To determine how many times bigger [tex]\( b \)[/tex] is than [tex]\( c \)[/tex], we can proceed step-by-step by first understanding the given ratios and then using them to find the relationship between [tex]\( b \)[/tex] and [tex]\( c \)[/tex].
1. Given Ratios:
- The ratio of [tex]\( a \)[/tex] to [tex]\( b \)[/tex] is [tex]\( 1 : 6 \)[/tex].
- The ratio of [tex]\( a \)[/tex] to [tex]\( c \)[/tex] is [tex]\( 3 : 1 \)[/tex].
2. Express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex]:
- The ratio [tex]\( a : b = 1 : 6 \)[/tex] implies that [tex]\( b \)[/tex] is 6 times [tex]\( a \)[/tex].
- Therefore, we can write [tex]\( b = 6a \)[/tex].
3. Express [tex]\( c \)[/tex] in terms of [tex]\( a \)[/tex]:
- The ratio [tex]\( a : c = 3 : 1 \)[/tex] implies that [tex]\( a \)[/tex] is 3 times [tex]\( c \)[/tex].
- Therefore, we can write [tex]\( a = 3c \)[/tex]. To express [tex]\( c \)[/tex] in terms of [tex]\( a \)[/tex], we rearrange this to [tex]\( c = \frac{a}{3} \)[/tex].
4. Finding the ratio [tex]\( b \)[/tex] to [tex]\( c \)[/tex]:
- We now have [tex]\( b = 6a \)[/tex] and [tex]\( c = \frac{a}{3} \)[/tex].
- To find how many times [tex]\( b \)[/tex] is bigger than [tex]\( c \)[/tex], we divide [tex]\( b \)[/tex] by [tex]\( c \)[/tex]:
- [tex]\( \frac{b}{c} = \frac{6a}{\frac{a}{3}} \)[/tex].
- Simplify the division:
[tex]\[
\frac{6a}{\frac{a}{3}} = 6a \times \frac{3}{a} = 6 \times 3 = 18.
\][/tex]
Thus, [tex]\( b \)[/tex] is 18 times bigger than [tex]\( c \)[/tex].
