Bunuel wrote:
If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number of integers that can be negative?
A. 2
B. 3
C. 4
D. 5
E. 6
Kudos for a correct solution.
To achieve negative result here with maximum number of negative integers, we can't use six. An even number of negative factors yields positive. Six negative factors "eat up" all the individual terms.
Five negative factors, however, will work.
Maximize and minimize1. Maximize the number of negative factors in (cdef): there can be three. Odd number of negative factors = negative product.
2. Maximize negative factors by two more. Make both a and b negative, but minimize their positive product's absolute value.
Make c, d, and e negative, and for ease, keep them small. Let f be the one positive number. Make it large so that |(cdef)| is > (ab).
Result is a small positive number plus a larger negative number, which yields a negative.
3. Let a, b, c, d, and e = -1
Let f = 20
(-1)(-1) + (-1)(-1)(-1)(20) =
1 + (-20) =
1 - 20 =
-19
4. Alternatively, make all but a or b negative, and one number large because:
-- if one term in (ab) is negative, and
-- the absolute value of (ab) is greater than (cdef)
-- the result is a very small negative number (far to the right on the number line) plus a smaller positive number, such that
-- sum of terms with those properties is less than 0.
Let b, c, d, e, and f = -1
Let a = 20
(20)(-1) + (-1)(-1)(-1)(-1) =
-20 + 1 = -19
The maximum number of negative factors possible for this expression is 5
Answer D
_________________
The humblest praise most, while cranks & malcontents praise least. Praise almost seems to be inner health made audible. ~ C. S. Lewis