If p, q, r and s are distinct single digit positive numbers, then what is the greatest value of (p + q) (r + s)?

Correct Option (B)

To determine the greatest value of (p + q)(r + s), where p, q, r, s are distinct single-digit positive numbers, the largest possible distinct single-digit positive numbers should be selected. These numbers are 9, 8, 7, and 6.

The sum of these four numbers is 9 + 8 + 7 + 6 = 30.

To maximize the product of two factors, (p + q) and (r + s), their values should be as close to each other as possible, given that their sum (p + q) + (r + s) is constant (30).

Consider the following pairing:

• Let p = 9 and q = 6. Then p + q = 15.
• Let r = 8 and s = 7. Then r + s = 15.

The product (p + q)(r + s) is (15)(15) = 225. This arrangement yields the maximum possible value because the two sums are equal.

Incorrect Options:

The other options represent smaller products resulting from different groupings of the largest distinct single-digit positive numbers, where the sums are not as close to each other.

• If the numbers are grouped as (9 + 8) and (7 + 6), the sums are 17 and 13. The product is 17 × 13 = 221. This corresponds to Option (4).
• If the numbers are grouped as (9 + 7) and (8 + 6), the sums are 16 and 14. The product is 16 × 14 = 224. This corresponds to Option (3).
• Option (1) 230 is not achievable with distinct single-digit positive numbers under the given conditions, as the maximum product is 225.