This Is Just Arithmetic!
a − d b + c + ( e + f ) ( g + h )
Let a , b , c , d , e , f , g , h = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , but not necessarily in that order, what's the greatest value this can have?
Answer to be accurate to 3 decimal places. No digit used more than once.
The answer is 172.000.
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4 solutions
Or alternatively, use AM-GM to see that ( e + f ) ( g + h ) ≤ ( 2 5 + 6 + 7 + 8 ) 2 = 1 6 9
This can be achieved by ( e , f , g , h ) = ( 5 , 8 , 6 , 7 ) .
I did check my work for 5 minutes wondering why I got an integer answer though.
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Nice, Daniel. You've just changed a loony arithmetic problem into a learning moment! Thank you.
While I did not think of using AM-GM, I also checked my work for 5 minutes thinking there must be a decimal.
I think the intuitive trick is to notice that multiplication grows much faster than addition therefore maximising that is first priority.
U should have put brackets around a- (b+c), because the question to me looked like a BIMDAS question.
Well.... Good question...i used am gm concept to solve this
oh boy almost got it right... got 169 cause I put 5+6 * 7+8 ugh Im mad >:P
As multiplication will clearly produce the highest output, 5 , 6 , 7 , 8 should be assigned within ( e + f ) ( g + h ) .
The highest value of x y when x + y = z when x = y , therefore pairs should be 5 + 8 , and 6 + 7 .
a is the next most influential, taking 4 . Leaving 1 , 2 , and 3 to fit into ( ( 1 + 2 ) / 3 ) , completing the problem.
(e+f)(g+h) is area of a rectangle whose perimeter is given. We all know the maximum area would be for the square with same perimeter. Perimeter is 26, so the sides are all 13. Rest is just simple.
using 5,6,7,8 for (e+f)(g+h), from the inequalites, as (e+f)+(g+h)=constant for this set of numbers fixed, then the terms of the product must be the same for maximum value, if possible. As it is, just set e+f=g+h=13=5+8=6+7 and (e+f)(g+h)=169. For the other numbers just set a=4 and (b+c)/d=1, cause (b+c)/d>=3/4 and using a value 0.25 lower will obligate using a value for a 1.0 lower, that does not pay. Sorry for the poor english.
4 − ( 1 + 2 ) / 3 + ( 5 + 8 ) ( 6 + 7 ) = 1 7 2
Keep the smallest digits on the left side, then play around with the largest on the right.