Akarat equals 1/24 part of gold in an alloy [ eg 9 karat gold is 9/24 gold]. how many grams of 9 karat gold must be mixed with
18 karat gold to make 200 grams of 14 karat gold alloy?

This is a mixture problem.

Let the 9 karat gold to be mixed be x. The 18 karat would weight (200 - x). Since the total = 200g.

mass1 * karat1 + mass2*karat = total mass * total carat

Let mass1 be x g.          mass 2 would be = (200 -x)
karat 1 = 9                    karat 2 = 18 

Total mass = 200g.
Total karat = 14.

9*x  + 18*(200 -x) = 14*200

9x + 18*200 - 18*x = 14*200

18*200 + 9x - 18x = 14*200

3600 - 9x = 2800

3600 - 2800 = 9x

800 = 9x

9x = 800

x = 800/9

x ≈ 88.89

Therefore 88.89 grams of 9 karat gold must be mixed.

Where are the