frac{\text{Star}}{\text{Circle}} \times \frac{\text{Circle}}{\text{Diamond}} $$ ### Step-by-Step Solution 1. Let $S$ represent a star, $C$ represent a circle, and $D$ represent a diamond. 2. Translate the first statement: "one star equals four circles" $1S = 4C \Rightarrow \frac{S}{C} = \frac{4}{1}$ 3. Translate the second statement: "three circles equal four diamonds" $3C = 4D \Rightarrow \frac{C}{D} = \frac{4}{3}$ 4. We need the ratio of star to diamond, which is $\frac{S}{D}$. 5. Multiply the two fractional ratios: $\frac{S}{D} = \left(\frac{S}{C}\right) \times \left(\frac{C}{D}\right)$ $\frac{S}{D} = \left(\frac{4}{1}\right) \times \left(\frac{4}{3}\right)$ 6. Perform the multiplication: $\frac{S}{D} = \frac{16}{3}$ 7. Convert back to ratio format: $S : D = 16 : 3$ ### Exam Strategy & Shortcut Treat the objects as variables immediately. You have $S/C = 4/1$ and $C/D = 4/3$. Multiplying them straight across to cancel $C$ gives $(4 \times 4) / (1 \times 3) = 16/3$. ### Common Pitfall A frequent error is incorrectly setting up the initial fractions. For instance, reading $1S = 4C$ and mistakenly writing the ratio $S/C$ as $1/4$ instead of $4/1$. Always double-check the isolated ratio. ### Final Answer Therefore, the correct answer is **$16 : 3$**.