To find the value of the expression A, we need to evaluate the sum of the terms given.

A = 1/2 * 15 + 1/15 * 3 + 1/3 * 21 + ... + 6/87 * 90

We can simplify each term before adding them up.

1/2 * 15 = 15/2
1/15 * 3 = 3/15 = 1/5
1/3 * 21 = 21/3 = 7
...

Now we can rewrite the expression A as:

A = 15/2 + 1/5 + 7 + ...

To find the common denominator for all the terms, we need to find the least common multiple (LCM) of 2, 5, 3, ... , 87.

LCM(2, 5, 3, ..., 87) = 1740

Now we can rewrite each term with the common denominator of 1740.

15/2 = (15/2) * (870/870) = 13050/1740
1/5 = (1/5) * (348/348) = 348/1740
7 = (7) * (248/248) = 1736/1740

Now the expression A becomes:

A = 13050/1740 + 348/1740 + 1736/1740 + ...

To add these fractions, we need to have a common denominator.

A = (13050 + 348 + 1736 + ...) / 1740

To find the sum of the numerators, we need to find the sum of the arithmetic series:

13050 + 348 + 1736 + ... = (n/2)(first term + last term)
= (n/2)(13050 + last term)

The last term can be found by multiplying the common difference (174) by the number of terms (90/3 = 30).

last term = 174 * 30 = 5220

Now we can find the sum of the numerators:

13050 + 348 + 1736 + ... = (90/2)(13050 + 5220)
= 45(18270)
= 822150

Substituting this back into the expression A:

A = 822150 / 1740

Now we can simplify the fraction:

A = 471 / 1

Therefore, the value of the expression A is 471.