Let's denote the number of red beads Ivan has as \( r \) and the number of blue beads as \( b \).
According to the problem, there are two key pieces of information:
-
The total number of red and blue beads is 49: \[ r + b = 49 \]
-
Ivan has six times as many blue beads as red beads: \[ b = 6r \]
Now we can substitute the second equation into the first equation:
\[ r + 6r = 49 \]
This simplifies to:
\[ 7r = 49 \]
Now, solve for \( r \):
\[ r = \frac{49}{7} = 7 \]
Now that we have the number of red beads, we can find the number of blue beads by substituting \( r \) back into the equation for \( b \):
\[ b = 6r = 6 \times 7 = 42 \]
Thus, Ivan has:
- 7 red beads
- 42 blue beads
So, Ivan has \( \boxed{42} \) blue beads.