To find the value of [tex]\( M : Q \)[/tex] given the ratios [tex]\( M : N : P = 3 : 7 : 5 \)[/tex] and [tex]\( P : Q = 1 : 2 \)[/tex], we can proceed step by step as follows:
1. Understand the given ratios:
- [tex]\( M : N : P \)[/tex] implies that if we assign specific values to [tex]\( M, N, \)[/tex] and [tex]\( P \)[/tex] based on their relative ratios, let's say [tex]\( M = 3k \)[/tex], [tex]\( N = 7k \)[/tex], and [tex]\( P = 5k \)[/tex] for some constant [tex]\( k \)[/tex].
2. Determine the value of [tex]\( Q \)[/tex] in terms of [tex]\( P \)[/tex]:
- The second ratio, [tex]\( P : Q = 1 : 2 \)[/tex], tells us that for every 1 part of [tex]\( P \)[/tex], there are 2 parts of [tex]\( Q \)[/tex].
- Given [tex]\( P = 5k \)[/tex], we need to find [tex]\( Q \)[/tex].
- Since [tex]\( P \)[/tex] is 5 parts, the value of [tex]\( Q \)[/tex] will be [tex]\( 5k \times 2 \)[/tex], which equals [tex]\( 10k \)[/tex].
3. Calculate the ratio [tex]\( M : Q \)[/tex]:
- We have [tex]\( M = 3k \)[/tex] and [tex]\( Q = 10k \)[/tex].
- To find the ratio [tex]\( M : Q \)[/tex], we simply divide the value of [tex]\( M \)[/tex] by [tex]\( Q \)[/tex]:
[tex]\[
M : Q = \frac{M}{Q} = \frac{3k}{10k} = \frac{3}{10}
\][/tex]
- This simplifies to [tex]\( M : Q = 3 : 10 \)[/tex].
Therefore, the value of [tex]\( M : Q \)[/tex] is:
[tex]\[
\boxed{3 : 10}
\][/tex]
The correct option is (c) [tex]\( 3 : 10 \)[/tex].
