Let's solve the problem step by step.

Problem:

We are given:

1. $a : b = 5 : 4$
2. $b : c = 2 : 5$

We need to find $a : b : c$.

Step-by-Step Solution:

1. Express the given ratios:

   • $a : b = 5 : 4$ implies $\frac{a}{b} = \frac{5}{4}$.
   • $b : c = 2 : 5$ implies $\frac{b}{c} = \frac{2}{5}$.

2. Combine the ratios using the common term $b$:

   • The value of $b$ must be the same in both ratios. For $a : b$, $b = 4k$, and for $b : c$, $b = 2m$. Let’s find a common multiple of 4 and 2. The least common multiple is 4.
   • Set $b = 4$ (common term).

3. Scale the ratios accordingly:

   • From $a : b = 5 : 4$, scale to make $b = 4$. This gives: $a = 5, \, b = 4$
   • From $b : c = 2 : 5$, scale to make $b = 4$. This gives: $c = \frac{5}{2} \cdot 4 = 10$

4. Combine $a : b : c$:

   • Combine all the terms: $a = 5$, $b = 4$, $c = 10$.
   • Thus, $a : b : c = 5 : 4 : 10$.

Final Answer:

The correct option is: $5 : 4 : 10$.

Would you like a detailed explanation of any step?

Related Questions:

1. How do you simplify ratios when three variables are involved?
2. Can ratios be represented as fractions? How does that help in solving problems?
3. What happens when one ratio is in decimals instead of integers? How do you handle it?
4. Why is finding the least common multiple (LCM) important when combining ratios?
5. How can you generalize this method for more than three terms (e.g., $a : b : c : d$)?

Tip:

Always make sure to equalize the common term in ratios before combining them.