x - the number of Karen's beads
y - the number of Patricia's beads
The ratio x:y=2:5.
[tex]\frac{x}{y}=\frac{2}{5} \\ 2y=5x \\ y=\frac{5}{2}x[/tex]
Then Patricia bought another 75 beads, so the number of her beads became y+75 and the ratio became 4:15.
[tex]\frac{x}{y+75}=\frac{4}{15} \\
4(y+75)=15x \\
y+75=\frac{15}{4}x \\
y=\frac{15}{4}x-75[/tex]
y from the first equation is equal to y from the second equation.
[tex]\frac{5}{2}x=\frac{15}{4}x-75 \\
\frac{5}{2}x-\frac{15}{4}x=-75 \\
\frac{10}{4}x-\frac{15}{4}x=-75 \\
-\frac{5}{4}x=-75 \\
x=-75 \times (-\frac{4}{5}) \\
x=-15 \times (-4) \\
x=60 \\ \\
y=\frac{5}{2}x=\frac{5}{2} \times 60=\frac{300}{2}=150[/tex]
Karen had 60 beads and Patricia had 150 beads.
