If $a : b = 7 : 9$ and $b : c = 15 : 7$, then what is $a : c$?

### Concept & Compounding Ratios To find the direct ratio between the first and last variables in a chain of ratios, you can multiply the corresponding fractions. $$ \frac{a}{c} = \frac{a}{b} \times \frac{b}{c} $$ ### Step-by-Step Solution 1. Convert the given ratios into fractions: $\frac{a}{b} = \frac{7}{9}$ $\frac{b}{c} = \frac{15}{7}$ 2. We want to find $a : c$, which is equivalent to $\frac{a}{c}$. 3. Multiply the two fractions together. The $b$ terms will cancel out: $\frac{a}{c} = \left(\frac{7}{9}\right) \times \left(\frac{15}{7}\right)$ 4. Simplify the expression by cancelling the common factor of $7$ in the numerator and denominator: $\frac{a}{c} = \frac{15}{9}$ 5. Reduce the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is $3$: $\frac{a}{c} = \frac{5}{3}$ 6. Convert back to ratio format: $a : c = 5 : 3$ ### Exam Strategy & Shortcut Write the numbers down and immediately cancel out common values diagonally. In $\frac{7}{9} \times \frac{15}{7}$, the $7$s visually cancel out immediately leaving $\frac{15}{9}$, which simplifies to $5 : 3$ in seconds. ### Common Pitfall Attempting to equalize the middle term $b$ in both ratios (finding a common multiple for $9$ and $15$) is a valid method but takes significantly more time than simple multiplication. ### Final Answer Therefore, the correct answer is **$5 : 3$**.