Xian and his cousin Kai both collect stamps. Xian has 64 stamps, and Kai has 68 stamps. The boys recently joined different stamp-collecting clubs. Xian's club will send him 20 new stamps per month. Kai's club will send him 16 new stamps per month. After how many months will Xian and Kai have the same number of stamps? How many stamps will each have?

After _ months Xian and his cousin will have _ stamps each.

We have been given that Xian's club will send him 20 new stamps per month. Number of stamps that Xian would have after x months would be [tex]20x[/tex].

Total number of stamps that Xian would have after x months would be [tex]20x+64[/tex].

We are also told that Kai's club will send him 16 new stamps per month. Number of stamps that Kai would have after x months would be [tex]16x[/tex].

Total number of stamps that Kai would have after x months would be [tex]16x+68[/tex].

Now, we will equate both expressions and solve for x.

[tex]20x+64=16x+68[/tex]

[tex]20x-16x+64=16x-16x+68[/tex]

[tex]4x+64=68[/tex]

[tex]4x+64-64=68-64[/tex]

[tex]4x=4[/tex]

[tex]\frac{4x}{4}=\frac{4}{4}[/tex]

[tex]x=1[/tex]

Therefore, after 1 month Xian and Kai will have same stamps.

[tex]16x+68[/tex]

[tex]16(1)+68[/tex]

[tex]16+68=84[/tex]

[tex]20x+64[/tex]

[tex]20(1)+64[/tex]

[tex]20+64=84[/tex]

Therefore, after one month, Xian and his cousin will have 84 stamps each.