To find the remaining ratio of orange beads in Vie's friendship bracelets based on the specified ratios of blue and yellow beads, we start by determining the parts represented by the blue and yellow beads.
- The ratio of blue beads is \( \frac{3}{10} \).
- The ratio of yellow beads is \( \frac{1}{10} \).
Next, we'll add the ratios of blue and yellow beads together:
\[ \frac{3}{10} + \frac{1}{10} = \frac{4}{10} \]
Now, we can simplify this sum:
\[ \frac{4}{10} = \frac{2}{5} \]
This means that the combined ratio of blue and yellow beads is \( \frac{2}{5} \). To find the remaining ratio that represents the orange beads, we subtract this value from 1 (the whole):
\[ 1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5} \]
Thus, the remaining ratio for orange beads is \( \frac{3}{5} \).
Now we have the ratios for the beads in the bracelet:
- Blue: \( \frac{3}{10} \)
- Yellow: \( \frac{1}{10} \)
- Orange: \( \frac{3}{5} \)
Next, we should express all of these ratios in a way that they add up to a single total of 1. To do this, the ratios must also be represented with a common denominator.
Here, the common denominator of 10 can be used for clarity:
- Blue: \( \frac{3}{10} \)
- Yellow: \( \frac{1}{10} \)
- Orange: \( \frac{6}{10} \) (as \( \frac{3}{5} = \frac{6}{10} \))
Finally, the complete ratio of the beads in the bracelet is:
- Blue: 3 parts
- Yellow: 1 part
- Orange: 6 parts
The complete simplified ratios for beads in the bracelets is \( 3:1:6 \).