1. Stating the problem: We are given three ratios involving four variables a, b, c, and d. - $a : b = 4 : 5$ - $b : c = 3 : 4$ - $c : d = 6 : 7$ We want to find the combined ratio $a : b : c : d$. 2. Start by expressing each ratio with variables in terms of a common variable, so we can combine them. 3. From $a : b = 4 : 5$, we have $\frac{a}{b} = \frac{4}{5}$ or $a = \frac{4}{5}b$. 4. From $b : c = 3 : 4$, we have $\frac{b}{c} = \frac{3}{4}$ or $b = \frac{3}{4}c$. 5. From $c : d = 6 : 7$, we have $\frac{c}{d} = \frac{6}{7}$ or $c = \frac{6}{7}d$. 6. We want to express all variables in terms of $d$ to find the combined ratio: - From step 5: $c = \frac{6}{7}d$ - From step 4: $b = \frac{3}{4}c = \frac{3}{4} \times \frac{6}{7}d = \frac{18}{28}d = \frac{9}{14}d$ - From step 3: $a = \frac{4}{5}b = \frac{4}{5} \times \frac{9}{14}d = \frac{36}{70}d = \frac{18}{35}d$ 7. Now we have: - $a = \frac{18}{35}d$ - $b = \frac{9}{14}d$ - $c = \frac{6}{7}d$ - $d = d$ 8. To get whole number ratios, multiply each by the least common multiple (LCM) of denominators 35, 14, 7 which is 70: - $a = \frac{18}{35}d \times 70 = 36d$ - $b = \frac{9}{14}d \times 70 = 45d$ - $c = \frac{6}{7}d \times 70 = 60d$ - $d = d \times 70 = 70d$ 9. Therefore, the combined ratio is: $$a : b : c : d = 36 : 45 : 60 : 70$$ 10. Simplify by dividing all terms by their greatest common divisor (GCD), which is 1 here, so the ratio remains the same. Final answer: $$a : b : c : d = 36 : 45 : 60 : 70$$