To solve this problem, we need to multiply the fractions and then perform the subtraction.

First, let's multiply the fractions:
(7/15) (3/23) = 21/345
(23/27) (1/6) = 23/162

Now, let's subtract the two products:
21/345 - 23/162
To subtract fractions, we need to find a common denominator. In this case, the common denominator is 345 * 162 = 55,890.

(21 162)/(345 162) - (23 345)/(162 345)
3422/55890 - 7935/55890
= (3422 - 7935)/55890
= -4513/55890

Therefore, the final answer is -4513/55890.