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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.

This problem explores how to determine how many times bigger b is than c using given ratios a:b and a:c. A step-by-step solution is provided, suitable for students in Grades 6-8.

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