\(S = ( C + \frac{1}{4} C)\) this is the exact way to rephrase the sentence.

\(C = ( T - \frac{2}{5} T )\)

So you do have \(\frac{5}{4} C\) and \(\frac{5}{3} T\)

Now the LCM between 4 and 3 is 12. Then substitute and you have the minimum quantity 35

Another approach is thinking real numbers, for instance: the number you are looking for must be integers.

If you think you do have 10 of C, 25% more is 2.5 so not possible. 11 you do have 2.75 BUT 12, 25% of it is 3. So it possible 12 + 3 = 15.

This is also an easy approach to follow.

Regards

Since the number of Cardigans, sweaters and turtlenecks should be integers.
Given: S = C + C/4 and C = T - 2T/5 --> S = 5C/4 and T = 5C/3

Now C/4 and 5C/3 have to be integers, we would take C as an LCM of 4 and 3 i.e 12 and work out the values of C and T

Let x be the no of Cardigans. Therefore, sweaters are 5x/4. (25% higher than Cardigans).
Now Cardigans are 2/3 fewer, so Cardigans = 3/5 of Turtlenecks. (1-2/5) No of Turtlenecks is 5x/3.
Ratio of Sweaters: Cardigans : Turtlenecks is 5x/4:x:5x/3
Equating, the ratio of S:C:T=15:12:20.
Therefore min no of S+T=35.