To solve the problem, let's use the given ratios to find out how many times bigger $b$ is than $c$.

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

• We are given the ratio $a : b = 1 : 7$. This means $b = 7a$.
• We are also given the ratio $a : c = 4 : 1$. This means $c = \frac{a}{4}$.

Step 2: Find the ratio $b : c$

To find how many times $b$ is bigger than $c$, we calculate the ratio $\frac{b}{c}$:

$\frac{b}{c} = \frac{7a}{\frac{a}{4}} = \frac{7a \times 4}{a} = \frac{28a}{a} = 28$

So, $b$ is 28 times bigger than $c$.

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:

1. What is the ratio $b : c$ when $a : b = 3 : 5$ and $a : c = 6 : 2$?
2. If $b = 3c$, what would be the ratio $b : c$?
3. How do ratios change if $a$ is doubled?
4. What is the relationship between three variables when two different ratios are given?
5. How would the solution change if $b = 2a$ instead of $b = 7a$?

Tip: When working with ratios, always express all variables in terms of one common variable to easily compare or find new ratios.