To solve for the ratio $a : b : c : d$ given the individual ratios:

1. Given Ratios:

   • $\frac{a}{b} = \frac{2/9}{1/3}$
   • $\frac{b}{c} = \frac{2/7}{5/14}$
   • $\frac{d}{c} = \frac{2/10}{3/5}$

2. Simplifying Each Ratio:

   • $a : b$: $\frac{a}{b} = \frac{\frac{2}{9}}{\frac{1}{3}} = \frac{2}{9} \times \frac{3}{1} = \frac{6}{9} = \frac{2}{3}$ So, $a : b = 2 : 3$.

   • $b : c$: $\frac{b}{c} = \frac{\frac{2}{7}}{\frac{5}{14}} = \frac{2}{7} \times \frac{14}{5} = \frac{28}{35} = \frac{4}{5}$ So, $b : c = 4 : 5$.

   • $d : c$: $\frac{d}{c} = \frac{\frac{2}{10}}{\frac{3}{5}} = \frac{2}{10} \times \frac{5}{3} = \frac{10}{30} = \frac{1}{3}$ So, $d : c = 1 : 3$.

3. Combining the Ratios:

   • Let's start by finding a common value for $b$ and $c$.

     • From $a : b = 2 : 3$ and $b : c = 4 : 5$, multiply the first ratio by 4 (the denominator of $b$ in the second ratio): $a : b = 8 : 12$
     • Multiply the second ratio by 3 (the denominator of $b$ in the first ratio): $b : c = 12 : 15$
     • Now $a : b : c = 8 : 12 : 15$.

   • Now include $d$:

     • We already have $c : d = 15 : 5 \times 3 = 15 : 5 \times 1 = 15 : 3$
     • $d$ corresponds to $1/3$ of $c$: $d = \frac{3 \times 15}{1} = 45$

4. Final Ratio: $a : b : c : d = 8 : 12 : 15 : 45$

Thus, the ratio $a : b : c : d$ is $8 : 12 : 15 : 45$.

Would you like to discuss any part of the solution in more detail or ask related questions?

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Here are some related questions:

1. How do we determine common multiples when combining ratios?
2. What is the significance of keeping consistent denominators while comparing ratios?
3. Can the method of proportion be applied to different types of ratios?
4. How would the solution change if one of the ratios was in decimal form?
5. What are other methods to solve for combined ratios?

Tip: When dealing with ratios, always simplify the fractions to the smallest terms for easier combination.