To solve this problem, let's break it down into manageable steps. We start by understanding the given ratio of red beads to white beads, and then we proceed to find the number of white beads, the total number of beads, and finally, the fraction of the beads that are white.
### Step 1: Understanding the Ratio
The ratio of red beads to white beads is given as 2:1. This means for every 2 red beads, there is 1 white bead.
### Step 2: Number of Red Beads
We are given that there are 42 red beads.
### Step 3: Calculating the Number of White Beads
Given the ratio of 2:1, we need to determine how many sets of 2 red beads there are among the 42 red beads.
- Each set of 2 red beads corresponds to 1 white bead.
- To find the number of such sets, we divide the number of red beads by 2:
[tex]\[ \text{Number of white beads} = \frac{42 \, \text{red beads}}{2} = 21 \, \text{white beads} \][/tex]
So, there are 21 white beads.
### Step 4: Calculating the Total Number of Beads
To find the total number of beads, we add the number of red beads and the number of white beads together:
[tex]\[ \text{Total number of beads} = 42 \, \text{red beads} + 21 \, \text{white beads} = 63 \, \text{beads} \][/tex]
### Step 5: Calculating the Fraction of Beads that Are White
To find the fraction of the beads that are white, we divide the number of white beads by the total number of beads:
[tex]\[ \text{Fraction of white beads} = \frac{21 \, \text{white beads}}{63 \, \text{total beads}} = \frac{1}{3} \approx 0.3333 \][/tex]
### Summary
- Number of red beads: 42
- Number of white beads: 21
- Total number of beads: 63
- Fraction of the beads that are white: [tex]\( \frac{1}{3} \)[/tex] or approximately 0.3333
So, the detailed solution yields:
- There are 42 red beads and 21 white beads, making a total of 63 beads.
- The fraction of the beads that are white is [tex]\( \frac{1}{3} \)[/tex] or approximately 0.3333.
