Let's solve this problem step-by-step.
1. Define the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
- Given that [tex]\( y \)[/tex] is [tex]\( 20\% \)[/tex] more than [tex]\( x \)[/tex],
[tex]\[
y = x + 0.2x = 1.2x
\][/tex]
2. Add the same number [tex]\( k \)[/tex] to both [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- New value of [tex]\( x \)[/tex] becomes [tex]\( x + k \)[/tex]
- New value of [tex]\( y \)[/tex] becomes [tex]\( y + k \)[/tex]
3. Given that the new value of [tex]\( x \)[/tex] is [tex]\(\frac{3}{2}\)[/tex] times its original value:
[tex]\[
x + k = \frac{3}{2}x
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
[tex]\[
x + k = \frac{3}{2}x
\][/tex]
[tex]\[
k = \frac{3}{2}x - x
\][/tex]
[tex]\[
k = \frac{3}{2}x - \frac{2}{2}x
\][/tex]
[tex]\[
k = \frac{1}{2}x
\][/tex]
5. Find the new value of [tex]\( y \)[/tex]:
- Original value of [tex]\( y \)[/tex] is [tex]\( 1.2x \)[/tex]
- New value of [tex]\( y \)[/tex] is [tex]\( y + k \)[/tex]:
[tex]\[
y + k = 1.2x + \frac{1}{2}x
\][/tex]
[tex]\[
y + k = 1.2x + 0.5x
\][/tex]
[tex]\[
y + k = 1.7x
\][/tex]
6. Calculate how many times the new value of [tex]\( y \)[/tex] is compared to its previous value:
[tex]\[
\text{Multiplier} = \frac{y + k}{y} = \frac{1.7x}{1.2x} = \frac{1.7}{1.2}
\][/tex]
7. Simplify the fraction:
[tex]\[
\frac{1.7}{1.2} = \frac{17}{12}
\][/tex]
Hence, the new value of [tex]\( y \)[/tex] is [tex]\(\frac{17}{12}\)[/tex] times its previous value.
[tex]\(\boxed{\frac{17}{12}}\)[/tex]
