The smallest possible positive value of 1−[(1/w)+(1/x)+(1/y)+(1/z)] where w, x, y, z are odd positive integers, has the form a/b, where a,b are coprime positive integers. Find a+b.
as we have to chose positive odd integers sp we cant choose maximum big numbers close to 1 its choose w=3 x=3 then y=5 z=7 then answer is minimum is 1/45 or 1+45=46
To find the smallest possible positive value of the expression 1−[(1/w)+(1/x)+(1/y)+(1/z)], we need to find the smallest possible values of w, x, y, and z.
Since w, x, y, and z are odd positive integers, let's consider the smallest odd numbers, which are 1, 3, 5, 7, ...
Let's start by choosing 1 for w. Then, the expression becomes:
1 - (1/w + 1/x + 1/y + 1/z)
= 1 - (1/1 + 1/x + 1/y + 1/z)
= 1 - (1 + 1/x + 1/y + 1/z)
= 1 - (x + y + z) / (xyz)
To minimize the expression, we should maximize the value of (x + y + z). Since x, y, and z are odd and positive, the smallest possible values for them are 1, 3, and 5, respectively.
So the expression becomes:
1 - (1 + 1/1 + 1/3 + 1/5)
= 1 - (1 + 1 + 1/3 + 1/5)
= 1 - (9/5 + 3/5 + 1/3 + 1/5)
= 1 - (54/30 + 18/30 + 10/30 + 6/30)
= 1 - (88/30)
= 1 - 44/15
= (15/15) - (44/15)
= -29/15
Therefore, the smallest possible value of the expression is -29/15, and a + b = (-29) + 15 = -14.
To find the smallest possible positive value of 1−[(1/w)+(1/x)+(1/y)+(1/z)], we need to consider the values of w, x, y, and z.
Since w, x, y, and z are odd positive integers, we can represent them as 2n+1, where n is a non-negative integer.
Substituting these values into the expression, we have:
1 - [(1/(2n+1)) + (1/(2n+1)) + (1/(2n+1)) + (1/(2n+1))].
Simplifying the expression, we have:
1 - (4/(2n+1)).
We need to minimize this expression, which means we need to maximize the value of 2n+1.
Since w, x, y, and z are positive integers, n must be greater than or equal to 0. Taking the smallest possible value of n, which is 0, we have:
1 - (4/(2(0)+1)) = 1 - (4/1) = 1 - 4 = -3.
However, since we are looking for a positive value, we can choose the next value of n, which is 1:
1 - (4/(2(1)+1)) = 1 - (4/3).
This value cannot be simplified further, so the smallest possible positive value is 1 - (4/3). The numerator and denominator are already coprime, so a = 4 and b = 3.
Therefore, a + b = 4 + 3 = 7.