  More than twenty-five years ago, Manickam, Mikl\'{o}s, and Singhi conjectured
that for positive integers $n,k$ with $n \geq 4k$, every set of $n$ real
numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element
subsets whose sum is also nonnegative. We verify this conjecture when $n \geq
8k^{2}$, which simultaneously improves and simplifies a bound of Alon, Huang,
and Sudakov and also a bound of Pokrovskiy when $k < 10^{45}$.
  Moreover, our arguments resolve the vector space analogue of this conjecture.
Let $V$ be an $n$-dimensional vector space over a finite field. Assign a
real-valued weight to each $1$-dimensional subspace in $V$ so that the sum of
all weights is zero. Define the weight of a subspace $S \subset V$ to be the
sum of the weights of all the $1$-dimensional subspaces it contains. We prove
that if $n \geq 3k$, then the number of $k$-dimensional subspaces in $V$ with
nonnegative weight is at least the number of $k$-dimensional subspaces in $V$
that contain a fixed $1$-dimensional subspace. This result verifies a
conjecture of Manickam and Singhi from 1988.
