Answer:
The fraction of the beads that are red is
Step-by-step explanation:
Algebraic Expressions
A bag contains red (r), yellow (y), and blue (b) beads. We are given the following ratios:
r:y = 2:3
y:b = 5:4
We are required to find r:s, where s is the total of beads in the bag, or
s = r + y + b
Thus, we need to calculate:
[tex]\displaystyle \frac{r}{r+y+b} \qquad\qquad [1][/tex]
Knowing that:
[tex]\displaystyle \frac{r}{y}=\frac{2}{3} \qquad\qquad [2][/tex]
[tex]\displaystyle \frac{y}{b}=\frac{5}{4}[/tex]
Multiplying the equations above:
[tex]\displaystyle \frac{r}{y}\frac{y}{b}=\frac{2}{3}\frac{5}{4}[/tex]
Simplifying:
[tex]\displaystyle \frac{r}{b}=\frac{5}{6} \qquad\qquad [3][/tex]
Dividing [1] by r:
[tex]\displaystyle \frac{r}{r+y+b}=\displaystyle \frac{1}{1+y/r+b/r}[/tex]
Substituting from [2] and [3]:
[tex]\displaystyle \frac{r}{r+y+b}=\displaystyle \frac{1}{1+3/2+6/5}[/tex]
Operating:
[tex]\displaystyle \frac{r}{r+y+b}=\displaystyle \frac{1}{\frac{10+3*5+6*2}{10}}[/tex]
[tex]\displaystyle \frac{r}{r+y+b}=\displaystyle \frac{10}{10+15+12}[/tex]
[tex]\displaystyle \frac{r}{r+y+b}=\displaystyle \frac{10}{37}[/tex]
The fraction of the beads that are red is [tex]\mathbf{\frac{10}{37}}[/tex]
