Question
Find all pairs of numbers such that when they are added they equal 100, and when
they are multiplied they are at least 2000.
they are multiplied they are at least 2000.
Answers
Steve
maximum product occurs when a=b=50
That makes the product 2500
The larger a gets, the smaller b must get, so try
a=40 b=60: ab=2400
a=30 b=70: ab=2100
a=20 b=80: ab=1600
and narrow in on it.
Or, algebraically,
a(100-a) >= 2000
a^2 - 100a + 2000 <= 0
10(5-√5) <= a <= 10(5+√5)
27.6 <= a <= 72.3
so, 28 < a <= 72
check:
28*72 = 2016
27*73 = 1971
That makes the product 2500
The larger a gets, the smaller b must get, so try
a=40 b=60: ab=2400
a=30 b=70: ab=2100
a=20 b=80: ab=1600
and narrow in on it.
Or, algebraically,
a(100-a) >= 2000
a^2 - 100a + 2000 <= 0
10(5-√5) <= a <= 10(5+√5)
27.6 <= a <= 72.3
so, 28 < a <= 72
check:
28*72 = 2016
27*73 = 1971