To find the ratio of [tex]\( x \)[/tex] to [tex]\( y \)[/tex] given the equation:
[tex]\[
\frac{2}{x-4} = \frac{5}{y-10}
\][/tex]
we can follow these steps:
1. Cross-multiply to eliminate the fractions:
[tex]\[
2(y - 10) = 5(x - 4)
\][/tex]
2. Distribute the constants on both sides:
[tex]\[
2y - 20 = 5x - 20
\][/tex]
3. Simplify by isolating the terms involving [tex]\( x \)[/tex] and [tex]\( y \)[/tex] on one side of the equation:
[tex]\[
2y - 20 = 5x - 20
\][/tex]
4. Add 20 to both sides to get rid of the [tex]\(-20\)[/tex] terms:
[tex]\[
2y = 5x
\][/tex]
5. Rearrange to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = \frac{2}{5} y
\][/tex]
6. From the equation [tex]\( x = \frac{2}{5} y \)[/tex], we can see that the ratio of [tex]\( x \)[/tex] to [tex]\( y \)[/tex] is:
[tex]\[
\frac{x}{y} = \frac{2}{5}
\][/tex]
Hence, the ratio of [tex]\( x \)[/tex] to [tex]\( y \)[/tex] is [tex]\( \frac{2}{5} \)[/tex] which is equivalent to [tex]\( 0.4 \)[/tex].
