The problem given is:

$\frac{a}{b} = \frac{2}{5} \quad \text{and} \quad \frac{b}{c} = \frac{3}{4}$

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

Step 1: Express $a$ in terms of $b$

From the first equation, we have: $\frac{a}{b} = \frac{2}{5} \implies a = \frac{2}{5}b$

Step 2: Express $b$ in terms of $c$

From the second equation, we have: $\frac{b}{c} = \frac{3}{4} \implies b = \frac{3}{4}c$

Step 3: Substitute $b$ in terms of $c$ into the expression for $a$

Now, substitute $b = \frac{3}{4}c$ into the expression $a = \frac{2}{5}b$: $a = \frac{2}{5} \times \frac{3}{4}c = \frac{6}{20}c = \frac{3}{10}c$

Step 4: Find the ratio $a : b : c$

Now we have the following:

• $a = \frac{3}{10}c$
• $b = \frac{3}{4}c$
• $c = c$

To express the ratio $a : b : c$, we need to make sure all the terms are proportional to $c$.

The least common denominator between 10 and 4 is 20. So we rewrite the terms as: $a = \frac{3}{10}c = \frac{6}{20}c, \quad b = \frac{3}{4}c = \frac{15}{20}c, \quad c = \frac{20}{20}c$

Thus, the ratio is: $a : b : c = 6 : 15 : 20$

Would you like further clarification or additional details?

Here are five related questions to deepen your understanding:

1. How do we solve ratios when more than two variables are involved?
2. What happens if the ratios involve square roots or exponents?
3. How can we simplify ratios with negative values or decimals?
4. Can we solve for individual values of $a$, $b$, or $c$ if given additional information?
5. How are proportions useful in real-life applications, such as scaling models or recipes?

Tip: When solving ratio problems, it's helpful to find the least common denominator to express all terms in the same form.